text:
\begin{document}
	\title{Geodesics of positive Lagrangians from special Lagrangians with boundary}
	\author[J. Solomon]{Jake P. Solomon}
	\address{Institute of Mathematics\\ Hebrew University, Givat Ram\\Jerusalem, 91904, Israel}
	\email{jake@math.huji.ac.il}
	\author[A. Yuval]{Amitai M. Yuval}
	\address{Institute of Mathematics\\ Hebrew University, Givat Ram\\Jerusalem, 91904, Israel}
	\email{amitai.yuval@mail.huji.ac.il}

	\keywords{geodesic, positive Lagrangian, special Lagrangian, elliptic, boundary value problem, unobstructed, deformation}
	\subjclass[2010]{53D12, 35J66 (Primary) 53C38, 35J70, 58B20 (Secondary)
	}
	\date{October 2020}

	\begin{abstract}
		Geodesics in the space of positive Lagrangian submanifolds are solutions of a fully non-linear degenerate elliptic PDE.
We show that a geodesic segment in the space of positive Lagrangians corresponds to a one parameter family of special Lagrangian cylinders, called the cylindrical transform. The boundaries of the cylinders are contained in the positive Lagrangians at the ends of the geodesic. The special Lagrangian equation with positive Lagrangian boundary conditions is elliptic and the solution space is a smooth manifold, which is one dimensional in the case of cylinders. A geodesic can be recovered from its cylindrical transform by solving the Dirichlet problem for the Laplace operator on each cylinder.

Using the cylindrical transform, we show the space of pairs of positive Lagrangian spheres connected by a geodesic is open. Thus, we obtain the first examples of strong solutions to the geodesic equation in arbitrary dimension not invariant under isometries. In fact, the solutions we obtain are smooth away from a finite set of points.
	\end{abstract}

\maketitle

\tableofcontents

	\section{Introduction}
\subsection{Overview}
	Let $(X,\omega,J,\Omega)$ be a \emph{Calabi-Yau manifold}. Namely, $X$ is a K\"ahler manifold with symplectic form $\omega$ and complex structure $J$, and $\Omega$ is a non-vanishing holomorphic volume form on $X$. We denote by $g$ the K\"ahler metric and by $n$ the complex dimension.

	An oriented Lagrangian submanifold $\Lambda\subset X$, possibly immersed, is said to be \emph{positive} if $\real \Omega|_\Lambda$ is a positive volume form. A positive Lagrangian submanifold is \emph{special} if $\imaginary \Omega|_\Lambda = 0.$ An oriented Lagrangian submanifold is called \emph{imaginary special} if $\real \Omega|_\Lambda = 0$ and $\imaginary \Omega|_\Lambda$ is a positive volume form.

	Let $\mathcal{O}$ be a Hamiltonian isotopy class of closed smoothly embedded positive Lagrangians diffeomorphic to a given manifold $L.$ Then $\mathcal{O}$ is naturally a smooth Fr\'echet manifold, and for $\Lambda\in\mathcal{O}$, there is a natural isomorphism
	\begin{equation}
	\label{equation: tangent space to O}
	T_\Lambda\mathcal{O}\cong\vmv(\Lambda):=\left\{h\in C^\infty(\Lambda)\left|\int_\Lambda h\real\Omega=0\right.\right\}.
	\end{equation}
	Following~\cite{solomon}, we define a Riemannian metric on $\mathcal{O}$ by
	\begin{equation}
	\label{equation: the metric}
	(h,k):=\int_\Lambda hk\real\Omega,\qquad h,k\in\vmv(\Lambda).
	\end{equation}
	It is shown in~\cite{solomon-curv} that the metric $(\cdot,\cdot)$ has a Levi-Civita connection and the associated sectional curvature is non-positive. The Levi-Civita connection, which we describe in detail in Section~\ref{subsection: calabi-yau manifolds}, gives rise to the notion of geodesics. If $\mathcal{O}$ is geodesically connected then there can exist at most one special Lagrangian in $\mathcal{O}$~\cite{solomon}. If two Lagrangians $\Lambda_0,\Lambda_1 \in \mathcal{O}$ are connected by a geodesic, the cardinality of $\Lambda_0 \cap \Lambda_1$ is bounded below by the number of critical points of a function on $\Lambda_0$ \cite{rubinstein-solomon}.

	The geodesic equation is a fully non-linear partial differential equation. It is shown in~\cite{rubinstein-solomon} that the geodesic equation is degenerate elliptic and the associated boundary value problem in the Euclidean setting has unique weak solutions. In~\cite{solomon-yuval} there are examples of smooth geodesics in arbitrary dimension, which are preserved by an isometric action of the orthogonal group $O(n).$ The group action allows the geodesic equation to be reduced to the one dimensional case, where it becomes an ODE. Further results on geodesics can be found in~\cite{dellatorre,xu}.

An analog to the space $\mathcal{O}$ under mirror symmetry is the space $\mathcal{H}$ of almost calibrated $(1,1)$-forms on a K\"ahler manifold. The geodesic equation in $\mathcal{H}$ is a degenerate form of the deformed Hermitian Yang-Mills equation. The space $\mathcal{H}$ and its geodesics have been studied in~\cite{chu2020space,collins2018moment}.

	In the present work, we establish a correspondence between geodesics of positive Lagrangians and one parameter families of imaginary special Lagrangian cylinders. See Theorems~\ref{theorem: family} and~\ref{theorem: geodesic-cylinder correspondence}. We call this correspondence the cylindrical transform.
By cylinders, we mean manifolds of the form $N \times [0,1],$ where $N$ is a manifold of dimension $n-1.$
The boundary components of the cylinders corresponding to a geodesic $(\Lambda_t)_{t \in [0,1]}$ are contained in $\Lambda_0$ and $\Lambda_1$ respectively. Positive Lagrangian submanifolds such as $\Lambda_0$ and $\Lambda_1$ are an elliptic boundary condition for the imaginary special Lagrangian equation. In Theorem~\ref{theorem: space of ISL cylinders is 1-dim} we show that the space of imaginary special Lagrangian cylinders with positive Lagrangian boundary conditions is a smooth $1$-dimensional manifold.

Using the cylindrical transform and the ellipticity of the imaginary special Lagrangian equation, we establish in Theorem~\ref{theorem: perturbation of geodesic} a perturbation result for solutions of the geodesic equation. Namely, the space of pairs of positive Lagrangian spheres intersecting transversally at two points that are connected by a geodesic is open. In particular, we show the existence of geodesics connecting positive Lagrangians of arbitrary dimension without any symmetry. The geodesics of Theorem~\ref{theorem: perturbation of geodesic} are smooth away from a finite number of points. In a sequel~\cite{spider}, we strengthen Theorem~\ref{theorem: perturbation of geodesic} so that it produces geodesics of positive Lagrangians that are $C^{1,1}$ submanifolds even at the non-smooth locus.

\subsection{Statement of results}
To set up the cylindrical transform in its natural generality, we consider geodesics of possibly immersed positive Lagrangians that are smooth away from a finite number of cone point singularities. For the rest of the paper, unless otherwise specified, the term geodesic allows such cone point singularities.
We define the \emph{critical locus} of a geodesic of positive Lagrangians $(\Lambda_t)_{t \in [0,1]}$ by
	\[
	\crit((\Lambda_t)_t):=\bigcap_{t\in[0,1]}\Lambda_t.
	\]
The non-smooth cone points of the geodesics in this paper are contained in their critical loci. The possibly limited regularity at the critical locus is consistent with the result of~\cite{rubinstein-solomon} that the symbol of the linearized geodesic equation has a 1-dimensional kernel except at the critical locus, where the kernel is $(n-1)$-dimensional.
A full account of our notion of geodesics is given in Definition~\ref{definition: geodesic}.

	Define a positive function $\rho : X \to \R$ by
	\begin{equation}
	\label{equation: rho}
	\rho^2\omega^n/n! = (-1)^{\frac{n(n-1)}2}\left(\frac{\ii}{2}\right)^n \Omega \wedge \overline \Omega.
	\end{equation}
	Let $f: L \to X$ be an immersion. Define
\[
\Delta_\rho: C^\infty(L)\to C^\infty(L)
\]
by $u\mapsto*d((\rho \circ f)*du)$, where $*$ is the Hodge star operator associated to the Riemannian metric $f^*g.$ Then $\Delta_\rho$ is elliptic (see Lemma~\ref{lemma: Delta rho is a cool operator}). A geodesic $(\Lambda_t)_{t\in[0,1]}$ has an associated \emph{Hamiltonian}, which is the family of functions $h_t \in \vmv(\Lambda_t)$ such that $h_t =  \deriv{t} \Lambda_t.$
The functions $h_t, t \in [0,1],$ are related by parallel transport of the Levi-Cevita connection on $\mathcal{O}$ and, in particular, have the same image and diffeomorphic level sets.

	\begin{thm}\label{theorem: family}
		Let $(\Lambda_t)_{t\in [0,1]}$ be a geodesic of positive Lagrangians and let $(h_t)_{t\in[0,1]}$ denote the associated Hamiltonian. For $c \in \R,$ let
		\[
		L_c := \{(p,t)| t \in [0,1],\; p \in h_t^{-1}(c)\setminus \crit((\Lambda_t)_t) \subset \Lambda_t\}.
		\]
		Then $L_c$ is a smooth immersed submanifold of $X \times [0,1]$ diffeomorphic to the cylindrical manifold $\left(h_0^{-1}(c)\setminus \crit((\Lambda_t)_t)\right)\times [0,1]$, and the map
		\[
		\Phi_c : L_c \to X
		\]
		given by $\Phi_c(p,t) = p$ is an imaginary special Lagrangian immersion mapping the boundary components of $L_c$ to $\Lambda_0$ and $\Lambda_1$. See Figure~\ref{fig:theorem family}. Moreover, the map
		\[
		\sigma_c : L_c \to [0,1]
		\]
		given by $\sigma_c(p,t) = t$ satisfies $\Delta_\rho \sigma_c =  0.$
	\end{thm}
\begin{figure}[ht]
\centering
\includegraphics[width=12cm]{picture.png}
\caption{The imaginary special Lagrangian cylinders corresponding to a geodesic $(\Lambda_t)_t$ according to Theorem~\ref{theorem: family}.}
\label{fig:theorem family}
\end{figure}

Let $\Lambda_0,\Lambda_1\subset X$ be smoothly embedded positive Lagrangians and let $N$ be a manifold of dimension $n-1$. We denote by $\mathcal{SLC}(N;\Lambda_0,\Lambda_1)$ the space of imaginary special Lagrangian submanifolds of $X,$ perhaps immersed, diffeomorphic to $N\times[0,1],$ such that the boundary corresponding to $N \times \{i\}$ is embedded in $\Lambda_i$ for $i = 0,1.$ We denote by $\mathcal{SLC}(\Lambda_0,\Lambda_1)$ the union of the spaces $\mathcal{SLC}(N;\Lambda_0,\Lambda_1)$ as $N$ varies.

	\begin{thm}
		\label{theorem: space of ISL cylinders is 1-dim}
		Given two smoothly embedded positive Lagrangians $\Lambda_0,\Lambda_1,$ and a connected closed $(n-1)$-manifold $N,$ the space of imaginary special Lagrangian cylinders $\mathcal{SLC}(N;\Lambda_0,\Lambda_1)$ is a smooth $1$-dimensional manifold.
	\end{thm}
	\begin{rem}
		More generally, positive Lagrangians are natural elliptic boundary conditions for imaginary special Lagrangians with boundary of arbitrary topology, and the associated deformation theory is unobstructed. See Remark~\ref{remark: not only cylinders}. The deformation theory of closed special Lagrangians was shown to be unobstructed in~\cite{mclean,salur}.
	\end{rem}

	The next result is a refinement and partial converse to Theorem~\ref{theorem: family} in the case where $\Lambda_i,\;i=0,1,$ are smoothly embedded spheres intersecting transversally at exactly two points.
To formulate the result we need the following definition.
	\begin{dfn}
		\label{definition: cylindrical transform}
		The \emph{cylindrical transform} of a geodesic of positive Lagrangians $(\Lambda_t)_{t \in [0,1]}$ is the subset of $\mathcal{SLC}(\Lambda_0,\Lambda_1)$ parameterized by the family of imaginary special Lagrangian immersions $\Phi_c : L_c \to X$ from Theorem~\ref{theorem: family} for $c \in \R$ such that $L_c \neq \emptyset$.
	\end{dfn}
We refer the reader to Definition~\ref{definition: regularity} for the notion of regularity of a connected component in $\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right).$
	\begin{thm}
		\label{theorem: geodesic-cylinder correspondence}
		Let $\Lambda_0,\Lambda_1\subset X$ be smoothly embedded positive Lagrangian spheres intersecting transversally at exactly two points. The cylindrical transform of a geodesic between $\Lambda_0$ and $\Lambda_1$ is a regular connected component in $\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right).$ Conversely, given a regular connected component $\mathcal{Z}\subset\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right),$ there exists a unique up to reparameterization geodesic between $\Lambda_0$ and $\Lambda_1$ with cylindrical transform $\mathcal{Z}.$
	\end{thm}

	Finally, we apply Theorem~\ref{theorem: geodesic-cylinder correspondence} to prove that geodesics of positive Lagrangian spheres with endpoints intersecting transversally at two points are stable under $C^{2,\alpha}$-small Hamiltonian perturbations. 		Let $\mathcal{O}$ be a Hamiltonian isotopy class of smoothly embedded positive Lagrangian spheres, and let $\mathfrak{G}_\mathcal{O}$ denote the space of geodesics $(\Lambda_t)_{t \in [0,1]}$ with $\Lambda_0,\Lambda_1 \in \mathcal{O}$ intersecting transversally at two points. We refer the reader to Definition~\ref{dfn:topology on geodesics} for the strong and weak $C^{1,\alpha}$ topologies on $\mathfrak{G}_\mathcal{O}.$ Roughly speaking, the strong topology controls closeness of all cylinders in the cylindrical transform of a geodesic while the weak topology controls closeness of a single cylinder.

	\begin{thm}
		\label{theorem: perturbation of geodesic}
Let $\Lambda_0,\Lambda_1\in\mathcal{O}$ intersect transversally at exactly two points. Suppose there exists a geodesic $(\Lambda_t)_{t\in[0,1]}$ between $\Lambda_0$ and $\Lambda_1.$ Let $\alpha\in(0,1).$ Then, there exists a $C^{2,\alpha}$-open neighborhood $\mathcal{Y}$ of $\Lambda_1$ in $\mathcal{O}$ and a weak $C^{1,\alpha}$-open neighborhood $\mathcal X$ of $(\Lambda_t)_{t \in [0,1]}$ in $\mathfrak{G}_\mathcal{O}$ such that for every $\Lambda\in\mathcal{Y}$ there exists a unique geodesic between $\Lambda_0$ and $\Lambda$ in $\mathcal{X}.$ This geodesic depends continuously on $\Lambda$ with respect to the $C^{2,\alpha}$ topology on $\mathcal{Y}$ and the strong $C^{1,\alpha}$ topology on $\mathcal{X}.$
	\end{thm}

	In the sequel~\cite{spider}, we strengthen Theorem~\ref{theorem: perturbation of geodesic} to show that if the geodesic $(\Lambda_t)_t$ is of regularity $C^{1,1}$ at the cone points, so are the geodesics connecting $\Lambda_0$ to any $\Lambda~\in~\mathcal{Y}.$
In~\cite{solomon-yuval}, there are examples of geodesics of positive Lagrangians of arbitrary dimension, many of which satisfy the conditions of Theorem~\ref{theorem: perturbation of geodesic}. However, they are all preserved by an isometric action of $O(n)$ on the ambient manifold $X.$ From Theorem~\ref{theorem: perturbation of geodesic} we obtain the following.
\begin{cor}
There exist geodesics of positive Lagrangians in arbitrary dimension that are not invariant under any isometries of the ambient manifold.
\end{cor}

	It should be possible to extend Theorems~\ref{theorem: geodesic-cylinder correspondence} and~\ref{theorem: perturbation of geodesic} to Lagrangians with more complicated topology and more critical points. Furthermore, it should be possible to extend the techniques of this paper to prove the existence of geodesics a priori. In fact, to show the existence of a geodesic and hence an isotopy between two positive Lagrangians, it may not be necessary to assume the existence of a Hamiltonian isotopy between them, but only an intersection point of Maslov index zero. In the sequel~\cite{spider}, we show that there exists a one parameter family of imaginary special Lagrangian cylinders near any such intersection point. It remains to identify situations in which this family of cylinders can be extended until it terminates at an intersection point of Maslov index $n.$ When $n = 2$ and $X$ is hyperk\"ahler, the relation between special Lagrangian submanifolds and holomorphic curves should simplify the analysis. It would also be interesting to study the analogy between the results of this paper and the work relating geodesics in the space of K\"ahler metrics with families of holomorphic disks~\cite{chen-tian,donaldson-holomorphicdiscs,semmes}. We plan to address these points in future work.

\subsection{Outline}
In Section~\ref{section:background}, we collect relevant background material on Lagrangian submanifolds, especially in Calabi-Yau manifolds. Particular attention is paid to a generalization of the Weinstein neighborhood theorem adapted to Lagrangians with boundary in a collection of Lagrangians.
Section~\ref{section: Lagrangians with Cone Points} develops a formalism for differential analysis on immersed submanifolds with cone-points.
In Section~\ref{section: lagrangian and special lagrangian cylinders}, we study spaces of Lagrangian and imaginary special Lagrangian cylinders and prove Theorem~\ref{theorem: space of ISL cylinders is 1-dim}. We define relative Lagrangian flux and several types of regular families of imaginary special Lagrangian cylinders.
Section~\ref{section: the relation between cylinders and geodesics} gives the proofs of Theorems~\ref{theorem: family} and~\ref{theorem: geodesic-cylinder correspondence}.
Finally, Section~\ref{section: proof of perturbation theorem} gives the proof of Theorem~\ref{theorem: perturbation of geodesic}.

\subsection{Acknowledgments}
The authors would like to thank T. Collins, I. Haran, Y. Rubinstein, P. Seidel, and G. Tian, for helpful conversations.
The authors were partially supported by ERC starting grant 337560 and ISF grant 569/18. The second author was partially supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. The authors would like to thank the Institute for Advanced Study for its hospitality in the initial stages of the research that led to this paper, funded by the Erik Ellentuck Fellowship and the IAS Fund for Math.

	\section{Background}
\label{section:background}

	\subsection{Weinstein neighborhoods and immersed Lagrangian submanifolds with boundary}
	\label{subsection: Weinstein neighborhoods}

Let $N,M$ be smooth manifolds, $M$ perhaps with boundary. Denote by $\diff(M)$ the diffeomorphisms of
$M$	preserving each boundary component. That is, if $\varphi \in \diff(M)$ and $B \subset \partial M$ is a component, then $\varphi(B) = B.$
\begin{dfn}
		\label{definition: immersed submanifold}
		An \emph{immersed (resp. embedded) submanifold of $N$ of type $M$} is an equivalence class of immersions (resp. embeddings),
		\[
		K=[f:M\to N],
		\]
		where the equivalence is with respect to the $\diff(M)$-action: The immersions $f$ and $f'$ are equivalent if there exists $\varphi\in\diff(M)$ such that
		\[
		f'=f\circ\varphi.
		\]
We say that $K=[f]$ is \emph{free} if $f$ has trivial isotropy subgroup. We say that $K$ has boundary if $M$ does. In this case, to each boundary component of $M$ we associate a boundary component of $K,$ which is itself an immersed submanifold.
	\end{dfn}

\begin{dfn}\label{definition:points}
	Let $K$ be an immersed submanifold of $N$ of type $M$. A \emph{point} $p\in K$ is an equivalence class of pairs, $p=[(f,q)]$, where $f:M\to N$ represents $K$ and $q\in M$. The pairs $(f,q)$ and $(f',q')$ are equivalent if there exists $\varphi\in\diff(M)$ such that
	\[
	f'=f\circ\varphi,\qquad\varphi(q')=q.
	\]
	For a point $p=[f,q]$ of  $K=[f]$, we let $p_0$ denote the image of $p$ in $N$,
	\[
	p_0:=f(q).
	\]
We say that $p$ is \emph{embedded} if $f^{-1}(p_0) = \{q\}.$
By abuse of notation, we abbreviate $p = f(q)$ when this does not lead to confusion. That is, we may consider $f(q)$ as a point of $K$ instead of as a point of $N.$

We define the \emph{tangent space} at $p$ by
	\[
	T_pK:=df_q(T_qM)\subset T_{p_0}N.
	\]
	The tangent bundle $TK$ is naturally an immersed submanifold of $TM$. A \emph{differential form} on $K$ is an equivalence class of pairs $\eta = [(f,\tau)]$ where $f$ is a representative of $K$ and $\tau \in \Omega^*(M).$ The pairs $(f,\tau)$ and $(f',\tau')$ are equivalent if there exists $\varphi \in \diff(M)$ such that $f' = f \circ \varphi$ and $\tau' = f^* \tau.$
An \emph{open subset} of $K$ is an immersed submanifold of $N$ of the form $[f|_U]$ where $f : M \to N$ represents $K$ and $U \subset M$ is open. For a differential form $\eta = [(f,\tau)]$ on $K,$ we may write $\tau = f^*\eta.$ When $\eta$ is a zero form, that is, a function, we also write $\tau = \eta\circ f.$

Let $K_i$ be immersed submanifolds of $N_i$ of type $M_i$ for $i = 0,1.$ A \emph{smooth map} $g : K \to K'$ is an equivalence class of triples $(f_0,f_1,h)$ where $h : M_0 \to M_1$ is smooth and $f_i : M_i \to N_i$ represents $K_i.$ Two such triples $(f_0,f_1,h)$ and $(f_0',f_1',h')$ are equivalent if there exist $\varphi_i \in \diff(M_i)$ such that $f_i' = f_i \circ \varphi_i$ and $h' = \varphi_1^{-1} \circ h \circ \varphi_0.$ We say $g$ is a diffeomorphism, embedding, and so on, if it is represented by a triple $(f_0,f_1,h)$ where $h$ is a diffeomorphism, embedding, and so on.
\end{dfn}

\begin{rem}\label{remark:tomakejakehappybutdeepdownalsoamitai}
Every point of an immersed submanifold has a neighborhood that is embedded. An embedded submanifold $K = [f : M \to N]$ is canonically diffeomorphic to the submanifold of type $f(M)$ represented by the inclusion $f(M) \hookrightarrow N.$ So, the usual notions of points, maps, and so on apply, and we do not need the added complexity of Definition~\ref{definition:points}.
\end{rem}

Lemma~\ref{lemma: michor's lemma} below is a rephrasing of~\cite[Lemma~1.4]{cervera-mascaro-michor}, which provides a sufficient condition for an immersed submanifold to be free. Lemma~\ref{lemma: embedded boundary point implies freeness} is a variant.

\begin{lemma}
	\label{lemma: michor's lemma}
	Let $K$ be a connected immersed submanifold of $N$ of type $M,$ and suppose $K$ has an embedded point. Then $K$ is free.
\end{lemma}

	 \begin{lemma}
	 	\label{lemma: embedded boundary point implies freeness}
	 	Let $K$ be an immersed submanifold of $N$ modeled on $M.$ Suppose that for each connected component $C \subset M$ there is a boundary component $B \subset \partial C$ such that the corresponding boundary component of $K$ has an embedded point. Then $K$ is free.
	 \end{lemma}

 	\begin{proof}
 	Let $f: M \to N$ be a representative of $K$ and let $\varphi \in \diff(M)$ be such that $f \circ \varphi = f.$ Let $C \subset M,$ and $B \subset \partial C$ be components. Suppose $q \in B$ is such that $p = [(f|_B,q)]$ is an embedded point of the immersed submanifold $[f|_B].$  As $\varphi$ preserves each boundary component of $M$, we have $\varphi(q)\in B$. Since $p$ is an embedded point of $[f|_B],$ the equality
 		\[
 		f(\varphi(q))=f(q)
 		\]
 		implies
 		\[
 		\varphi(q)=q.
 		\]
 		Now, by local injectivity of $f$, continuity of $\varphi$ and the equality $f\circ\varphi=f,$ the subset
 		\[
 		U:=\{r\in M\;|\varphi(r)=r\}\subset M
 		\]
 		is open. As $U$ is also closed and $q\in U,$ we deduce $\varphi|_C=\id_{C}$ and complete the proof.
 	\end{proof}

	Let $(X,\omega)$ be a symplectic manifold of dimension $2n,$ and let $L$ be a smooth manifold of dimension $n,$ perhaps with boundary.

	\begin{dfn}
		\label{definition: Lagrangian submanifold}
		An immersion $f:L\to X$ is said to be \emph{Lagrangian} if it satisfies $f^*\omega=0$. Let $f,f':L\to X$ be immersions such that, for some $\varphi\in\diff(L),$ we have $f'=f\circ\varphi.$ Then $f$ is Lagrangian if and only if $f'$ is. We say the associated immersed submanifold $[f]$ is \emph{Lagrangian} if $f$ is Lagrangian.
	\end{dfn}

\begin{ntn}	\label{notation:space of Lagrangians}
	If $L$ has no boundary, we let $\mathcal{L}(X,L)$ denote the space of free immersed Lagrangian submanifolds of $X$ of type $L$. Suppose now that $L$ has $k$ boundary components $B_1,\ldots,B_k,$ for some $k\in\mathbb{N}$, and let $\Lambda_1,\ldots,\Lambda_k\subset X$ be fixed Lagrangians. Let $\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)$ denote the space of free immersed Lagrangian submanifolds $Z\subset X$ of type $L$ with boundary components $C_1,\ldots,C_k,$ corresponding to the boundary components $B_1,\ldots,B_k,$ of $L$ such that the following hold.
	\begin{enumerate}[label=(\alph*)]
		\item For $i=1,\ldots,k,$ the boundary component $C_i$ is an immersed submanifold of $\Lambda_i$.
		\item \label{not tangent} For $i=1,\ldots,k,$ and $p\in C_i,$ we have $T_pZ\ne T_{p_0}\Lambda_i$.
	\end{enumerate}
\end{ntn}
	The spaces $\mathcal{L}(X,L)$ and $\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)$ are Fr\'echet manifolds locally parameterized by spaces of closed $1$-forms. The former is treated in~\cite{akveld-salamon} for embedded Lagrangians, and the generalization to free immersed Lagrangians is performed by means of~\cite[Theorem~1.5]{cervera-mascaro-michor}. In order to study the latter, we formulate the Lagrangian boundary condition in terms of differential forms. This is done below by employing the following version of the Weinstein neighborhood theorem. For a smooth manifold $M$ and a submanifold $Q\subset M$ we let $\nu_Q$ denote the conormal bundle of $Q$ in $T^*M$.

	\begin{lemma}
		\label{lemma: Weinstein with boundary}
		Suppose $L$ is compact with $k$ boundary components, $C_1,\ldots,C_k.$ Let $\Lambda_1,\ldots,\Lambda_k\subset X$ be fixed embedded Lagrangian submanifolds and let $Z=[f:L\to X]\in\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)$. Then there exist an open neighborhood of the zero section, $V\subset T^*L$, and a local symplectomorphism $\varphi:V\to X$, such that the following hold.
		\begin{enumerate}[label=(\alph*)]
			\item Identifying $L$ with the zero section in $T^*L,$ we have
			\[
			\varphi|_L=f.
			\]
			\item For $i=1,\ldots,k$, a point $p\in C_i$ and a covector $\xi\in T^*_pL,$ we have
			\[
			\varphi(p,\xi)\in\Lambda_i\Leftrightarrow(p,\xi)\in \nu_{C_i}.
			\]
		\end{enumerate}
	\end{lemma}

	\begin{proof}
		We follow the lines of the argument by Moser presented in~\cite[Section~3.2]{babymcduff-salamon}, making the necessary adaptations. First we construct a smooth subbundle $E\subset f^*TX\to L$ satisfying the following.
		\begin{enumerate}
			\item\label{fiber is Lag} For $p\in L,$ the fiber $E_p\subset T_{f(p)}X$ is Lagrangian.
			\item\label{direct sum} We have $f^*TX=df(TL)\oplus E,$ where $df : TL \to f^*TX$ is the differential of the immersion $f.$
			\item\label{compatible with boundary} For $p\in C_i,\;i=1,\ldots,k,$ we have
			\[
			E_p\cap T_{f(p)}\Lambda_i\ne\{0\}.
			\]
			It then follows that $\dim\left(E_p\cap T_{f(p)}\Lambda_i\right)=1.$
		\end{enumerate}
		To do so, let $J$ be an $\omega$-compatible almost complex structure. Note that, if the Lagrangian $Z$ were closed, the bundle $JTZ\subset f^*TX$ would be a good choice for~$E.$ In our case, however, $JTZ$ does not necessarily satisfy condition~\eqref{compatible with boundary}. We thus perform the following perturbation. For $p\in L$ and any number $a\in\R,$ the linear space
		\[
		E_{p,a}:=(J+a)df(T_{p}L)\subset T_{f(p)}X
		\]
		satisfies conditions~\eqref{fiber is Lag} and~\eqref{direct sum}. Condition~\eqref{compatible with boundary} determines $a$ uniquely for $p \in \partial L$. Here, we use condition~\ref{not tangent} in the definition of $\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k).$ Hence, constructing $E$ amounts to extending the smooth assignment $p\mapsto a(p)$ from $\partial L$ to all~$L$.

		Contraction with $\omega$ along the immersion $f$ yields an isomorphism of vector bundles $E\cong T^*L.$ We thus think of $T^*L$ as a subbundle of $f^*TX$. Condition~\eqref{compatible with boundary} in the construction of $E$ implies that for $p\in C_i,i=1,\ldots,k,$ the annihilator $(T_pC_i)^0\subset T_p^*L$ is identified with a line in $T_{f(p)}\Lambda_i.$

		For $i=1,\ldots,k,$ let $\nabla^i$ be a connection on $TX,$ with respect to which the Lagrangian $\Lambda_i$ is totally geodesic. Let $\left(\alpha_i:L\to\mathbb{R}\right)_{i=1}^k$ be a smooth partition of unity with
		\[
		\alpha_i|_{C_i}\equiv1,\quad i=1,\ldots,k.
		\]
		For a connection $\nabla$ on $TX,$ let $\exp^\nabla$ denote the exponential map of $\nabla.$ To each $p\in L$ we assign a connection $\nabla^p$ on $TX$ given by
		\[
		\nabla^p:=\sum_i\alpha_i(p)\nabla^i.
		\]
		Set
		\[
		\widetilde{V}:=\left\{(p,\xi)\in T^*L\,\left|\,\exp^{\nabla^p}_{f(p)}(\xi)\;\mathrm{exists}\right.\right\}
		\]
		and define
		\[
		\widetilde{\varphi}:\widetilde{V}\to X,\qquad(p,\xi)\mapsto\exp^{\nabla^p}_{f(p)}(\xi).
		\]
		Shrinking $\widetilde{V}$ if necessary, $\widetilde{\varphi}$ is an immersion and one-to-one on each fiber. By construction, we have
\begin{equation}
\label{equation: conormal mapped into Lambda}
\widetilde{\varphi}\left(\nu_{C_i}\cap\widetilde{V}\right)\subset\Lambda_i,\quad i=1,\ldots,k.
\end{equation}

Let $\omega_0$ denote the canonical symplectic form on $\widetilde{V}\subset T^*L$. Write
		\[
		\omega_1:=\widetilde{\varphi}^*\omega.
		\]
		Then $\omega_0$ and $\omega_1$ coincide on $T\widetilde{V}|_L$. Also, by~\eqref{equation: conormal mapped into Lambda}, the conormal bundle $\nu_{C_i}$ is Lagrangian with respect to both forms for $i=1,\ldots,k.$ We now construct the desired $V\subset T^*L$ and an open embedding $\chi:V\hookrightarrow\widetilde{V},$ satisfying the following:
		\begin{enumerate}
			\item $\chi^*\omega_1=\omega_0$.
			\item For $q\in V$ and $i=1,\ldots,k,$ we have
			\[
			\chi(q)\in\nu_{C_i}\Leftrightarrow q\in\nu_{C_i}.
			\]
			\item For $q\in L\subset V$ we have $\chi(q)=q$.
		\end{enumerate}
		This will complete the proof since we can take $\varphi:=\widetilde{\varphi}\circ\chi$. The embedding $\chi$ is obtained as the flow of a time-dependent vector field as follows. Let
		\[
		H:\widetilde{V}\times[0,1]\to\widetilde{V},\qquad((p,\xi),t)\mapsto(p,t\xi)
		\]
		and
		\[
		\pi:\widetilde{V}\times[0,1]\to\widetilde{V},\qquad((p,\xi),t)\mapsto(p,\xi).
		\]
		Let $\eta$ be the 1-form on $\widetilde{V}$ given by
		\[
		\eta:=\pi_*H^*(\omega_1-\omega_0),
		\]
		where $\pi_*$ denotes pushing forward by integration along the fibers of $\pi$. For a discussion of the properties of integration along the fiber, see~\cite[Section~3.1]{kupferman-solomon}. Then $\eta$ satisfies
		\begin{equation}
		\label{equation: properties of eta}
		d\eta=\omega_1-\omega_0,\quad\eta|_{T_L\widetilde{V}}=0,\quad\eta|_{\nu_{C_i}}=0,\;i=1,\ldots,k.
		\end{equation}
		Shrinking $\widetilde{V}$ again, we may assume that, for $t\in[0,1],$ the closed 2-form $\omega_t:=t\omega_1+(1-t)\omega_0$ is non-degenerate. For $t\in[0,1],$ define a vector field $u_t$ on $\widetilde{V}$ by
		\[
		i_{u_t}\omega_t=-\eta.
		\]
		Then $u_t$ vanishes on $L$ and is tangent to $\nu_{C_i}$ for $i=1,\ldots,k.$ Let $\chi_t,\;t\in[0,1],$ denote the flow of $u_t$. That is,
		\[
		\chi_0=\id,\qquad\deriv{t}\chi_t=u_t\circ\chi_t.
		\]
		Let $V$ be the domain of $\chi:=\chi_1$. By the Cartan formula we have
		\begin{align*}
		\deriv{t}\chi_t^*\omega_t
		&=\chi_t^*\left(\deriv{t}\omega_t+i_{u_t}d\omega_t+di_{u_t}\omega_t\right)\\
		&=\chi_t^*((\omega_1-\omega_0)-d\eta)\\
		&=0.
		\end{align*}
		Hence we have $\chi^*\omega_1=\omega_0,$ as desired.
	\end{proof}

	\begin{dfn}
		We call the pair $(V,\varphi)$ of Lemma~\ref{lemma: Weinstein with boundary} an \emph{immersed Weinstein neighborhood of $Z$ compatible with $\Lambda_1,\ldots,\Lambda_k$}.
	\end{dfn}

	We now show the space $\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)$ is a Fr\'echet manifold. For a manifold $M$ with boundary, we let $\Omega^1(M)$ denote the space of smooth 1-forms on $M$ and $\Omega_B^1(M)\subset\Omega^1(M)$ the Fr\'echet subspace consisting of closed forms annihilating the boundary.

	\begin{cor}
		\label{corollary: frechet}
		Suppose $L$ is compact with $k$ boundary components and let $\Lambda_i\subset X,\; i=1,\ldots,k,$ be fixed embedded Lagrangians. Then $\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)$ is a Fr\'echet manifold. In fact, for $Z \in \mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)$ an immersed Weinstein neighborhood of $Z$ compatible with $\Lambda_1,\dots,\Lambda_k,$ gives rise to a local parameterization
		\[ \mathbf{X}:\mathcal{U}\subset\Omega_B^1(L)\to\widetilde{\mathcal{U}}\subset\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k).
		\]
	\end{cor}

	\begin{proof}
		Let $Z\in\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)$ and choose a Lagrangian immersion $f:L\to X$ representing $Z.$ Let $(V,\varphi)$ be a compatible immersed Weinstein neighborhood. Set
		\[
		\mathcal{U}:=\left\{\left.\alpha\in\Omega_B^1(L)\;\right|\;\mathrm{Graph}(\alpha)\subset V\right\}.
		\]
		Then $\mathcal{U}$ is open in $\Omega_B^1(L).$ For $\alpha\in\mathcal{U},$ write
		\[
		\mathcal{G}_\alpha:=\varphi\left(\mathrm{Graph}(\alpha)\right)\in\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k).
		\]
		One verifies that the map $\mathbf{X}$ given by $\alpha\mapsto\mathcal{G}_\alpha$ is one-to-one and onto an open subset in $\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)$ equipped with the quotient topology (compare with~\cite[Theorem~1.5]{cervera-mascaro-michor}). Here we rely on freeness of $Z.$ Repeating the above procedure for a Lagrangian $Z'\in\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)$ close to $Z,$ the induced transition map $\mathcal{U}\to\mathcal{U}'\subset\Omega_B^1(L)$ is smooth.
	\end{proof}

	Lemma~\ref{lemma: tangent space} and Remark~\ref{remark: tangent space hands on} below describe tangent vectors in spaces of Lagrangian submanifolds explicitly. Lemma~\ref{lemma: tangent space} \ref{tangent space of closed lag} is stated and proved in~\cite{akveld-salamon}, while part \ref{tangent space of lag with boundary} is the analogue for Lagrangians with boundary. Remark~\ref{remark: tangent space hands on} provides a hands-on approach.

	\begin{lemma}
		\label{lemma: tangent space}$\;$
		\begin{enumerate}[label=(\alph*)]
			\item\label{tangent space of closed lag} Suppose $L$ is closed. For $Z\in\mathcal{L}(X,L)$, the tangent space $T_Z\mathcal{L}(X,L)$ is canonically isomorphic to the space of closed $1$-forms on $Z$.
			\item\label{tangent space of lag with boundary} Suppose $L$ has $k$ boundary components. For $Z\in\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)$, the tangent space $T_Z\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)$ is canonically isomorphic to the space $\Omega_B^1(Z).$
		\end{enumerate}
	\end{lemma}

	\begin{rem}
		\label{remark: tangent space hands on}
		The canonical isomorphism of Lemma~\ref{lemma: tangent space} can be understood as follows. Let $(Z_t)_{t\in(-\epsilon,\epsilon)}$ be a smooth path of Lagrangians diffeomorphic to~$L,$ which may or may not have boundary. Let $\Psi_t:L\to X$ for $t\in(-\epsilon,\epsilon)$ be a smooth family of immersions such that $\Psi_t$ represents $Z_t.$ Then, recalling Definition~\ref{definition:points}, we have
		\[
		\deriv{t}Z_t=\left[\left(\Psi_t,i_{\deriv{t}\Psi_t}\omega\right)\right] \in \Omega_B^1(Z_t).
		\]
	\end{rem}

	\begin{dfn}$\;$
		\begin{enumerate}[label=(\alph*)]
			\item Suppose $L$ is closed. Let $\Lambda=(\Lambda_t)_{t\in[0,1]}\subset\mathcal{L}(X,L)$ be a smooth path. We say $\Lambda$ is \emph{exact} if $\deriv{t}\Lambda_t$ is exact for $t\in[0,1]$.
			\item Suppose $L$ has $k$ boundary components. Let
\[
Z=(Z_t)_{t\in[0,1]}\subset\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)
\]
be a smooth path. We say $Z$ is \emph{exact relative to the boundary} if $\deriv{t}Z_t$ is exact relative to the boundary for $t\in[0,1]$, that is, if there exists a function $h_t\in C^\infty(Z_t)$ vanishing on $\partial Z_t$ such that $\deriv{t}Z_t=dh_t.$
		\end{enumerate}
	\end{dfn}

	\begin{rem}
		\label{remark: exact is Hamiltonian}
		Recall that a smooth path of embedded closed Lagrangians is exact if and only if it is induced by a Hamiltonian flow on $X$~\cite{akveld-salamon}.
	\end{rem}

	\subsection{Calabi-Yau manifolds, special Lagrangians and the space of positive Lagrangians}
	\label{subsection: calabi-yau manifolds}

	\begin{dfn}
		A \emph{Calabi-Yau manifold} is a quadruple $(X,\omega,J,\Omega)$, where $(X,\omega)$ is a symplectic manifold, $J$ is an $\omega$-compatible integrable complex structure, and $\Omega$ is a nowhere-vanishing holomorphic $(n,0)$ form (with respect to $J$). In particular, $X$ is a K\"ahler manifold with the metric
		\[
		g=g_J=\omega(\cdot,J\cdot).
		\]
	\end{dfn}

	\begin{rem}
		In the literature, various definitions of Calabi-Yau manifolds can be found. In previous work by the authors, the notion used here would be called an \emph{almost Calabi-Yau manifold}. A more restrictive definition adds the requirement $\rho\equiv1,$ where $\rho$ is the function defined in~\eqref{equation: rho}.
	\end{rem}

	In what follows, we fix a Calabi-Yau manifold $(X,\omega,J,\Omega)$. Recall the following observation of~\cite{harvey-lawson}.

	\begin{lemma}
		\label{lemma: Omega does not vanish on Lags}
		Let $p\in X$ and let $S\subset T_pX$ be an oriented Lagrangian subspace. Then $\Omega$ does not vanish on $S$. In fact, for an oriented basis $v_1,\ldots,v_n\in S,$ we have
		\[
		|\Omega(v_1,\ldots,v_n)|=\rho\vol(v_1,\ldots,v_n),
		\]
		where $\vol$ denotes the Riemannian volume form.
	\end{lemma}

	In particular, it follows from Lemma~\ref{lemma: Omega does not vanish on Lags} that if $\imaginary \Omega|_\Lambda = 0,$ then $\real\Omega|_\Lambda$ is non-vanishing. If $\real\Omega|_\Lambda = 0,$ then $\imaginary\Omega|_\Lambda$ is non-vanishing.

\begin{dfn}
Let $p \in X$ and let $S \subset T_pX$ be an oriented Lagrangian subspace. The \emph{phase} $\theta_S \in S^1$ of $S$ is the argument of $\Omega(v_1,\ldots,v_n)$ for $v_1,\ldots,v_n \in S$ an oriented basis. We say that $S$ is positive if $\theta_S \in \left(-\frac{\pi}{2},\frac{\pi}{2}\right).$ An oriented Lagrangian submanifold $\Lambda\subset X$ has a \emph{phase function} $\theta_\Lambda : \Lambda \to S^1$ given by $\theta_\Lambda(p) = \theta_{T_p\Lambda}.$ The Lagrangian $\Lambda$ is positive if all its tangent spaces are positive.
\end{dfn}

	Let $\mathcal{O}$ be a Hamiltonian isotopy class of closed embedded positive Lagrangians in $X$ of type $L$, and let $\Lambda\in\mathcal{O}$. By Remark~\ref{remark: exact is Hamiltonian}, the tangent space $T_\Lambda\mathcal{O}$ consists of exact 1-forms on $\Lambda$. Positivity of $\Lambda$ yields the isomorphism~\eqref{equation: tangent space to O}, which in turn gives rise to the Riemannian metric $(\cdot,\cdot)$ defined in~\eqref{equation: the metric}. As shown in~\cite{solomon-curv}, the metric $(\cdot,\cdot)$ has a Levi-Civita connection.

	\begin{dfn}
		Let $\Lambda=(\Lambda_t)_{t\in[0,1]}$ be a smooth path in $\mathcal{O}$. A \emph{lifting} of $\Lambda$ is a smooth path of embeddings $\Psi_t:L\to X,\;t\in[0,1],$ such that $\Psi_t$ represents $\Lambda_t$. A lifting $(\Psi_t)$ is \emph{horizontal} if it satisfies
		\[
		i_{\deriv{t}\Psi_t}\real\Omega=0,\quad t\in[0,1].
		\]
	\end{dfn}

	It is shown in~\cite{solomon} that, for a path $\Lambda$ as above, every embedding representing $\Lambda_0$ extends uniquely to a horizontal lifting. This allows us to describe the Levi-Civita connection of $(\cdot,\cdot)$ as follows. Let $u_t,\;t\in[0,1],$ be a vector field along $\Lambda$. That is,
	\[
	u_t\in\vmv(\Lambda_t),\qquad t\in[0,1].
	\]
	Let $(\Psi_t)$ be a horizontal lifting of $\Lambda$. Then the covariant derivative of $u_t$  is given by
	\[
	\left(\frac{D}{dt}u_t\right)\circ\Psi_t=\deriv{t}(u_t\circ\Psi_t).
	\]
	Thus, the path $\Lambda$ is a \emph{geodesic} if and only if it admits a lifting $(\Psi_t)_t$ and a family of functions $h_t:\Lambda_t\to\R$ satisfying the equations
	\begin{equation}
	\label{equation: geodesic}
	i_{\deriv{t}\Psi_t}\omega = d(h_t\circ\Psi_t), \qquad i_{\deriv{t}\Psi_t} \real \Omega = 0, \qquad \deriv{t}(h_t \circ \Psi_t) = 0.
	\end{equation}
	This equivalent definition allows us to extend the notion of geodesics to a larger class of Lagrangian paths. Indeed, it makes sense for paths of immersed Lagrangians which are not necessarily embedded or closed. In the present article we also consider geodesics of non-smooth Lagrangians with singular locus consisting of finitely many cone points. We provide an explicit definition of geodesics in a more general form after setting up the necessary theory of submanifolds with cone points.

	\section{Lagrangians with cone points}
	\label{section: Lagrangians with Cone Points}

	For the purposes of this article it is natural to consider Lagrangians with \emph{cone points} and paths thereof. In Section~\ref{subsection: oriented blowups and cone-immersed submanifolds} below we discuss differentiability on general submanifolds with cone points, where no additional geometric structure is assumed. In Section~\ref{subsection: cone-immersed Lagrangians} we discuss spaces of positive Lagrangians with cone points in a Calabi-Yau manifold.

	\subsection{Oriented blowups and cone-smooth differential topology}
	\label{subsection: oriented blowups and cone-immersed submanifolds}

	\begin{dfn}
		\label{definition: blowup}
		$\;$
		\begin{enumerate}
			\item Let $n\in\mathbb{N}$ and let $V$ be a real vector space of dimension $n.$ For $0\ne v\in V,$ the \emph{ray} spanned by $v$ is the subset $\{\lambda v\;|\;\lambda\ge0\}\subset V.$ The \emph{oriented projective space} $\orientedprojective(V)$ is defined to be the set of rays in $V.$ As a set, $\orientedprojective(V)$ is naturally identified with the quotient $(V\setminus\{0\})/\mathbb{R}^+.$ We equip $\orientedprojective(V)$ with the smooth structure making this identification a diffeomorphism. In particular, $\orientedprojective(V)$ is diffeomorphic to the sphere $S^{n-1}.$ The \emph{oriented blowup} of $V$ is defined by
			\[
			\widetilde{V}:=\left\{\left.(q,r)\in V\times\orientedprojective(V)\;\right|\;q\in r\right\}.
			\]
			The \emph{blowup projection} $\pi:\widetilde{V}\to V$ is given by $(q,r)\mapsto q.$ We call $E=\pi^{-1}(0)$ the \emph{exceptional sphere}. Then $\widetilde{V}$ is a smooth manifold with boundary $E.$ The restricted blowup projection $\pi|_{\widetilde{V}\setminus E}:\widetilde{V}\setminus E\to V\setminus\{0\}$ is a diffeomorphism.
			\item \label{blowup at a single point} Let $M$ be an $n$-dimensional smooth manifold and let $p\in M.$ Let $p\in U\subset M$ be open with a diffeomorphism $\varphi:U\to\mathbb{R}^n$ carrying $p$ to $0.$ Let $\widetilde{U}$ denote the oriented blowup of $U$ with respect to the vector space structure induced by $\varphi$, let $\pi_U$ denote the blowup projection and let $E$ denote the exceptional sphere. We define the \emph{oriented blowup of $M$ at $p$} by gluing $\widetilde{U}$ and $M\setminus\{p\}$ along $\pi_U:$
			\[
			\widetilde{M}_p:=\left(\widetilde{U}\cup (M\setminus\{p\})\right)/(q,r)\sim q,\;(q,r)\in\widetilde{U}\setminus E.
			\]
			The projection $\pi_U$ then extends to the global blowup projection $\pi:\widetilde{M}_p\to M.$ We call $E_p=\pi^{-1}(p)$ the \emph{exceptional sphere over $p$.} One verifies that everything defined here is independent of $U$ and $\varphi,$ and the exceptional sphere $E_p$ is naturally identified with $\orientedprojective(T_pM).$ Once again, the blowup projection $\pi$ restricts to a diffeomorphism
			\[
			\pi|_{\widetilde{M}_p\setminus E}:\widetilde{M}_p\setminus E\to M\setminus\{p\}.
			\]
			\item\label{blowup at discrete set} Let $M$ be as in~\eqref{blowup at a single point}, and let $S\subset M$ be finite. For $p\in S,$ let $\widetilde{M}_p$ denote the oriented blowup of $M$ at $p.$ The \emph{oriented blowup of $M$ at $S$,} denoted by $\widetilde{M}_S,$ is obtained by gluing all the oriented blowups $\widetilde{M}_p\setminus\{S\setminus\{p\}\},\;p\in S,$ in the obvious manner. For $p\in S,$ the exceptional sphere over $p$ is naturally identified with $\orientedprojective(T_pM).$ The oriented blowup $\widetilde{M}_S$ is a smooth manifold with boundary equal to the disjoint union of the exceptional spheres. It comes with a blowup projection $\pi : \widetilde M_S \to M$ as before. We write \[
\widetilde{M}_S^\circ = \widetilde{M}_S\setminus\partial\widetilde{M}_S
\]
for the \emph{interior} of $\widetilde{M}_S.$
\item\label{cone coordinates}
Let $M$ and $S$ be as in \eqref{blowup at discrete set}. Let $p \in S$ and let $\widetilde{p} \in E_p.$ \emph{Cone coordinates} at $\widetilde{p}$ are a triple $(U,\mathbf{X},\alpha)$ where $\widetilde{p} \in U \subset E_p$ is open,
\[
\mathbf{X} = (x^1,\ldots,x^m): U \to \R^m
\]
are local coordinates, and $\alpha : U \times [0,\epsilon) \to \widetilde{M}_S$ is a smooth open embedding such that $\alpha(\widetilde q,0) = \widetilde q$ for $\widetilde q \in U.$ We generally denote the coordinate on $[0,\epsilon)$ by $s.$ Given a map $f : M \to N,$ we abbreviate
\[
\qquad\qquad f\circ \pi \circ \alpha(\widetilde q,s) = f(\widetilde q,s) = f(x_1,\ldots,x_m,s), \qquad  (x_1,\ldots,x_m) = \mathbf{X}(\widetilde q).
\]
		\end{enumerate}
	\end{dfn}

\begin{rem}\label{remark:polar coordinates}
Let $V$ be an $n$-dimensional real vector space. Given a smooth section~$\sigma$ of the quotient map $V\setminus\{0\} \to \orientedprojective(V),$ we identify
\[
\orientedprojective(V) \times [0,\infty) \overset{\sim}{\longrightarrow} \widetilde{V}
\]
by $(r,s) \mapsto (s \sigma(r),r)$. Since $\orientedprojective(V) \cong S^{n-1},$ we identify $S^{n-1} \times [0,\infty) \cong \widetilde{V}.$
\end{rem}

\begin{dfn}\label{definition:polar coordinates}
Let $M$ be a smooth manifold, let $p \in M.$ Let $\sigma$ be a smooth section of the quotient map $T_pM\setminus\{0\} \to \orientedprojective(T_pM) \cong S^{n-1}.$ A \emph{polar coordinate map} centered at $p$ associated with $\sigma$ is a smooth map
\[
\kappa : S^{n-1} \times [0,\epsilon) \to M
\]
satisfying the following.
\begin{enumerate}
\item
$\kappa|_{S^{n-1}\times\{0\}}$ is the constant map to $p.$
\item
$\kappa|_{S^{n-1}\times (0,\epsilon)}$ is an open embedding.
\item\label{item:sigma}
For $r \in S^{n-1},$
\[
\pderiv[\kappa]{s}(r,0) = \sigma(r).
\]
\end{enumerate}
An elementary argument shows that for every $\sigma$ there exist many polar coordinate maps. We may sometimes speak of a polar-coordinate map $\kappa$ without mentioning $\sigma$ explicitly. In this case, we refer to the section $\sigma$ determined by condition~(\ref{item:sigma}) as the section associated with $\kappa.$
\end{dfn}

	The oriented blowup gives rise to the following notions associated with differentiability, which are weaker than the usual ones.

	\begin{dfn}
		\label{definition: cone differentiability}
		Let $M$ and $N$ be smooth manifolds, let $p\in M,$ and let $\Psi:M\to N$ be continuous. Let $\widetilde{M}_p,E_p$ and $\pi:\widetilde{M}_p\to M$ be as in Definition~\ref{definition: blowup}.
		\begin{enumerate}
			\item The map $\Psi$ is said to be \emph{cone-smooth} at $p$ if there exists an open $E_p\subset\widetilde{U}\subset\widetilde{M}_p$ such that the composition $\Psi\circ\pi|_{\widetilde{U}}:\widetilde{U}\to N$ is smooth.
			\item Suppose $\Psi$ is cone-smooth at $p.$ The \emph{cone-derivative} of $\Psi$ at $p$ is the unique map
			\[
			d\Psi_p:T_pM\to T_{\Psi(p)}N
			\]
			satisfying the equality
			\[
			d(\Psi\circ\pi)_{\widetilde{p}}=d\Psi_p\circ d\pi_{\widetilde{p}}
			\]
			for $\widetilde{p}\in E_p.$ One verifies that the cone-derivative is well-defined and homogeneous of degree 1. Also, the restricted map $d\Psi_p|_{T_pM\setminus\{0\}}$ is smooth. Nevertheless, the cone-derivative is not linear in general.
			\item Suppose $\Psi$ is cone-smooth at $p.$ We say $\Psi$ is \emph{cone-immersive} at $p$ if the restricted map
			\[
			d\Psi_p|_{T_pM\setminus\{0\}}:T_pM\setminus\{0\}\to T_{\Psi(p)}N
			\]
			is a smooth immersion. In particular, in this case we have $d\Psi_p(v)\ne0$ for $0\ne v\in T_pM.$
		\end{enumerate}
	\end{dfn}

\begin{rem}
Let $\Psi : M \to N$ be cone-smooth at $p \in M,$ and let $\kappa : S^{n-1}\times [0,\epsilon) \to M$ be a polar coordinate map centered at $p.$ It follows from Remark~\ref{remark:polar coordinates} and the definition of the oriented blowup that, possibly after shrinking $\epsilon,$ the composition $\Psi \circ \kappa$ is smooth.
\end{rem}

	Recall that the \emph{Euler vector field} on a real vector space is the radial vector field that integrates to rescaling by $e^t.$ We let $\varepsilon$ denote the Euler vector field.
\begin{lemma}\label{lemma:immersion criterion}
Let $\Psi : M \to N$ be cone-smooth at $p$ and let $\kappa: S^{n-1}\times [0,\epsilon) \to M$ be a polar-coordinate map at $p.$ Then $\Psi$ is cone-immersive at $p$ if and only if
\[
\left.\pderiv{s}(\Psi \circ \kappa)\right|_{s = 0} : S^{n-1} \to T_{\Psi(p)}N
\]
is an immersion nowhere tangent to the Euler vector field.
\end{lemma}
\begin{proof}
Let $\sigma$ be the section associated with $\kappa.$ Let $r \in S^{n-1}.$ Observe that
\[
\pderiv{s}(\Psi \circ \kappa)(r,0) = d\Psi_p(\sigma(r)).
\]
So, by the chain rule,
\[
d(d\Psi_p)_{\sigma(r)}\circ d\sigma_r = d\left(\left.\pderiv{s}(\Psi \circ \kappa)\right|_{s=0}\right)_r.
\]
Also,
\[
d(d\Psi_p)_{\sigma(r)}(\varepsilon(\sigma(r))) = \varepsilon(d\Psi_p(\sigma(r))).
\]
Finally, observe that
\[
T_{\sigma(r)}T_pM  = \R\langle\varepsilon(\sigma(r))\rangle \oplus d\sigma_r\left(T_rS^{n-1}\right).
\]
The lemma follows.
\end{proof}

	\begin{lemma}
		\label{lemma: cone-immersions}
		In the setting of Definition~\ref{definition: cone differentiability}, suppose $\Psi$ is cone-immersive at $p.$
		\begin{enumerate}[label=(\alph*)]
			\item \label{sort of locally one-to-one} There exists an open neighborhood $p\in V\subset M$ such that, for $p\ne q\in V,$ we have $\Psi(q)\ne\Psi(p).$
			\item \label{existence of lifting} Let $p\in V\subset M$ as in \ref{sort of locally one-to-one} and let $\widetilde{V}:=\pi^{-1}(V)\subset\widetilde{M}_p.$ Then there exists a unique continuous $\widetilde{\Psi}:\widetilde{V}\to\widetilde{N}_{\Psi(p)}$ satisfying
			\begin{equation}
			\label{equality: strict transform}
			\pi_N\circ\widetilde{\Psi}=\Psi\circ\pi|_{\widetilde{V}}\;,
			\end{equation}
			where $\widetilde{N}_{\Psi(p)}$ denotes the oriented blowup of $N$ at $\Psi(p)$ and $\pi_N$ is the associated blowup projection. Moreover, for some open subset $E_p\subset \widetilde{V}'\subset\widetilde{V},$ the restricted map $\widetilde{\Psi}|_{\widetilde{V}'}$ is a smooth immersion carrying $E_p$ into $E_{\Psi(p)}$ and satisfying, for $\widetilde{p}\in E_p$ and $v\in T_{\widetilde{p}}\widetilde{M}_p,$
			\begin{equation}\label{implication:notangent}
			d\widetilde{\Psi}_{\widetilde{p}}(v)\in T_{\widetilde{\Psi}\left(\widetilde{p}\right)}E_{\Psi(p)}\Leftrightarrow v\in T_{\widetilde{p}}E_p.
			\end{equation}
		\end{enumerate}
	\end{lemma}

	\begin{proof}
		As the lemma is local, we may assume $M=\mathbb{R}^{m+1}$ and $N=\mathbb{R}^{n+1}.$ Keeping in mind Remark~\ref{remark:polar coordinates}, we have
		\[
		\Psi\circ\pi:S^m\times[0,\infty)\to\mathbb{R}^{n+1},\quad\Psi\circ\pi|_{S^m\times\{0\}}=0.
		\]
		Let $s$ denote the $[0,\infty)-$coordinate. Since $\Psi$ is cone-immersive, for $r \in S^m,$ the one-sided directional derivative $\pderiv[(\Psi\circ\pi)]{s}|_{(r,0)}$ is non-vanishing. So, there exist $\epsilon~>~0,$ an open half space $H \subset \R^{n+1},$ and a neighborhood $r \in U \subset S^m,$ such that $\pderiv[(\Psi\circ\pi)]{s}|_{(r',s')} \in H$ for $(r',s') \in U  \times [0,\epsilon).$ By the mean-value theorem,
$\Psi\circ\pi(r',s') \neq 0$
for $(r', s') \in U\times(0,\epsilon).$  Part \ref{sort of locally one-to-one} now follows from compactness of~$S^m.$

		Again by Remark~\ref{remark:polar coordinates} we have $\widetilde{N}_{\Psi(p)}=\orientedprojective\left(\mathbb{R}^{n+1}\right)\times[0,\infty).$ Let
\[
\orientedprojective(d\Psi_0):\orientedprojective\left(\mathbb{R}^{m+1}\right)\to \orientedprojective\left(\mathbb{R}^{n+1}\right)
\]
denote the oriented projectivization of $d\Psi_0.$ By assumption, $\orientedprojective(d\Psi_0)$ is a well-defined smooth immersion. The equality~\eqref{equality: strict transform} determines the desired $\widetilde{\Psi}$ of part \ref{existence of lifting} uniquely away from the exceptional sphere $\orientedprojective\left(\mathbb{R}^{m+1}\right)\times\{0\}.$ We set
		\[
		\widetilde{\Psi}(c,0):=\left(\orientedprojective(d\Psi_0)(c),0\right),\quad c\in \orientedprojective\left(\mathbb{R}^{m+1}\right).
		\]
		One verifies by a straightforward computation that this is indeed a smooth continuation of $\widetilde{\Psi}|_{\orientedprojective\left(\mathbb{R}^{m+1}\right)\times(0,\infty)}.$ Also, the derivative $d\widetilde{\Psi}_{(c,0)}$ is one-to-one for $c\in \orientedprojective\left(\mathbb{R}^{m+1}\right).$  Since being an immersion is an open property, we can find $\widetilde{V}'$ as desired. One direction of implication~\eqref{implication:notangent} is trivial. To see the other direction, suppose $v\in T_{\widetilde{p}}\widetilde{M}_p \setminus T_{\widetilde{p}}E_p.$ Then, $d\pi(v) \neq 0,$ so $d\Psi_p \circ d\pi(v) \neq 0.$ It follows that $d\widetilde{\Psi}_{\widetilde{p}}(v)\notin T_{\widetilde{\Psi}\left(\widetilde{p}\right)}E_{\Psi(p)}$ as desired.
	\end{proof}

	\begin{dfn}
		\label{definition: strict transform}
		The map $\widetilde{\Psi}$ of Lemma~\ref{lemma: cone-immersions} \ref{existence of lifting} is called the \emph{strict transform} of $\Psi|_V.$
	\end{dfn}

	\begin{dfn}\label{definition:cone immersed submanifold}
		Let $M$ and $N$ be smooth manifolds. Let $S\subset M$ be a finite subset and $\Psi:M\to N$ a continuous map.
		\begin{enumerate}[label=(\alph*)]
        \item
    We say the pair $(\Psi,S)$ is \emph{cone-smooth} if $\Psi$ is smooth away from $S$ and cone-smooth at every element of $S.$
			\item We say the pair $(\Psi, S)$ is a \emph{cone-immersion} from $(M,S)$ to $N$ if $\Psi$ is a smooth immersion away from $S$ and cone-immersive at every element of $S.$
    \item
    Suppose $N = M$ and $\Psi(p) = p$ for all $p \in S.$ The pair $(\Psi,S)$ is a \emph{cone-diffeomorphism} of $(M,S)$ if $\Psi$ is a smooth diffeomorphism away from $S$ and for $p \in S,$ the cone derivative $d\Psi_p|_{T_pM\setminus \{0\}} : T_pM\setminus \{0\} \to T_pM$ is a diffeomorphism onto $T_pM \setminus \{0\}.$ We let $\diff(M,S)$ denote the group of cone-diffeomorphisms of $(M,S)$ that act trivially on the set of connected components.
			\item Let the diffeomorphism group $\diff(M,S)$ act on cone-immersions from $(M,S)$ to $N$ by composition. A \emph{cone-immersed submanifold of $N$ of type $(M,S)$} is an orbit of the $\diff(M,S)$-action.
    \item\label{item:orientation} Suppose $M$ is orientable and let $\diff^+(M,S)\triangleleft \diff(M,S)$ denote the normal subgroup of orientation preserving cone-smooth diffeomorphisms. An \emph{orientation} on a cone-immersed submanifold $K$ of $N$ of type $(M,S)$ is an equivalence class of pairs $(O,C)$ where $O$ is an orientation of $M$ and $C$ is a $\diff^+(M,S)$ orbit inside the $\diff(M,S)$ orbit $K.$ There is a natural $\diff(M,S)/\diff^+(M,S)$ action on such pairs and this gives rise to the desired equivalence relation.
		\end{enumerate}
	\end{dfn}

\begin{rem}\label{remark: cone-diffeomorphism lifting}
It follows from Lemma~\ref{lemma: cone-immersions} that a map $\Psi : (M,S) \to (M,S)$ is a cone-diffeomorphism if and only if it lifts to a diffeomorphism $\widetilde\Psi : \widetilde{M}_S \to \widetilde{M}_S$ such that $\widetilde{\Psi}(E_p) = E_p$ for $p \in S.$
\end{rem}

\begin{rem}
		\label{remark: embedded cone locus}
		In this article, all cone-immersions are assumed to have embedded cone locus. That is, for a cone-immersion $(\Psi,S)$ and points $c\ne c'\in S$ we have $\Psi(c)\ne\Psi(c').$
	\end{rem}

\begin{cor}\label{cor:cone-immersion open mapping}
Let $M,N,$ be smooth manifolds of equal dimension, let $S \subset M$ be finite and let $(\Psi : M \to N,S)$ be a cone-immersion. Then $\Psi$ is an open map.
\end{cor}
\begin{proof}
It suffices to prove that for $p \in S$ and every open $p \in V \subset M$ as in Lemma~\ref{lemma: cone-immersions}~\ref{sort of locally one-to-one}, the image $\Psi(V)$ contains an open neighborhood of $\Psi(p).$ Indeed, consider the strict transform $\widetilde \Psi : \widetilde V \to \widetilde N_{\Psi(p)}$ as in Lemma~\ref{lemma: cone-immersions}~\ref{existence of lifting}. Then, $\widetilde \Psi(E_p) \subset E_{\Psi_p}$ is open and closed, so $\widetilde \Psi(E_p) = E_{\Psi_p}.$ By the inverse function theorem, there is an open $E_p \subset U$ with $E_{\Psi(p)} \subset \widetilde \Psi(U)$ and $\widetilde \Psi(U)$ open. So, the image under the blowup projection $\pi_N\left(\widetilde \Psi(U)\right)$ is open, which implies the claim.
\end{proof}

Now we are ready to define cone-immersed submanifolds.
\begin{dfn}
	Let $K=[(\Psi:M\to N,S)]$ be a cone-immersed submanifold of type $(M,S).$ As in Definition~\ref{definition:points}, a \emph{point} $p$ in $K$ is an equivalence class of pairs $((\chi,S),q),$ where $(\chi: M \to N,S)$ is a representative of $K$ and $q\in M.$ We let $p_0$ denote the image of $p$ in $N.$ That is, $p_0 = \chi(q)$ for $((\chi,S),q)$ a representative of $p.$ The cone-immersed submanifold $K$ thus has a well-defined \emph{cone locus}
	\[
	K^C:=\{[((\Psi,S),c)]\;|\;c\in S\}.
	\]
A \emph{cone point} is an element of the cone locus. We define the \emph{tangent cone} of $K$ at a point $p = [((\Psi,S),q)]$ to be the cone-immersed submanifold
\[
TC_pK := [(d\Psi_c : T_cM \to T_{p_0}N,\{0\})]
\]
of $T_{p_0}N.$ The tangent cone $TC_pK$ is indeed a cone, that is, invariant under scalar multiplication. Moreover, it is independent of the choice of $\Psi.$ If $\Psi$ is smooth at $q,$ then $TC_pK$ is smoothly embedded and recovers the usual notion of tangent space. We define the \emph{projective tangent cone} of $K$ at $p$ by
	\[
	\orientedprojective(TC_pK):=\left[\orientedprojective(d\Psi_c) :\orientedprojective(T_cM)\to \orientedprojective(T_{p_0}N)\right].
	\]
	This is a smooth immersed sphere in $\orientedprojective(T_{p_0}N).$

A \emph{function} on $K$ is an equivalence class of pairs $((\chi,S),f),$ where $(\chi,S)$ is a representative of $K$ and $f$ is a function on $M.$ We say the function $h=[((\Psi,S),f)]$ is \emph{cone-smooth} at the point $p=[((\Psi,S),q)]$ if $f$ is cone-smooth at $q.$ In this case $h$ has a well-defined \emph{cone-derivative} $dh_p : TC_pK \to \R,$ which is a degree-1 homogeneous function.
\end{dfn}

	\begin{lemma}
		\label{lemma: induced tangent frame}
Let $M,S,\pi : \widetilde M_S \to M,p$ and $\tilde p$ be as in Definition~\ref{definition: blowup}(\ref{blowup at discrete set}) and~(\ref{cone coordinates}).
Let $(U,\mathbf{X},\alpha)$ be cone coordinates at $\widetilde{p}$ and abbreviate $V = \mathrm{Im}(\alpha) \subset \widetilde M_S.$ Define sections $e_i$ of the vector bundle $\pi^*TM|_V$ by
		\[
		e_i(\tilde q, s) :=
		\left\{
		\begin{array}{ll}
		\frac{1}{s}\pderiv[\pi]{x^i}(\tilde q, s),
		&s\neq 0,\\
		\nabla_s \pderiv[\pi]{x^i}(\tilde q, s),
		&s =0,
		\end{array}
		\right.\quad i=1,\ldots,m,
		\]
where $\nabla$ is an arbitrary connection.
		Define $e_0:= \pderiv[\pi]{s}.$ Then $e_0,\ldots,e_m,$ are independent of the choice of $\nabla,$ smooth, and everywhere linearly independent.
	\end{lemma}

	The proof of Lemma~\ref{lemma: induced tangent frame} relies on the following elementary observation (compare with~\cite[Lemma~2.1]{milnor}).

	\begin{lemma}
		\label{lemma: milnor lemma}
		Let $f:\mathbb{R}^k\times\mathbb{R}\to\mathbb{R}$ be smooth with $f(x,0)=0,\;x\in\mathbb{R}^k.$ Then we have $f(x,s)=s\cdot g(x,s)$ for $(x,s)\in\mathbb{R}^k\times\mathbb{R},$ where $g:\mathbb{R}^k\times\mathbb{R}\to\mathbb{R}$ is smooth and satisfies
		\[
		g(x,0)=\pderiv[f]{s}(x,0),\;\pderiv[g]{s}(x,0)=\frac{1}{2}\frac{\partial^2f}{\partial s^2}(x,0),\quad x\in\mathbb{R}^k.
		\]
	\end{lemma}

	\begin{proof}[Proof of Lemma~\ref{lemma: induced tangent frame}]
		As the blowup projection $\pi$ maps the exceptional sphere $E_{p}$ to the point $p,$ we have
		\[
		 \pderiv[\pi]{x^i}(\tilde q, 0) = 0, \qquad \tilde q \in U, \quad i=1,\ldots,m.
		\]
		The sections $e_i,\;i=1,\ldots,m,$ are thus well-defined on $U$ independently of the choice of connection $\nabla.$ They are smooth by Lemma~\ref{lemma: milnor lemma}. One shows the sections $e_0,\ldots,e_m,$ are everywhere linearly independent by using the definition of the blowup projection $\pi.$
	\end{proof}

Let $(\Psi : M \to N, S)$ be a cone-smooth map and let $p \in S.$ Recall that the cone derivative $d\Psi_p : T_pM \to T_{\Psi(p)}N$ is homogeneous of degree $1$ and $d\Psi_p|_{T_pM \setminus\{0\}}$ is smooth. It follows that for $0 \neq v \in T_pM$ and $\lambda > 0,$ we have $d(d\Psi_p)_{v} = d(d\Psi_p)_{\lambda v}$ under the canonical identification $T_v T_pM \simeq T_pM \simeq T_{\lambda v}T_pM.$ Thus, for $\widetilde p = [v] \in \orientedprojective(T_pM),$ we define
\[
d(d\Psi_p)_{\widetilde p} : = d(d\Psi_p)_v : T_pM \to T_{\Psi(p)}N.
\]
\begin{lemma}\label{lemma:blowupdifferential}
Let $(\Psi : M \to N, S)$ be a cone-smooth map. Consider the map $d\left(\Psi|_{M\setminus S}\right) : TM|_{M \setminus S} \to \Psi^* TN|_{M \setminus S}.$ Pulling back by $\pi$ gives a map
\[
\pi^* d\left(\Psi|_{M\setminus S}\right) : \pi^* TM|_{\widetilde M_S^\circ} \to \pi^* \Psi^* TN|_{\widetilde M_S^\circ}.
\]
This map extends uniquely to a map of bundles $\widetilde{d\Psi}:\pi^* TM \to \pi^* \Psi^* TN.$ Moreover, for $p \in S$ and $\widetilde p \in E_p,$ we have
\begin{equation}\label{equation:dderivatives}
\widetilde{d\Psi}_{\widetilde p} = d(d\Psi_p)_{\widetilde p}.
\end{equation}
In particular, if $\Psi$ is a cone-immersion, then $\widetilde{d\Psi}$ is an injective map of vector bundles.
\end{lemma}
\begin{proof}
By Lemma~\ref{lemma: induced tangent frame} and using the notation therein, it suffices for the first claim to show that the sections $\pi^* d\Psi(e_i|_{\{s > 0\}})$ extend to smooth sections of $\pi^*\Psi^* TN$ for $i = 0,\ldots,m$ and any choice of cone coordinates. Linearity of $\widetilde{d\Psi}$ follows by continuity. Consider the differential of the blowup projection $d\pi|_{\widetilde M_S\setminus E_S} : T\widetilde M_S|_{\widetilde M_S^\circ} \to \pi^* TM|_{\widetilde M_S^\circ}.$ We have
\[
\pi^* d\left(\Psi|_{M\setminus S}\right)  = \pi^* d\left(\Psi|_{M\setminus S}\right) \circ d\pi \circ d \pi^{-1}|_{\widetilde M_S^\circ}  = d(\Psi \circ \pi) \circ d \pi^{-1}|_{\widetilde M_S^\circ} .
\]
By definition of cone-smooth, $d(\Psi \circ \pi) : T\widetilde{M}_S \to \Psi^* \pi^* TN$ is a smooth bundle map. For $p \in S,$ since $\Psi\circ \pi$ maps $E_p$ to a point, it follows that $d(\Psi \circ \pi)|_{\{s = 0\}} = 0.$ On the other hand, $d \pi^{-1}(e_i|_{\{s > 0\}}) = \frac{1}{s} \pderiv{x^i}$ for $i = 1,\ldots, m,$ and $d \pi^{-1}(e_0|_{\{s > 0\}}) = \pderiv{s}.$ So, it follows from Lemma~\ref{lemma: milnor lemma} that $\pi^*\left( d\Psi(e_i|_{\{s>0\}})\right)$ extends to a smooth section of $\pi^*\Psi^* TN,$ which is $\widetilde{d\Psi}(e_i).$  Thus, the map $\widetilde{d\Psi}$ is well-defined.

To prove equation~\eqref{equation:dderivatives}, we claim that for $\widetilde q \in U,$
\begin{gather*}
\widetilde{d\Psi}(e_i)(\widetilde q,0) = \frac{\partial^2\Psi}{\partial x^i \partial s}(\widetilde q, 0), \qquad i = 1,\ldots, m, \\
\widetilde{d\Psi}(e_0)(\widetilde q,0) = \pderiv[\Psi]{s} (\widetilde q, 0).
\end{gather*}
Indeed,
\begin{equation*}
\frac{\partial^2\Psi}{\partial x^i \partial s}(\widetilde q, 0) = \frac{\partial^2\Psi}{\partial s \partial x^i}(\widetilde q, 0) =\pderiv{s}\left(\widetilde{d\Psi}\left(\pderiv[\pi]{x^i}\right)\right)(\widetilde q, 0) = \widetilde{d\Psi}(e_i)(\widetilde q, 0).
\end{equation*}
Similarly,
\begin{equation*}
\pderiv[\Psi]{s}(\widetilde q, 0) =\widetilde{d\Psi}\left(\pderiv[\pi]{s}\right)(\widetilde q, 0) = \widetilde{d\Psi}(e_0)(\widetilde q, 0).
\end{equation*}
On the other hand, writing $v(\widetilde q) = e_0(\widetilde q,0) \in T_pM,$ recalling the definition of the cone derivative $d\Psi_p,$ and identifying $T_vT_pM \simeq T_pM,$ we have
\begin{gather*}
\frac{\partial^2\Psi}{\partial x^i \partial s} (\widetilde q, 0) =  \pderiv{x^i}d\Psi_p(v(\widetilde q)) = d(d\Psi_p)_{v(\widetilde q)}(e_i(\widetilde q,0)), \quad i = 1,\ldots, m,\\
\pderiv[\Psi]{s}(\widetilde q, 0) = d\Psi_p(v) = d(d\Psi_p)_{v(\widetilde q)}(e_0(\widetilde q,0)).
\end{gather*}
Equation~\eqref{equation:dderivatives} follows.
\end{proof}

\begin{dfn}\label{definition: many tangent spaces}
Let $S \subset M$ be a finite subset. The \emph{blowup tangent bundle} of $(M,S)$ is the bundle $\widetilde{TM}_S : = \pi^*TM \to \widetilde{M}_S.$ When clear from the context, we may omit the subscript $S.$ Let $(\Psi : M \to N, S)$ be a cone-smooth map. The \emph{blowup differential} of $\Psi$ is the map
\[
\widetilde{d \Psi} : \widetilde{TM}_S \to \pi^* \Psi^* TN
\]
given by Lemma~\ref{lemma:blowupdifferential}.
	\end{dfn}

Let $M$ be a smooth manifold and let $S \subset M$ be a finite subset and let $\pi : \widetilde{M}_S \to M$ denote the blowup projection. Given a differential form $\alpha$ on $M$ we can pull-back $\alpha$ as a section of $\Lambda^*(T^*M)$ to obtain a section of $\pi^*\Lambda^*\left(T^*M\right) \simeq \Lambda^*\left(\widetilde{TM}^*_S\right).$ We denote this pull-back by $\pi^{-1}\alpha.$ Observe that this pull-back is different from the pull-back of $\alpha$ as a differential form, $\pi^*\alpha,$ which would be a section of $\Lambda^*(T^*\widetilde{M}_S).$ A similar distinction applies to pull-backs of metrics.
\begin{dfn}\label{definition: cone-smooth differential form etc}
A \emph{cone-smooth differential form} on $(M,S)$ is a smooth differential form $\alpha$ on $M \setminus S$ such that $\pi^{-1} \alpha$ extends to a smooth section $\widetilde{\alpha}$ of $\Lambda^*\left(\widetilde{TM}_S^*\right).$ We call $\widetilde{\alpha}$ the \emph{blowup form}. We say that $\alpha$ and $\widetilde{\alpha}$ are \emph{closed} if $\alpha$ is closed as a differential form on $M\setminus S.$

A \emph{cone-smooth Riemannian metric} on $(M,S)$ is a smooth Riemannian metric $g$ on $M \setminus S$ such that $\pi^{-1}g$ extends to a smooth metric $\widetilde g$ on the blowup tangent bundle $\widetilde{TM}_S.$ We call $\widetilde g$ the \emph{blowup metric}.

Let $(\Psi:M \to N,S)$ be a cone-smooth map. A \emph{cone-smooth vector field along} $\Psi$ is a smooth section $\xi$ of $\left(\Psi|_{M\setminus S}\right)^*TN$ such that $\left(\pi|_{\widetilde{M}_S^\circ}\right)^*\xi$ extends to a smooth section $\widetilde \xi$ of $(\Psi \circ \pi)^*TN.$ We call $\widetilde \xi$ the \emph{blowup vector field.}
\end{dfn}

\begin{rem}\label{remark:differential of cone-smooth function}
Let $S \subset M$ be a finite subset and let $(f : M \to \R,S)$ be a cone-smooth function. Then, the blowup differential $\widetilde{df}$ is a smooth section of the blowup cotangent bundle $\widetilde{TM}_S^* \to \widetilde{M}_S,$ so $df$ is a cone-smooth 1-form. For $p \in S$ and $\widetilde p \in E_p,$ we use the notation $df_{\widetilde p} = \widetilde{df}|_{\widetilde p}.$ Given a cone-smooth Riemannian metric on $(M,S),$ we can define the cone-smooth vector field $\nabla f$ on $(M,S)$ in the usual way.
\end{rem}

\begin{rem}\label{remark:cone-smooth de Rham}
By Remark~\ref{remark:differential of cone-smooth function} the differential of a cone-smooth function is a cone-smooth $1$-form. However, a cone-smooth $0$-form need not be a cone-smooth function as it may not extend continuously to $M.$ Furthermore, the differential of a cone-smooth $0$-form need not be a cone-smooth $1$-form. More generally, the exterior derivative of a cone-smooth differential form may not be cone-smooth. In order to define the cone-smooth de Rham complex of $(M,S),$ one should consider only cone-smooth differential forms with cone-smooth exterior derivative. Since this is not used in the present paper, we will not discuss it further.
\end{rem}
\begin{lemma}\label{lemma:cone-smooth pullback}
Let $(\Psi: M \to N, S)$ be a cone-smooth map, let $\alpha$ be a smooth differential form on $N$ and let $\xi$ be a cone-smooth vector field along $\Psi.$ Then $\left(\Psi|_{M\setminus S}\right)^*\alpha$ and $i_{\xi}\alpha$ are cone-smooth differential forms on $(M,S).$ If the blowup vector field $\widetilde \xi$ vanishes on $\partial \widetilde{M}_S,$ then so does the blowup form $\widetilde{i_{\xi}\alpha}.$ If $(\Psi,S)$ is a cone-immersion, and $g$ is a Riemannian metric on $N,$ then $\left(\Psi|_{M\setminus S}\right)^*g$ is a cone-smooth Riemannian metric on $(M,S).$
\end{lemma}
\begin{proof}
This follows from Lemma~\ref{lemma:blowupdifferential}.
\end{proof}
In light of the preceding lemma, we abbreviate
\begin{equation}\label{eq:pullbackabbrev}
\Psi^*\alpha : = \left(\Psi|_{M\setminus S}\right)^*\alpha, \qquad \Psi^*g : = \left(\Psi|_{M\setminus S}\right)^*g.
\end{equation}
\begin{lemma}\label{lemma:pullbackform}
Let $\alpha$ be a cone-smooth differential form on $(M,S).$ Then, the pull-back differential form $\left(\pi|_{\widetilde{M}_S^\circ}\right)^*\alpha$ extends to a smooth differential form on $\widetilde{M}_S.$ If the blowup form $\widetilde \alpha$ vanishes on $\partial\widetilde{M}_S,$ then so does the extension of $\left(\pi|_{\widetilde{M}_S^\circ}\right)^*\alpha$ considered as a section of $\Lambda^*\left(T^*\widetilde{M}_S\right)|_{\partial\widetilde{M}_S}.$
\end{lemma}
\begin{proof}
The dual of the differential of $\pi$ gives a map of vector bundles
\[
d \pi^* : \widetilde{TM}_S^* = \pi^*T^*\!M \to T^*\widetilde{M}_S,
\]
which induces a map $\Lambda^*(d\pi^*) : \Lambda^*(\widetilde{TM}_S^*) \to \Lambda^*(T^*\widetilde{M}_S).$
Since
\[
\left(\pi|_{\widetilde{M}_S^\circ}\right)^*\alpha = \left.\Lambda^*(d\pi^*)\circ \widetilde \alpha\right|_{\widetilde{M}_S^\circ},
\]
the required extension is given by $\Lambda^*(d\pi^*)\circ \widetilde \alpha.$ The vanishing claim is immediate.
\end{proof}
We write $\pi^*\alpha$ for the extension of $\left(\pi|_{\widetilde{M}_S^\circ}\right)^*\alpha$ given by the preceding lemma.
\begin{lemma}\label{lemma:exactness of extension}
Let $\alpha$ be a cone-smooth differential 1-form on $(M,S)$ such that there exists a smooth function $f^\circ : M\setminus S \to \R$ with  $\alpha|_{M\setminus S} = df^\circ$ and the blowup form $\widetilde{\alpha}$ vanishes on $\partial\widetilde{M}_S$. Then $f^\circ$ extends to a cone-smooth function $f$ on $(M,S),$ and $S$ is contained in the critical locus of $f.$
\end{lemma}
\begin{proof}
Since $d\left(\alpha|_{M\setminus S}\right) = d(df^\circ)= 0,$ it follows that $d(\pi^*\alpha) = 0.$ Lemma~\ref{lemma:pullbackform} gives $\pi^*\alpha|_{\partial \widetilde{M}_S}=0,$ so $\pi^*\alpha$ is exact. Let $\widetilde f: \widetilde{M}_S \to \R$ be smooth with $d\widetilde{f} = \pi^*\alpha.$ After possibly adding a constant to $\widetilde f,$ we may assume that $\widetilde f|_{\widetilde{M}_S^\circ} = f^\circ \circ \pi|_{\widetilde{M}_S^\circ}.$ Again invoking the vanishing of $\pi^*\alpha|_{\widetilde M_S},$ it follows that $\widetilde{f}|_{\partial\widetilde{M}_S}$ is locally constant. So, we take $f: M \to \R$ to be the unique function such that $f \circ \pi = \widetilde f,$ and $f$ is cone-smooth by definition. The vanishing of the cone-derivative of $f$ at $S$ follows from the vanishing of $\pi^*\alpha|_{\partial\widetilde{M}_S}$ as a section of $T^*\widetilde{M}_S|_{\partial\widetilde{M}_S}$ given by Lemma~\ref{lemma:pullbackform}.
\end{proof}
\begin{dfn}
Suppose $M$ is oriented and let $\alpha$ be a cone-smooth differential form on $(M,S)$ such that $\pi^*\alpha$ has compact support. We define the integral of $\alpha$ by
\[
\int_M \alpha := \int_{\widetilde M_S} \pi^*\alpha.
\]
\end{dfn}

\begin{lemma}\label{lemma:cone smooth vanishing tangent}
Let $\xi$ be a cone-smooth vector field on $(M,S)$ with blowup vector field $\widetilde \xi$ vanishing on $\partial \widetilde{M}_S.$ Then, there exists a unique smooth vector field $\widehat \xi$ on $\widetilde{M}_S$ such that
$
d\pi\left(\widehat \xi|_{\widetilde M_S^\circ}\right) = \xi.
$
Moreover, $\widehat \xi$ is tangent to $\partial \widetilde{M}_S.$
\end{lemma}
\begin{proof}
Since $\pi|_{\widetilde M_S^\circ}$ is a diffeomorphism, there exists a unique vector field $\overline \xi$ on $\widetilde M_S^\circ$ such that
\begin{equation}\label{equation:oxi}
d\pi(\overline \xi) = \xi.
\end{equation}
To prove the lemma, it suffices to show that $\overline \xi$ extends smoothly to a vector field $\widetilde \xi$ on $\widetilde M_S$ that is tangent to $\partial \widetilde M_S.$
To this end, we use Lemma~\ref{lemma: induced tangent frame} and the notation therein. Abbreviate $V^\circ = V \cap \widetilde M_S^\circ.$ It suffices to show that $\overline \xi|_{V^\circ}$ extends to $V.$ Write
$\widetilde \xi|_V = \sum_{i = 0}^m \widetilde \xi^i e_i.$ So,
\[
\widetilde \xi|_{V^\circ} = \widetilde \xi^0|_{V^\circ} \pderiv[\pi]{s} + \sum_{i = 1}^m \widetilde \xi^i|_{V^\circ} \frac{1}{s}\pderiv[\pi]{x^i}.
\]
Furthermore, write
\[
\overline \xi|_{V^\circ} = \overline\xi^0 \pderiv{s} + \sum_{i = 1}^m \overline \xi^i \pderiv{x^i}.
\]
Equation~\eqref{equation:oxi} gives
\[
\overline{\xi}^0 = \widetilde{\xi}^0|_{V^\circ}, \qquad\qquad \overline{\xi}^i =\frac{1}{s}\widetilde{\xi}^i|_{V^\circ}, \quad i = 1,\ldots,m.
\]
Since $\widetilde \xi$ vanishes on $\partial \widetilde M_S,$ it follows that the functions $\widetilde \xi^i$ vanish on $\partial \widetilde M_S.$ By Lemma~\ref{lemma: milnor lemma}, the functions $\overline \xi^i, \, i = 1,\ldots,m,$ extend smoothly to functions $\widehat \xi^i$ on~$V.$ Take $\widehat \xi^0 = \widetilde \xi^0.$ Define $\widehat \xi$ on $V$ by
\[
\widehat \xi|_V = \widehat \xi^0 \pderiv{s} + \sum_i \widehat \xi^i \pderiv{x^i}.
\]
Since $\widehat \xi^0 = \widetilde \xi^0$ vanishes on $\partial \widetilde M_S,$ it follows that $\widehat \xi$ is tangent to $\partial \widetilde{M}_S.$
\end{proof}

\begin{rem}\label{remark: blowup differential lifting}
Let $\Psi \in \diff(M,S).$ Then Remark~\ref{remark: cone-diffeomorphism lifting} and the fact that
\[
\pi^*\Psi^* TM = \widetilde \Psi^*\pi^*TM = \widetilde \Psi^*\widetilde{TM}_S
\]
imply that the blowup differential gives an isomorphism of vector bundles
\[
\widetilde{d\Psi} : \widetilde{TM}_S \overset{\sim}{\longrightarrow} \widetilde{\Psi}^*\widetilde{TM}_S.
\]
In particular, cone-smooth diffeomorphisms act by pull-back on cone-smooth differential forms.
\end{rem}

\begin{dfn}\label{definition:immersed cone smooth differential form}
Let $K = [(\Psi: M \to N,S)]$ be a cone-immersed submanifold of type $(M,S).$ A \emph{cone-smooth differential form} on $K$ is an equivalence class $\tau = [((\chi,S),\alpha)]$ where $(\chi,S)$ represents $K$ and $\alpha$ is a cone-smooth differential form on $(M,S).$ Two pairs are equivalent if they belong to the same orbit of the $\diff(M,S)$ action given by Remark~\ref{remark: blowup differential lifting}. We may write $\alpha = \Psi^*\tau.$ Given a smooth form $\eta$ on $N,$ the \emph{restriction} to $K$ is the cone-smooth form given by
\[
\eta|_K : = [((\Psi,S),\Psi^*\eta)].
\]
We say that $\tau$ is \emph{closed} if $\alpha$ is. We say that $\tau$ \emph{vanishes at the cone locus} if $\widetilde{\alpha}$ vanishes on $\partial \widetilde{M}_S.$ Given an orientation on $K,$ if $\pi^*\alpha$ has compact support, we define
\[
\int_K \tau := \int_M \alpha.
\]
\end{dfn}

\begin{dfn}
Let $K= [(\Psi:M \to N,S)]$ be a cone-immersed submanifold, let $p = [((\Psi,S),q)]$ be a cone point, and let $\widetilde{p} = \left[\left(\orientedprojective(d\Psi_q),\widetilde q\right)\right]\in \orientedprojective(TC_pK)$. The \emph{tangent space of $K$ at $\widetilde{p}$} is defined by
\[
T_{\widetilde p} K := \widetilde{d\Psi}_{\widetilde q}\left(\widetilde{TM}_{\widetilde q}\right)\subset T_{p_0}N,
\]
which is independent of the choice of representatives. At a smooth point $p$ of $K,$ we define the tangent space $T_pK$ as in Definition~\ref{definition:points}.

Let $h = [((\Psi,S),f)]$ be a cone-smooth function on $K.$ Then the differential of $h$ at $\widetilde p$ is defined by
\[
dh_{\widetilde p} := df_{\widetilde p}\circ \widetilde{d\Psi}^{-1}_{\widetilde q} : T_{\widetilde p} K \to \R,
\]
which is independent of choices of representatives.
\end{dfn}

\begin{rem}
An orientation on a cone immersed submanifold $K$ as in Definition~\ref{definition:cone immersed submanifold}~\ref{item:orientation} is equivalent to a continuously varying orientation on its tangent spaces.
\end{rem}

\begin{dfn}
		\label{definition: critical cone point}\mbox{}
\begin{enumerate}
\item\label{item:critical cone point intrinsic}
Let $(f : M \to \R,S)$ be a cone-smooth function, and let $p \in S.$ The point $p$ is said to be a \emph{critical point} of $f$ if the cone-derivative $df_p : T_pM \to \R$ vanishes identically.
\item\label{item:critical cone point submanifold}
Let $K = [(\Psi : M \to N,S)]$ be a cone-immersed submanifold, let $h = [((\Psi,S),f)]$ be a cone-smooth function on $K,$ and let $p  = [((\Psi,S),q)]$ be a cone point. The point $p$ is a \emph{critical point} of $h$ if $q$ is a critical point of $f.$
\end{enumerate}
	\end{dfn}
\begin{rem}\label{remark: critical cone point}
It follows from Lemma~\ref{lemma:blowupdifferential} that in the situation of part~(\ref{item:critical cone point intrinsic}) of the preceding definition, if $p$ is a critical point of $f,$ then $df_{\widetilde p} = 0$ for all $\widetilde p \in E_p.$ The analogous statement holds in the situation of part~(\ref{item:critical cone point submanifold}).
\end{rem}

\begin{dfn}\label{dfn:cone-Hessian}
Let $(f : M \to \R,S)$ be a cone-smooth function, and let $p \in S$ be a critical point of $f.$
\begin{enumerate}
\item
The \emph{cone-Hessian} of $f$ at $p$ is the map
\[
\nabla df : T_pM \to T_pM^*,
\]
smooth away from $0$ and homogeneous of degree $1$, defined as follows.
By Remark~\ref{remark: critical cone point}, the blowup differential $\widetilde{df}$ vanishes on the exceptional sphere $E_p \subset \widetilde{M}_S.$ So, the restriction of the second covariant derivative $\nabla \widetilde{df} \in Hom\left(T\widetilde{M}_S, \widetilde{TM}_S^*\right)$ to $E_p$ is independent of the choice of connection. Moreover, $\nabla \widetilde{df}$ vanishes on $TE_p \subset T\widetilde{M}_S|_{E_p}.$ Recall that a vector $0 \neq v \in T_pM$ gives rise to a point $[v] \in \orientedprojective(T_pM) \simeq E_p.$ For $v \in T_pM,$ and $\widetilde v \in T_{[v]} \widetilde{M}_S$ such that $d \pi_{[v]}(\widetilde v) = v,$  we define
\[
\nabla_v df := \nabla_{\widetilde v} \widetilde{df} \in \left(\widetilde{TM}^*_S\right)_{[v]} = T_p^*M.
\]
\item
\label{definition: degenerate critical cone point}
The critical point $p$ is said to be \emph{degenerate} if there exists a tangent vector $0\ne v\in T_pM$ with $\nabla_vdf=0.$
\end{enumerate}
\end{dfn}

\begin{lemma}\label{lemma: coordinate expression of cone-second derivative}
Continue with the notation of Definition~\ref{dfn:cone-Hessian}. Let $0 \neq v \in T_pM$ and write $\widetilde p = [v].$ Take local cone coordinates with $\pderiv[\pi]{s}\left(\widetilde{p},0\right)=v$ and let $e_0,\ldots,e_m,$ denote the induced local frame of the blowup tangent bundle as in Lemma~\ref{lemma: induced tangent frame}. The cone-Hessian is given by
	\begin{equation*}
	(\nabla_vdf)(e_0(\widetilde p, 0))=\frac{\partial^2f}{\partial s^2}\left(\widetilde{p},0\right),\qquad(\nabla_vdf)(e_i(\widetilde p, 0))=\frac{1}{2}\frac{\partial^3f}{\partial s^2\partial x^i}\left(\widetilde{p},0\right).
	\end{equation*}
\end{lemma}
\begin{proof}
Since
\[
\pderiv[f]{x^i}(\widetilde{q},0) = 0, \qquad \widetilde q \in U,
\]
Lemma~\ref{lemma: milnor lemma} gives
$\pderiv[f]{x^i} = s g_i,$ where $g_i$ is smooth and
\[
\pderiv[g_i]{s}(\widetilde q, 0) = \frac{1}{2}\frac{\partial^3 f}{\partial s^2 \partial x^i}(\widetilde q, 0).
\]
Observe that for $s > 0,$ we have
\[
df(e_i) =
\frac{1}{s}\pderiv[f]{x^i} = g_i.
\]
By continuity, we have $df(e_i) = g_i$ everywhere. Use the connection on $\widetilde{TM}_S$ with respect to which the frame $e_0,\ldots,e_m,$ is parallel.
Then, for $i = 1,\ldots,m,$
\[
(\nabla_vdf)(e_i(\widetilde p, 0)) = \left.\pderiv{s} df(e_i) \right|_{(\widetilde p, 0)}= \pderiv[g_i]{s}(\widetilde p, 0) =  \frac{1}{2}\frac{\partial^3 f}{\partial s^2 \partial x^i}(\widetilde p, 0).
\]
The proof of the left-hand equality is similar but easier.
\end{proof}

\begin{dfn}
Let $K = [(\Psi: M \to N,S)]$ be a cone-immersed submanifold, let $h = [((\Psi,S),f)]$ be a cone-smooth function and let $p = [((\Psi,S),q)]$ be a critical cone point. Let $\widehat TC_pK = TC_pK \setminus\{0\}$ denote the punctured tangent cone. The \emph{cone-Hessian} of $h$ at $p$ is the section $\nabla dh$ of the cotangent bundle of the punctured tangent cone
\[
T^*\widehat TC_pK \to \widehat TC_pK
\]
defined as follows. Let $v = [(d\Psi_q,w)]$ be a point of the punctured tangent cone. Observe that $T_v \widehat TC_pK \simeq T_{[v]} K$ by equation~\eqref{equation:dderivatives}. Recall that $(\widetilde {T M}_S)_{[w]} \simeq T_q M$ and by definition $T_{[v]}K= \widetilde{d\Psi}_{[w]}((\widetilde {T M}_S)_{[w]}).$ We define
\[
\nabla_v dh := \nabla_{w} df \circ \widetilde{d\Psi}^{-1}_{[w]} : T_{[v]}K \to \R.
\]
The critical cone point $p$ is \emph{degenerate} if $q$ is a degenerate critical point of $f.$
\end{dfn}

	The following lemma is well-known for extrema of smooth functions. We show it is true also for cone-smooth functions.

	\begin{lemma}
		\label{lemma: vanishing second derivative at extremum point}
		Let $M$ be a smooth manifold, let $p\in M,$ let $0\ne v\in T_pM$ and let $h:M\to\mathbb{R}$ be cone-smooth at $p$ with $dh_p=0.$ Assume further that $p$ is an extremum point of $h.$ If the equality $(\nabla_vdh)(v)=0$ holds, then we have $\nabla_vdh=0.$ In particular, $p$ is a degenerate critical point of $h$ in this case.
	\end{lemma}

	\begin{proof}
		Without loss of generality we assume $p$ is a minimum of $h.$ Let $\widetilde{p}:=[v]\in\orientedprojective(T_pM).$ Let $(U,\mathbf{X},\alpha)$ be local cone coordinates with $\pderiv[\pi]{s}\left(\widetilde{p},0\right)=v$ and let $e_0,\ldots,e_m,$ denote the induced local frame of the blowup tangent bundle as in Lemma~\ref{lemma: induced tangent frame}. By the assumption and Lemma~\ref{lemma: coordinate expression of cone-second derivative}, we have
		\begin{equation}
		\label{second s derivative vanishes}
		\frac{\partial^2h}{\partial s^2}\left(\widetilde{p},0\right)=0.
		\end{equation}
		Since $p$ is a critical point where $h$ attains a minimum, we have
		\begin{equation}
		\label{all second derivatives are non-negative}
		\frac{\partial^2h}{\partial s^2}\left(\widetilde{q},0\right)\ge0,\quad\widetilde{q}\in U.
		\end{equation}
		For $i=1,\ldots,m,$ Lemma~\ref{lemma: coordinate expression of cone-second derivative} and equality~\eqref{second s derivative vanishes} together with inequality~\eqref{all second derivatives are non-negative} yield
		\begin{align}
		\label{another derivative vanishes}
		(\nabla_vdh)\left(e_i\left(\widetilde{p},0\right)\right)
		&=\frac{1}{2}\frac{\partial^3h}{\partial s^2\partial x^i}\left(\widetilde{p},0\right)\\
		&=\frac{1}{2}\frac{\partial^3h}{\partial x^i\partial s^2}\left(\widetilde{p},0\right)\nonumber\\
		&=0.\nonumber
		\end{align}
		Since $e_0\left(\widetilde{p},0\right) = v,$ the assumption and~\eqref{another derivative vanishes} give $\nabla_vdh=0,$ as desired.
	\end{proof}

Recall the meaning of a polar coordinate map from Definition~\ref{definition:polar coordinates}.
	\begin{lemma}
		\label{lemma: nice level sets}
		Let $M$ be a smooth manifold of dimension $m+1$ and let $p\in M.$ Let $h:M\to\mathbb{R}$ be cone-smooth at $p$ such that $p$ is a non-degenerate critical point and an extremum point of $h.$ Then there exist a positive $\epsilon$ and a polar coordinate map $\kappa:S^m\times[0,\epsilon)\to M$ centered at $p$ such that
for each $s\in(0,\epsilon)$ the restricted map $\kappa|_{S^m\times\{s\}}$ parameterizes a level set of $h.$
	\end{lemma}

	\begin{proof}
		Without loss of generality we suppose $h(p)=0$ is a minimum. Let $\pi:\widetilde{M}_p\to M$ denote the blowup projection. For simplicity, we write $h$ instead of $h\circ\pi$ and think of $h$ as a function on $\widetilde{M}_p.$ Identify a neighborhood $E_p\subset V\subset\widetilde{M}_p$ with $\orientedprojective(T_pM)\times[0,\epsilon)$ and let $r$ denote the $[0,\epsilon)$-coordinate. Then we have
		\[
		\pderiv[h]{r}\left(\widetilde{p},0\right)=0,\quad\widetilde{p}\in\orientedprojective(T_pM).
		\]
		By Lemmas~\ref{lemma: coordinate expression of cone-second derivative} and~\ref{lemma: vanishing second derivative at extremum point}, we have
		\[
		\frac{\partial^2h}{\partial r^2}\left(\widetilde{p},0\right)>0,\quad\widetilde{p}\in\orientedprojective(T_pM).
		\]
		Applying Lemma~\ref{lemma: milnor lemma} twice and diminishing $\epsilon$ if necessary, we write
		\[
		h\left(\widetilde{p},r\right)=r^2f\left(\widetilde{p},r\right),
		\]
		where $f:\orientedprojective(T_pM)\times[0,\epsilon)\to\mathbb{R}$ is smooth and positive. It follows that the function $\sqrt{h}:\orientedprojective(T_pM)\times[0,\epsilon) \to \R$ is smooth. Diminishing $\epsilon$ again if necessary, $\sqrt{h}$ has no critical points. By the implicit function theorem, there exists a diffeomorphism
		\[
		\widetilde{\kappa}:S^m\times[0,\epsilon)\to\orientedprojective(T_pM)\times[0,\epsilon)
		\]
		satisfying
		\[
		\sqrt{h}\left(\widetilde{\kappa}(q,s)\right)=s,\quad(q,s)\in S^m\times[0,\epsilon).
		\]
We claim that the map $\kappa:=\pi\circ\widetilde{\kappa}$ has all the desired properties. Let
\[
\varpi : T_pM \setminus\{0\} \to \orientedprojective(T_pM)
\]
denote the projection. To show that $\kappa$ satisfies property~(\ref{item:sigma}) of a polar coordinate map, it suffices to show that $\pderiv[\kappa]{s}(q,0) \neq 0$ for $q \in S^m$ and the composition $\varpi \circ \pderiv[\kappa]{s}(\cdot,0) : S^m \to \orientedprojective(T_pM)$ is a diffeomorphism. Indeed,
\[
\pderiv[\kappa]{s}(q,0) = d\pi \circ \pderiv[\widetilde\kappa]{s}(q,0).
\]
Since $\widetilde \kappa$ is a diffeomorphism, $\pderiv[\widetilde{\kappa}]{s}(q,0)$ is not tangent to $E_p.$ So, $\pderiv[\kappa]{s}(q,0) \neq 0.$ Finally, $\varpi \circ \pderiv[\kappa]{s}(\cdot,0) = \widetilde{\kappa}(\cdot,0),$ which is a diffeomorphism by construction. The remaining properties of $\kappa$ are immediate.
	\end{proof}

	\subsection{Cone-immersed Lagrangians}
	\label{subsection: cone-immersed Lagrangians}

	Let $(X,\omega)$ be a $2n$-dimensional symplectic manifold and let $L$ be a connected $n$-dimensional smooth manifold. A cone-immersion $(\Psi:L\to X,S)$ is said to be \emph{Lagrangian} if it is Lagrangian away from its cone locus. The cone-immersed submanifold represented by a Lagrangian cone-immersion is also said to be Lagrangian. Suppose $\Lambda=[(\Psi:L\to X,S)]$ is Lagrangian, let $p\in\Lambda$ be a cone point, and let $\widetilde{p}\in\orientedprojective(TC_p\Lambda).$ As the Lagrangian Grassmannian bundle of $X$ is a closed subset of the $n$-Grassmannian bundle, it follows that $T_{\widetilde{p}}\Lambda$ is a Lagrangian subspace of $T_{p_0}X.$ Thus, we have a well-defined phase function $\theta_\Psi : \widetilde{L}_S \to S^1$ given by $\theta_\Psi(q) = \theta_{\widetilde{d\Psi}_q((\widetilde{TL}_S)_q)}.$

	We wish to study paths of cone-immersed Lagrangians with static cone locus. For a finite subset $C_0\subset X,$ we let $\mathcal{L}(X,L;S,C_0)$ denote the space of oriented cone-immersed Lagrangians in $X$ of type $(L,S)$ with cone locus image equal to $C_0.$ For a path $\Lambda=(\Lambda_t)_{t\in[0,1]}$ in $\mathcal{L}(X,L;S,C_0),$ a lifting of $\Lambda$ is a family of cone-immersions, $((\Psi_t:L\to X,S))_{t\in[0,1]},$ such that $(\Psi_t,S)$ represents $\Lambda_t$ for $t\in[0,1].$ The path $\Lambda$ is smooth if it admits a smooth lifting, that is, if the family of maps $\Psi_t\circ\pi$ is smooth, where $\pi:\widetilde{L}_S\to L$ is the blowup projection. Given a smooth path $\Lambda = (\Lambda_t)_{t \in [0,1]},$ we define a family of $1$-forms $\sigma_t$ on $\Lambda_t$ as follows. Let $((\Psi_t:L \to X,S))_{t}$ be a smooth lifting. We abbreviate
\[
\deriv{t}\Psi_t : = \deriv{t}\Psi_t|_{L\setminus S},
\]
which is a cone-smooth vector field along $\Psi_t.$ Moreover, the blowup vector field $\widetilde{\deriv{t}\Psi_t}$ vanishes on $\partial \widetilde{L}_S.$ We define
\[
\sigma_t := \left[\left((\Psi_t,S),i_{\deriv{t}\Psi_t}\omega\right)\right].
\]
\begin{lemma}\label{lemma:derivative 1-form of cone smooth path}
The form $\sigma_t$ is independent of the choice of $\Psi_t,$ closed, and vanishes at the cone locus of $\Lambda_t.$
\end{lemma}
\begin{proof}
The proof that $\sigma_t$ is independent of $\Psi_t$ and closed is analogous to the proof of~\cite[Lemma 2.1]{akveld-salamon}.   Lemma~\ref{lemma:cone-smooth pullback} implies that $\sigma_t$ vanishes at the cone locus of $\Lambda_t.$
\end{proof}
We call $\sigma_t$ the time derivative and write
\[
\deriv{t}\Lambda_t := \sigma_t.
\]
A path of cone-immersed Lagrangians is said to be exact if its time-derivative is the differential of a cone-smooth function $\deriv{t}\Lambda_t = dh_t.$ In this case, it follows that every cone point is a critical point of $h_t.$

	Suppose $(X,\omega,J,\Omega)$ is Calabi-Yau. An oriented cone-immersed Lagrangian $\Lambda\in\mathcal{L}(X,L;S,C_0)$ is \emph{positive} if the tangent space $T_p\Lambda$ is positive for each smooth point $p$, and for each cone point $p$ and $\widetilde p \in \orientedprojective(TC_p\Lambda)$ the tangent space $T_{\widetilde p}\Lambda$ is positive. Note that this is stronger than positivity at the smooth locus. Assume now that $L$ is closed. Let $\mathcal{O}\subset\mathcal{L}(X,L;S,C_0)$ be an exact isotopy class of positive cone-immersed Lagrangians and let $\Lambda = [(\Psi:L \to X,S)]\in\mathcal{O}.$ Recall Definition~\ref{definition:immersed cone smooth differential form}. As the blowup $\widetilde{L}_S$ is compact, the volume form $\real\Omega|_\Lambda$ is integrable. Let $C^\infty(\Lambda)$ denote the space of cone-smooth functions on $\Lambda.$ Set
	\[
	\overline{C^\infty}(\Lambda):=\left\{h\in C^\infty(\Lambda)\left|\int_\Lambda h\real\Omega=0,\;\forall c\in \Lambda^C,\;dh_c=0\right.\right\}.
	\]
	Then the isomorphism~\eqref{equation: tangent space to O} and Riemannian metric~\eqref{equation: the metric} make sense as in the smooth case. Let $(\Lambda_t)_t$ be a smooth path in $\mathcal{O}$ and let $(\Psi_t)_t$ be a smooth lifting. We say $(\Psi_t)_t$ is \emph{horizontal} if it satisfies $i_{\deriv{t}\Psi_t}\real\Omega=0.$ It is shown in~\cite{solomon} that every compactly supported path of \emph{smooth} positive Lagrangians admits horizontal liftings. We show the same for cone-immersed positive Lagrangians.

	\begin{lemma}
		\label{lemma: path of cone-Lags admits horizontal liftings}
		Let $(\Lambda_t)_{t\in[0,1]}$ be a smooth path in $\mathcal{O},$ and let $(\Psi,S)$ be a representative of $\Lambda_0.$ Then $(\Psi,S)$ extends uniquely to a horizontal lifting of $(\Lambda_t)_t.$
	\end{lemma}

	\begin{proof}
		We imitate the argument presented in~\cite{solomon}. Let $\left(\left(\Psi_t,S\right)\right)_t$ be a smooth lifting of $(\Lambda_t)_t$ with $\Psi_0=\Psi.$ For $t\in[0,1],$ let $w_t$ denote the unique vector field on $L\setminus S$ satisfying
		\[ i_{w_t}\Psi_t^*\real\Omega=-i_{\deriv{t}\Psi_t}\real\Omega.
		\]
We claim that $w_t$ is cone-smooth and the blowup vector field $\widetilde w_t$ vanishes on $\partial \widetilde L_S.$ Indeed, Lemma~\ref{lemma:cone-smooth pullback}
implies that the differential forms $\Psi_t^*\real\Omega$ and $i_{\deriv{t}\Psi_t}\real\Omega$ are cone-smooth and the blowup differential form $\widetilde{i_{\deriv{t}\Psi_t}\real\Omega}$ vanishes on $\partial \widetilde L_S.$ Moreover, since $\Lambda_t$ is positive, it follows from Lemma~\ref{lemma:blowupdifferential} that the blowup form $\widetilde{\Psi_t^*\real \Omega}$ is non-vanishing. Let $\widetilde w_t$ be the unique section of $\widetilde{TL}_S$ such that
\[
i_{\widetilde w_t}\widetilde{\Psi_t^*\real\Omega}=-\widetilde{i_{\deriv{t}\Psi_t}\real\Omega}.
\]
Then, $\widetilde w_t|_{\widetilde L_S^\circ} = \pi|_{\widetilde L_S^\circ}^*w_t.$ So, $w_t$ is cone-smooth and $\widetilde w_t$ is the blowup vector field. Furthermore, $\widetilde w_t$ vanishes on $\partial \widetilde L_S$ since $\widetilde{i_{\deriv{t}\Psi_t}\real\Omega}$ vanishes on $\partial \widetilde L_S.$

By Lemma~\ref{lemma:cone smooth vanishing tangent} there exists a unique vector field $\widehat w_t$ on $\widetilde L_S$ such that
\[
d\pi\left(\widehat w_t|_{\widetilde L_S^\circ}\right) = w_t.
\]
Moreover, $\widehat w_t$ is tangent to $\partial \widetilde L_S.$
Let $(\widehat \varphi_t)_t$ denote the flow of $(\widehat w_t)_t.$ Then, $\widehat\varphi_t(\partial\widetilde L_S) = \partial \widetilde L_S,$  and $\widehat \varphi_t$ descends to a map $\varphi_t : M \to M.$
Remark~\ref{remark: cone-diffeomorphism lifting} implies that $\varphi_t$ is cone-smooth.
Thus, the family of compositions $(\Psi_t\circ\varphi_t)_t$ gives the desired horizontal lifting.
	\end{proof}

	By virtue of Lemma~\ref{lemma: path of cone-Lags admits horizontal liftings}, the Levi-Civita connection described in Section~\ref{subsection: calabi-yau manifolds} extends naturally to exact isotopy classes of cone-immersed positive Lagrangians. We use horizontal lifts to define geodesics in such classes. The definition below is more general than that used in previous works, as it allows the Lagrangians in question to be non-closed or non-smooth. Note, however, that it is equivalent to the old definition when the Lagrangians in question are smoothly embedded and closed.

	\begin{dfn}
		\label{definition: geodesic}
		Let $(X,\omega,J,\Omega)$ be a Calabi-Yau manifold, let $L$ be a connected smooth manifold, not necessarily closed, and let $S \subset L$ be a finite subset. Let $C_0\subset X$ be finite, let $\mathcal{O}\subset\mathcal{L}(X,L;S,C_0)$ be an exact isotopy class of cone-immersed Lagrangians, and let $(\Lambda_t)_{t\in[0,1]}$ be a path in $\mathcal{O}.$ The path $(\Lambda_t)_t$ is a \emph{geodesic} if it admits a horizontal lifting $((\Psi_t,S))_{t\in[0,1]}$ and a family of functions $h_t\in C^\infty(\Lambda_t)$ satisfying
		\[
		\deriv{t}\Lambda_t=dh_t,\quad\deriv{t}(h_t\circ\Psi_t)=0.
		\]
		We call the family $(h_t)_{t\in[0,1]}$ the \emph{Hamiltonian} of the geodesic. We also call the time independent function $h = h_t \circ \Psi_t : L \to \R$ the \emph{Hamiltonian} with respect to the horizontal lifting $(\Psi_t)_t.$ Observe that $h_t = [(\Psi_t,h)].$  If $L$ is not compact, the Hamiltonian is only well-defined up to a time independent constant. If $C_0$ is empty, we say $(\Lambda_t)_t$ is a \emph{smooth geodesic} or \emph{geodesic of smooth Lagrangians}.
	\end{dfn}
From now on, unless otherwise specified, the term geodesic will be used in the sense of the preceding definition.

\begin{lemma}\label{lemma:extension of geodesic to cone point}
Let $(\Lambda_t)_t$ be a path in $\mathcal{L}(X,L;S,C_0)$ and let $(\Lambda_t^\circ)_t$ be the path in $\mathcal{L}(X,L\setminus S)$ obtained by removing the cone points. If $(\Lambda_t^\circ)_t$ is a geodesic, then so is~$(\Lambda_t)_t.$
\end{lemma}
\begin{proof}
This follows from Lemma~\ref{lemma:exactness of extension} and Lemma~\ref{lemma:derivative 1-form of cone smooth path}.
\end{proof}

	\section{Lagrangian and special Lagrangian cylinders}
	\label{section: lagrangian and special lagrangian cylinders}

	\subsection{The space of Lagrangian cylinders}
	\label{subsection: the space of lagrangian cylinders}

	We start this section with the definition of its main objects.

	\begin{dfn}\label{definition: Lagrangian cylinders}$\;$
		\begin{enumerate}[label=(\alph*)]
			\item \label{item:just Lagrangian} Let $(X,\omega)$ be a symplectic manifold, and let $\Lambda_0,\Lambda_1\subset X$ be smoothly embedded Lagrangians. A \emph{Lagrangian cylinder} between $\Lambda_0$ and $\Lambda_1$ is a smooth immersed Lagrangian submanifold with boundary, $Z=[f:L\to X]\in\mathcal{L}(X,L;\Lambda_0,\Lambda_1)$, where $L=N\times[0,1]$ for some smooth manifold $N,$ and the restricted immersion $f|_{N\times\{i\}}$ is an embedding into $\Lambda_i$ for $i=0,1.$ We let $\mathcal{LC}(N;\Lambda_0,\Lambda_1)$ denote the space of Lagrangian cylinders between $\Lambda_0$ and $\Lambda_1$ of type $N\times[0,1]$. We let $\mathcal{LC}(\Lambda_0,\Lambda_1)$ denote the space of Lagrangian cylinders between $\Lambda_0$ and $\Lambda_1$ of general topological type.
			\item\label{item:special} If $X$ is Calabi-Yau and $\Lambda_0$ and $\Lambda_1$ are positive, we let $\mathcal{SLC}(N;\Lambda_0,\Lambda_1)\subset\mathcal{LC}(N;\Lambda_0,\Lambda_1)$ denote the subspace consisting of imaginary special Lagrangian cylinders. The space $\mathcal{SLC}(\Lambda_0,\Lambda_1)$ is defined analogously.
		\end{enumerate}
	\end{dfn}

\begin{rem}\label{remark:automatic}\mbox{}
\begin{enumerate}
  \item \label{item:free}
Let $Z = [f : N \times [0,1] \to X]$ as in Definition~\ref{definition: Lagrangian cylinders}~\ref{item:just Lagrangian}. By Lemma~\ref{lemma: embedded boundary point implies freeness} the requirement that $f|_{N \times \{i\}}$ is an embedding implies that $f$ is free as required in Notation~\ref{notation:space of Lagrangians}.
  \item \label{item:not tangent}
Let $X, \Lambda_0,\Lambda_1,$ be as in Definition~\ref{definition: Lagrangian cylinders}~\ref{item:special}. Let $Z = [f: N \times [0,1] \to X]$ be an immersed imaginary special Lagrangian with the boundary component corresponding to $N\times \{i\}$ in $\Lambda_i.$  Then $Z$ automatically satisfies condition~\ref{not tangent} in Notation~\ref{notation:space of Lagrangians}.
\end{enumerate}
\end{rem}

	It is well-known that special Lagrangian submanifolds can be described as solutions to an elliptic PDE (see~\cite{caffarelli-nirenberg-spruck,harvey-lawson} and the references therein or~\cite{mclean,salur} for a geometric approach). We show below that any pair of positive Lagrangian submanifolds, $\Lambda_0$ and $\Lambda_1$, provides an elliptic boundary condition to the imaginary special Lagrangian equation. In particular, we shall see in Proposition~\ref{proposition: space of special lag cylinders is one dimensional} that, if $N$ is compact, $\mathcal{SLC}(N;\Lambda_0,\Lambda_1)$ is a smooth $1$-dimensional submanifold of the Fr\'echet manifold $\mathcal{LC}(N;\Lambda_0,\Lambda_1)$. Our approach is similar to that of McLean~\cite{mclean} with some necessary adaptations.

	Fix two Lagrangian submanifolds, $\Lambda_0,\Lambda_1\subset X,$ and a compact $n-1$-dimensional manifold $N.$ By Corollary~\ref{corollary: frechet}, the space $\mathcal{LC}(N;\Lambda_0,\Lambda_1)$ is a Fr\'echet manifold modeled locally on $\Omega_B^1(N\times[0,1]),$ the space of closed 1-forms on $N\times[0,1]$ annihilating the boundary. Moreover, by Lemma~\ref{lemma: tangent space} \ref{tangent space of lag with boundary}, for $Z\in\mathcal{LC}(N;\Lambda_0,\Lambda_1)$ we have a canonical isomorphism
	\[
	T_Z\mathcal{LC}(N;\Lambda_0,\Lambda_1)\cong\Omega_B^1(Z).
	\]
	The following observation allows us to replace the spaces $\Omega_B^1(N\times[0,1])$ and $\Omega_B^1(Z)$ with spaces of functions.

	\begin{lemma}
		\label{lemma: closed forms on cylinder}
		Let $N$ be a smooth manifold without boundary, and let $\sigma\in\Omega^1(N\times[0,1])$ be closed with pullback to the boundary component $N\times\{0\}$ zero. Then $\sigma$ is exact.
	\end{lemma}

	\begin{proof}
		Let $\pi_0:N\times[0,1]\to N\times[0,1]$ be given by $(p,t)\mapsto(p,0)$. Let $\gamma:S^1\to N\times[0,1]$ be a smooth loop, and write $\gamma_0:=\pi_0\circ\gamma$. Then $\gamma_0$ is homotopic to $\gamma$. As $\sigma$ is closed and annihilates the boundary component $N\times\{0\}$, we have
		\[
		\int_{S^1}\gamma^*\sigma=\int_{S^1}\gamma_0^*\sigma=0,
		\]
		and the lemma follows.
	\end{proof}

	In the rest of this section, we assume $N$ is connected. By Lemma~\ref{lemma: closed forms on cylinder}, every differential form in $\Omega_B^1(N\times[0,1])$ is exact with primitive constant on each boundary component. So, by Corollary~\ref{corollary: frechet} we obtain the following lemma. For a smooth cylinder $Z$ with boundary components $C_0$ and $C_1$, we let $\cob{\infty}(Z)$ denote the space of smooth functions on $Z$ which vanish on $C_0$ and are constant on $C_1$.

	\begin{lemma}
		\label{lemma: local description of space of cylinders}
		Let $(X,\omega)$ be a symplectic manifold, let $\Lambda_0,\Lambda_1\subset X$ be fixed smoothly embedded Lagrangians, let $N$ be a connected closed $n-1$-dimensional manifold and let $Z\in\mathcal{LC}(N;\Lambda_0,\Lambda_1)$. Then any immersed Weinstein neighborhood of $Z$ compatible with $\Lambda_0$ and $\Lambda_1$ gives rise to a local parameterization
		\[
		\mathbf{X}:\mathcal{U}\subset\cob{\infty}(Z)\to\widetilde{\mathcal{U}}\subset\mathcal{LC}(N;\Lambda_0,\Lambda_1).
		\]
		Moreover, we have a canonical isomorphism
		\[
		T_Z\mathcal{LC}(N;\Lambda_0,\Lambda_1)\cong\cob{\infty}(Z).
		\]
	\end{lemma}

	Continuing with the same setting, we now assume further that $(X,\omega,J,\Omega)$ is Calabi-Yau, $\Lambda_0$ and $\Lambda_1$ are positive and $Z\in\mathcal{SLC}(N;\Lambda_0,\Lambda_1).$ Abbreviate
\[
L = N \times [0,1].
\]
Fix an immersion
\[
f: L\to X
\]
representing $Z.$
By Lemma~\ref{lemma: Weinstein with boundary}, choose an immersed Weinstein neighborhood $(V,\varphi)$ of $Z$ compatible with $\Lambda_0$ and $\Lambda_1,$ where $V \subset T^*L$ and $\varphi : V\to X$ with $\varphi|_L = f.$ Let $\pi_L : T^*L \to L$ denote the projection. For $u\in\cob{\infty}(L),$ let $\mathrm{Graph}(du) \subset T^*L$ denote the graph. Write
	\[
	\mathcal{U}:=\left\{\left.u\in\cob{\infty}(L)\right|\mathrm{Graph}(du)\subset V\right\}.
	\]
For $u\in\mathcal{U},$ let $j_u : L \to X$ be given by
\[
j_u = \varphi \circ \left( \pi_L|_{\mathrm{Graph}(du)}\right)^{-1}.
\]

Define
	\begin{equation}
	\label{SL operator}
	F:\mathcal{U}\to C^\infty(L),\qquad u\mapsto*j_u^*\real\Omega,
	\end{equation}
	where $*$ denotes the Hodge star operator of $f^*g.$ Then, for $u\in\mathcal{U}$, the cylinder $[j_u]$ is imaginary special Lagrangian if and only if the function $u$ satisfies $F(u)=0$. We have thus established a local characterization of $\mathcal{SLC}(N;\Lambda_0,\Lambda_1)$ as the zero set of a differential operator. Since $Z\in\mathcal{SLC}(N;\Lambda_0,\Lambda_1),$ we have $F(0)=0$. We compute the linearization of $F$ in Lemma~\ref{lemma: linearized operator} below. By Lemma~\ref{lemma: Omega does not vanish on Lags} we have
	\begin{equation}
	\label{equation:rho on imaginary cylinder}
	f^*\imaginary\Omega=\rho_f\vol_f,
	\end{equation}
	where $\vol_f$ denotes the Riemannian volume form of $f^*g$ and $\rho_f := \rho \circ f$ with $\rho$ the positive function defined in~\eqref{equation: rho}.

	\begin{lemma}
		\label{lemma: linearized operator}
		Consider the above setting.
		\begin{enumerate}[label=(\alph*)]
			\item\label{more general lemma} Let $f_t:L\to X,\;t\in(-\epsilon,\epsilon),$ be a smooth family of Lagrangian immersions with $f_0=f.$ Write $v:=\left.\deriv{t}\right|_{t=0}f_t$ and suppose we have $i_v\omega=du$ for some $u\in C^\infty\left(L\right).$ Then
			\[ \left.\deriv{t}\right|_{t=0}\left(*f_t^*\real\Omega\right)=*d(\rho_f*du).
			\]
			\item\label{less general lemma} The linearization of the operator $F$ of~\eqref{SL operator} at $0 \in \mathcal{U}$ is given by
			\[
			dF_0(u)=*d(\rho_f *du).
			\]
		\end{enumerate}
	\end{lemma}

	\begin{proof}
		We prove part \ref{more general lemma}. Decompose $v$ into
		\[
		v=v^T+v^\perp,
		\]
		where $v^T$ is tangent to $Z$ and $v^\perp$ is orthogonal to $Z$ with respect to the K\"ahler metric $g$. As $Z$ is Lagrangian, we have
		\[
		i_{v^\perp}\omega=i_{v}\omega=du,
		\]
		which implies
		\begin{equation}
		\label{orthogonal component}
		v^\perp=-Jdf(\nabla u),
		\end{equation}
		where $\nabla$ denotes the gradient with respect to $f^*g$. By~\eqref{equation:rho on imaginary cylinder} and~\eqref{orthogonal component}, as $Z$ is imaginary special Lagrangian and $\Omega$ is complex-linear, we have
		\begin{align}
		\label{gradient contraction}
		i_{v}\real\Omega
		&=i_{v^\perp}\real\Omega\\
		&=i_{\nabla u}f^*\imaginary\Omega\nonumber\\
		&=\rho_f \,i_{\nabla u}\vol_f\nonumber\\
		&=\rho_f*du.\nonumber
		\end{align}
		Finally, as $\Omega$ is closed, from the Cartan formula and~\eqref{gradient contraction} we deduce
		\begin{align*}
		*\left(\left.\deriv{t}\right|_{t=0}f_t^*\real\Omega\right)
		&=*di_{v}\real\Omega\\
		&=*d(\rho_f*du),
		\end{align*}
		as desired. Part \ref{less general lemma} is a particular case of \ref{more general lemma}.
	\end{proof}

	In light of Lemma~\ref{lemma: linearized operator} we define the linear second-order operator
	\begin{equation}
	\label{equation: definition of rho-Laplacian}
	\Delta_\rho:\cob{\infty}(L)\to C^\infty(L),\qquad u\mapsto*d(\rho_f*du).
	\end{equation}
For $Z  \in \mathcal{SLC}(N;\Lambda_0,\Lambda_1),$ and $f : L \to X$ an immersion with $Z = [f],$ we have a canonical identification $C^\infty(Z) = C^\infty(L),$ so we obtain an operator
\[
\Delta_\rho : \cob{\infty}(Z) \to C^\infty(Z).
\]
This operator does not depend on the choice of $f.$
	The operator $\Delta_\rho$ is similar to the usual Riemannian Laplacian in a manner made precise in Lemma~\ref{lemma: Delta rho is a cool operator} below.

For $k\ge0$ and $\alpha\in(0,1)$ we let $\cob{k,\alpha}(L)$ denote the completion of $\cob{\infty}(L)$ with respect to the H\"older $C^{k,\alpha}$-norm. We let $C^{k,\alpha}(L;\partial L)$ denote the space of functions on $L$ of regularity $C^{k,\alpha}$ which vanish on the boundary. For $P,Q,$ smooth manifolds, we let $C^{k,\alpha}(P,Q)$ denote the smooth Banach manifold of maps $P \to Q$ of regularity $C^{k,\alpha}.$

	\begin{lemma}
		\label{lemma: Delta rho is a cool operator}
		Let $k$ be a non-negative integer and let $\alpha\in(0,1)$. Then the naturally extended linear operator $\Delta_\rho:\cob{k+2,\alpha}(L)\to C^{k,\alpha}(L)$ is surjective with a 1-dimensional kernel consisting of $C^\infty$ functions.
	\end{lemma}

	\begin{proof}
		We note first that the principal symbol of $\Delta_\rho$ differs from that of the usual Riemannian Laplacian by the positive function $\rho_f$. Hence $\Delta_\rho$ is elliptic. Also, $\Delta_\rho$ annihilates constants. Equivalently, expressing $\Delta_\rho$ in local coordinates as
		\[
		\Delta_\rho u=a^{ij}u_{ij}+b^iu_i+cu,
		\]
		the coefficient $c$ vanishes. As $L$ is compact, it now follows from standard arguments (see, for example,~\cite[Chapter~6]{gilbarg-trudinger} and~\cite[Section~5.1]{taylor1}) that
\[
\Delta_\rho:C^{k+2,\alpha}(L;\partial L)\to C^{k,\alpha}(L)
\]
is one-to-one and surjective. Finally, for $a\in\R,$ there exists a unique $u\in\cob{k+2,\alpha}(L)$ satisfying
		\[
		\Delta_\rho u=0,\quad u|_{N\times \{1\}}=a,
		\]
		which is in fact $C^\infty$. The lemma follows.
	\end{proof}

	\begin{prop}
		\label{proposition: space of special lag cylinders is one dimensional}
		Let $\Lambda_0,\Lambda_1\subset X$ be smoothly embedded positive Lagrangians, let $N$ be a connected closed smooth manifold, and abbreviate $L = N\times [0,1].$ Then the space $\mathcal{SLC}(N;\Lambda_0,\Lambda_1)$ is a smoothly embedded $1$-dimensional submanifold of $\mathcal{LC}(N;\Lambda_0,\Lambda_1)$. Moreover, for $Z = [f : L \to X]\in\mathcal{SLC}(N;\Lambda_0,\Lambda_1)$, we have
		\[
		T_Z\mathcal{SLC}(N;\Lambda_0,\Lambda_1)=\ker\Delta_\rho,
		\]
		where $\Delta_\rho:\cob{\infty}(L)\to C^\infty(L)$ is defined as in~\eqref{equation: definition of rho-Laplacian}.
	\end{prop}

	\begin{proof}
		Let $Z = [f : L \to X]\in\mathcal{SLC}(N;\Lambda_0,\Lambda_1)$. Recall the definition of the operator $F : \mathcal{U} \to C^\infty(L)$ from~\eqref{SL operator}. Pick $\alpha\in(0,1)$ and let $\mathcal{U}^{2,\alpha} \subset \cob{2,\alpha}(L)$ be an open set with $\mathcal{U}^{2,\alpha} \cap \cob{\infty}(L) = \mathcal{U}.$ Extend the operator $F$ to an operator
\[
F : \mathcal{U}^{2,\alpha} \to C^{\alpha}(L).
\]
Since $\Omega$ is smooth, and the map
\[
\mathcal{U}^{2,\alpha} \to C^{1,\alpha}(L,X), \qquad u \mapsto j_u,
\]
is smooth, it follows that
the extended operator $F$ is smooth. Since $Z$ is imaginary special Lagrangian, $F(0)=0$. By Lemmas~\ref{lemma: linearized operator} and~\ref{lemma: Delta rho is a cool operator}, the linearization
\[
dF_0=\Delta_\rho:\cob{2,\alpha}(L)\to C^\alpha(L)
\]
is onto with a $1$-dimensional kernel. By the implicit function theorem, there exist $\epsilon>0$, an open $0\in\mathcal{W}\subset\cob{2,\alpha}(L)$ and a smooth embedding $\gamma:(-\epsilon,\epsilon)\to\mathcal{W}$ with $\gamma(0)=0,$ such that for $f\in\mathcal{W}$, we have $F(f)=0$ if and only if $f=\gamma(t)$ for some $t\in(-\epsilon,\epsilon)$. The path $\gamma$ satisfies
		\[
		\Delta_\rho(\dot{\gamma}(0))=\left.\deriv{t}\right|_{t=0}F(\gamma(t))=0.
		\]
		By elliptic regularity (e.g.\ \cite[Chapter 17]{gilbarg-trudinger}), for $t\in(-\epsilon,\epsilon)$ the function $\gamma(t)$ is in fact in $\cob{\infty}(L)$. Moreover, for every $k\ge2$, the above argument with $2$ replaced by $k$ shows that $\gamma$ is smooth as an embedding into the space $\cob{k,\alpha}(L)$. It follows that $\gamma$ is smooth as a map into $\cob{\infty}(L)$.
	\end{proof}

	\begin{proof}[Proof of Theorem~\ref{theorem: space of ISL cylinders is 1-dim}]
		Follows from Proposition~\ref{proposition: space of special lag cylinders is one dimensional}.
	\end{proof}

	For a Lagrangian cylinder $Z \in\mathcal{LC}(N;\Lambda_0,\Lambda_1)$, let $Z^\mathcal{H}\subset\mathcal{LC}(N;\Lambda_0,\Lambda_1)$ denote the space of Lagrangian cylinders exact isotopic to $Z$ relative to the boundary. Then we have
	\[
	T_ZZ^\mathcal{H}=C^\infty(Z;\partial Z)\subset\cob{\infty}(Z),
	\]
	the space of smooth functions vanishing on the boundary. As we have
	\[
	\cob{\infty}(Z)=C^\infty(Z;\partial Z)\oplus\ker\Delta_\rho,
	\]
	the following is a consequence of Proposition~\ref{proposition: space of special lag cylinders is one dimensional}.

	\begin{cor}
		\label{corollary: Hamiltonian isotopy class is descrete}
		Let $\Lambda_0,\Lambda_1\subset X$ be smoothly embedded positive Lagrangians and let $Z\in\mathcal{LC}(N;\Lambda_0,\Lambda_1)$. Then the intersection $Z^\mathcal{H}\cap\mathcal{SLC}(N;\Lambda_0,\Lambda_1)$ is a discrete subset of $\mathcal{LC}(N;\Lambda_0,\Lambda_1)$.
	\end{cor}

	\begin{rem}
		\label{remark: not only cylinders}
		Using the above technique, one can generalize Proposition~\ref{proposition: space of special lag cylinders is one dimensional} to spaces of imaginary special Lagrangians modeled by an arbitrary manifold $L$ with any number of boundary components. Namely, recalling Notation~\ref{notation:space of Lagrangians}, for $\Lambda_1,\ldots,\Lambda_k \subset X$ positive Lagrangians, let
\[
\mathcal{SL}(X,L;\Lambda_1,\ldots,\Lambda_k) \subset \mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)
\]
denote the subspace of imaginary special Lagrangians. As Lemma~\ref{lemma: closed forms on cylinder} fails to hold for arbitrary manifolds $L,$ one considers the space $\Omega_B^1(L)$ as a local model of $\mathcal{L}(X,L;\Lambda_1,\ldots,\Lambda_k)$ as in Corollary~\ref{corollary: frechet}. Special Lagrangian submanifolds near $Z \in \mathcal{SL}(X,L;\Lambda_1,\ldots,\Lambda_k)$ are then parameterized by $\Delta_\rho$-harmonic $1$-forms on $L$, where $\Delta_\rho$ in this case is a modified Hodge Laplacian. It follows from the Hodge decomposition for manifolds with boundary (see~\cite{gunterschwartz},~\cite[Section~5.9]{taylor1} and the references therein) that $\mathcal{SL}(X,L;\Lambda_1,\ldots,\Lambda_k)$ has the dimension of the real relative cohomology space $H^1(L,\partial L)$. Similarly to Corollary~\ref{corollary: Hamiltonian isotopy class is descrete}, the intersection $Z^\mathcal{H} \cap \mathcal{SL}(X,L;\Lambda_1,\ldots,\Lambda_k)$ is discrete.
	\end{rem}

	\subsection{Relative Lagrangian flux}
	\label{subsection: relative Lagrangian Flux}

	Let $(X,\omega)$ be a symplectic manifold. Recall the notion of \emph{Lagrangian flux}~\cite{calabi, fukaya, solomon}. Roughly speaking, given a path of closed Lagrangian submanifolds, $\Lambda_t \subset X,\;t\in[0,1],$ its Lagrangian flux is a linear functional on $H_1(\Lambda_0)$ which measures the path's deviation from being exact. Suppose now that $\Lambda_0,\Lambda_1\subset X$ are fixed Lagrangians and $Z_s,\;s\in[0,1],$ is a path in $\mathcal{LC}(N;\Lambda_0,\Lambda_1).$ Then the \emph{relative} Lagrangian flux of the path is a linear functional on $H_1(Z_0,\partial Z_0)$ which measures the deviation of the path from being exact relative to the boundary. As the relative homology group $H_1(Z_0,\partial Z_0)$ is generated by a single element, one can think of the relative Lagrangian flux of a path of cylinders as a number. The precise definition is as follows.

Let $N$ be a closed connected $n-1$-dimensional smooth manifold and let $Z_s,\;s\in[s_0,s_1],$ be a smooth path in $\mathcal{LC}(N;\Lambda_0,\Lambda_1)$. Let $\Phi:N\times[0,1]\times[s_0,s_1]\to X$ be a smooth parameterization of the path $(Z_s)$. Namely, for $s\in[s_0,s_1],$ the restricted map $\Phi_s:=\Phi|_{N\times[0,1]\times\{s\}}$ is a parameterization of the cylinder $Z_s$, and for $(p,s)\in N\times[s_0,s_1]$ we have
\[
	\Phi(p,i,s)\in\Lambda_i,\quad i=0,1.
\]
For $s\in[s_0,s_1],$ write
\[
	h_s:=\deriv{s}Z_s\in\cob{\infty}(Z_s).
\]
	For $i = 0,1,$ let $C_{i,s}$ denote the boundary component of the cylinder $Z_s$ corresponding to $N \times \{i\},$ let $A_s \in \R$ satisfy
	\[
	h_s|_{C_{1,s}} \equiv A_s.
	\]
	Let $\gamma:[0,1]\to N\times[0,1]$ be a smooth path representing the fundamental class in the relative homology group $H_1(N\times[0,1],\partial(N\times[0,1])).$ Namely, $\gamma$ satisfies
	\[
	\gamma(i)\in N\times\{i\},\quad i=0,1.
	\]
	Set
	\[
	\overline{\gamma}:[0,1]\times[s_0,s_1]\to X,\quad(t,s)\mapsto\Phi(\gamma(t),s).
	\]

	\begin{lemma}
		\label{lemma: relative Lagrangian flux}
		In the above setting we have
		\begin{equation}
		\label{equality: relative Lagrangian flux}
		\int_{[0,1]\times[s_0,s_1]}\overline{\gamma}^*\omega=-\int_{s_0}^{s_1}A_sds.
		\end{equation}
	\end{lemma}

	\begin{proof}
		For $s\in[s_0,s_1],$ let
		\[
		Y_s:=\deriv{s}(\Phi_s\circ\gamma)\in\Gamma([0,1],(\Phi_s\circ\gamma)^*TX).
		\]
		By definition of the derivative of a Lagrangian path (see Remark~\ref{remark: tangent space hands on}), we have
		\[
		i_{Y_s}\omega=d(h_s\circ\Phi_s\circ\gamma).
		\]
		By Fubini's theorem,
		\begin{align*}
		\int_{[0,1]\times[s_0,s_1]}\overline{\gamma}^*\omega
		&=-\int_{s_0}^{s_1}\left(\int_{[0,1]}i_{Y_s}\omega\right)ds\\
		&=-\int_{s_0}^{s_1}A_sds,
		\end{align*}
		as desired.
	\end{proof}

	\begin{dfn}
		\label{definition:relative Lagrangian flux}
		The quantity in equality~\eqref{equality: relative Lagrangian flux} is called the \emph{relative Lagrangian flux} of the path $(Z_s)_{s\in[s_0,s_1]}.$  We let $\mathrm{RelFlux}\left((Z_s)_{s\in[s_0,s_1]}\right)$ denote the relative Lagrangian flux. More generally, if $I \subset \R$ is an interval, possibly open or half open, with endpoints $a < b,$ and $(Z_s)_{s \in I}$ is a path in $\mathcal{LC}(N;\Lambda_0,\Lambda_1),$ we write
\[
\mathrm{RelFlux}\left((Z_s)_{s\in I}\right) = \lim_{s_0 \to a}\lim_{s_1 \to b}\mathrm{RelFlux}\left((Z_s)_{s\in[s_0,s_1]}\right)
\]
whenever the limit exists.
	\end{dfn}

\begin{rem}\label{remark:invariance of relative flux}
	A straightforward modification of the arguments in~\cite{solomon} shows that $\mathrm{RelFlux}\left((Z_s)_{s\in I}\right)$ is independent of the choices of the parameterization $\Phi$ and the path $\gamma.$ Moreover, when $I$ is a closed interval, $\mathrm{RelFlux}\left((Z_s)_{s\in I}\right)$ depends only on the homotopy class of the path $(Z_s)_s$ relative to its endpoints.
\end{rem}

	\subsection{Regular families of special Lagrangian cylinders}

	\label{subsection: regular families of special Lagrangian cylinders}

	Let $(X,\omega,J,\Omega)$ be a Calabi-Yau manifold of real dimension $2n,$ let $\Lambda_0,\Lambda_1\subset X$ be smoothly embedded positive Lagrangians, and let $N$ be a closed smooth manifold of dimension~$n-1.$

	\begin{dfn}
		\label{definition: fundamental harmonic and regular harmonics}
		Let $Z\in\mathcal{SLC}(N;\Lambda_0,\Lambda_1),$ and let $C_i,\;i=0,1,$ denote the boundary component of $Z$ corresponding to $N \times \{i\}.$ The \emph{fundamental harmonic} on $Z$ is the unique function $\sigma\in\cob{\infty}(Z)$ satisfying
		\[
		\Delta_\rho(\sigma)=0,\quad\sigma|_{C_1}\equiv 1.
		\]
		If $\sigma$ has no critical points, we say $Z$ has \emph{regular harmonics}.
	\end{dfn}

For the remainder of this section, we assume that $N$ is connected. Let
\[
\gamma:(-\epsilon,\epsilon)\to\mathcal{SLC}(N;\Lambda_0,\Lambda_1)
\]
be a smooth path. A smooth lifting of $\gamma$ is a smooth  map $\Phi:N\times[0,1]\times(-\epsilon,\epsilon)\to X$ such that for $s \in (-\epsilon,\epsilon)$ the restriction
\[
\Phi_s : = \Phi|_{N\times [0,1]\times\{s\}} : N \times [0,1] \to X
\]
represents the immersed submanifold $\gamma(s)\in \mathcal{SLC}(N;\Lambda_0,\Lambda_1).$
	\begin{lemma}
		\label{lemma:immersive cylinder has regular harmonics}
		Let $Z\in\mathcal{SLC}\left(N;\Lambda_0,\Lambda_1\right).$ Then the following are equivalent.
\begin{enumerate}
\item\label{item:regular harmonics}
$Z$ has regular harmonics.
\item\label{item:all}
For every embedded curve $\gamma : (-\epsilon,\epsilon) \to \mathcal{SLC}\left(N;\Lambda_0,\Lambda_1\right)$ with $\gamma(0) = Z,$ and for every smooth lifting of $\gamma,$
		\[
		\Phi:N\times[0,1]\times(-\epsilon,\epsilon)\to X,
		\]
after possibly diminishing $\epsilon,$ the map $\Phi$ is an immersion.
\item\label{item:one}
For one embedded curve $\gamma : (-\epsilon,\epsilon) \to \mathcal{SLC}\left(N;\Lambda_0,\Lambda_1\right)$ with $\gamma(0) = Z,$ and one smooth lifting of $\gamma,$
\[
		\Phi:N\times[0,1]\times(-\epsilon,\epsilon)\to X,
		\]
the map $\Phi$ is an immersion.
\end{enumerate}
\end{lemma}

	\begin{proof}
First, we prove that condition~(\ref{item:one}) implies condition~(\ref{item:regular harmonics}).		Let $\gamma$ and $\Phi$ be as in condition~(\ref{item:one}).
		Let $\sigma$ denote the fundamental harmonic on $Z$. By Proposition~\ref{proposition: space of special lag cylinders is one dimensional}, we have
		\[
		\dot\gamma(0)=a\sigma
		\]
		for some $a\in\R.$ Let
		\begin{equation}
		\label{this is Y}
		Y:=\left.\deriv{s}\right|_{s=0}\Phi(\cdot,\cdot,s)\in\Gamma\left(N\times[0,1], \Phi_0^*TX\right).
		\end{equation}
		By Remark~\ref{remark: tangent space hands on} we have
		\begin{equation}
		\label{equation: Y and a and sigma}
		d(a\sigma\circ\Phi_0)=i_Y\omega.
		\end{equation}
		As $\Phi$ is an immersion, the section $Y$ is nowhere tangent to $Z$. As $Z$ is Lagrangian, the right-hand side of~\eqref{equation: Y and a and sigma} is nowhere vanishing. It follows that $a\sigma$ does not have critical points and neither does $\sigma.$

		Next, we prove that condition~(\ref{item:regular harmonics}) implies condition~(\ref{item:all}). Indeed, suppose $Z$ has regular harmonics. Let $\gamma$ and $\Phi$ be as in condition~(\ref{item:all}). By Proposition~\ref{proposition: space of special lag cylinders is one dimensional} we have
		\[
		\dot{\gamma}(0)=a\sigma
		\]
		for some $0\ne a\in\mathbb{R}.$ Defining $Y$ as in~\eqref{this is Y}, the equality~\eqref{equation: Y and a and sigma} continues to hold true. By assumption, the left-hand side of~\eqref{equation: Y and a and sigma} is non-vanishing. It follows that $Y$ is nowhere tangent to $Z,$ which implies that, diminishing $\epsilon$ if necessary, $\Phi$ is an immersion.

It is immediate that condition~(\ref{item:all}) implies condition~(\ref{item:one}).
	\end{proof}

	The following lemma is, in fact, a particular case of Lemma~\ref{lemma:immersive cylinder has regular harmonics}. We provide an independent proof for clarity.

	\begin{lemma}
		\label{lemma: cylinder with regular harmonics in Euclidean space}
		Equip $\mathbb{C}^n$ with the standard Calabi-Yau structure, let $\Lambda_0,\Lambda_1\subset\mathbb{C}^n$ be positive Lagrangian linear subspaces, and let $Z\in\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right).$ Then $Z$ has regular harmonics if and only if $Z$ is nowhere tangent to the Euler vector field.
	\end{lemma}

	\begin{proof}
		Let $\Phi_1:S^{n-1}\times[0,1]\to\mathbb{C}^n$ be a representative of $Z,$ and set
		\[
		\Phi:S^{n-1}\times[0,1]\times(0,\infty)\to\mathbb{C}^n,\quad(p,t,s)\mapsto s\cdot\Phi_1(p,t).
		\]
		Then for $s\in(0,\infty),$ the map $\Phi_s:=\Phi|_{S^{n-1}\times[0,1]\times\{s\}}$ is an immersion representing an element in $\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right).$ Hence, we have
		\[
		\left.\deriv{s}\right|_{s=1}[\Phi_s]=a\sigma
		\]
		for some $0\ne a\in\mathbb{R},$ where $\sigma$ denotes the fundamental harmonic on $Z.$ By Remark~\ref{remark: tangent space hands on}, a point $q\in Z$ is a critical point of $\sigma$ if and only if the Euler vector at $q$ is tangent to $Z.$ The lemma follows.
	\end{proof}

	The following lemma, which is in fact an elementary observation in functional analysis, shows that the fundamental harmonic depends smoothly on the geometry of a cylinder. In particular, the property of having regular harmonics is stable under small perturbations.

	\begin{lemma}
		\label{lemma: regular harmonics is open property}
		Let $N$ be a closed connected smooth manifold, and write $Z:=N\times[0,1].$ Let $C_i:=N \times\{i\},i=0,1,$ denote the boundary components of $Z.$ Let $\alpha\in(0,1),$ let $k\in\mathbb{N},$ and let $L:\cob{k+2,\alpha}(Z)\to C^{k,\alpha}(Z)$ be a bounded linear operator, such that $L_0:=L|_{C^{k+2,\alpha}(Z;\partial Z)}:C^{k+2,\alpha}(Z;\partial Z)\to C^{k,\alpha}(Z)$ is an isomorphism. Let $u\in\cob{k+2,\alpha}(Z)$ be the unique function satisfying
		\[
		Lu=0,\quad u|_{C_1}\equiv 1.
		\]
		Then, letting $\mathcal{B}\left(\cob{k+2,\alpha}(Z),C^{k,\alpha}(Z)\right)$ denote the space of bounded linear operators $\cob{k+2,\alpha}(Z)\to C^{k,\alpha}(Z)$ endowed with the operator norm, there is an open neighborhood
\[
L\in \mathcal{V}\subset\mathcal{B}\left(\cob{k+2,\alpha}(Z),C^{k,\alpha}(Z)\right)
\]
such that for $\widetilde{L}\in\mathcal{V},$ there exists a unique $\widetilde{u}=\widetilde{u}_{\widetilde{L}}$ satisfying
		\[
		\widetilde{L}\widetilde{u}=0,\quad\widetilde{u}|_{C_1}\equiv 1.
		\]
		Moreover, the assignment $\widetilde{L}\mapsto\widetilde{u}_{\widetilde{L}}$ is smooth.
	\end{lemma}

	\begin{proof}
		Let $L\in\mathcal{V}\subset\mathcal{B}\left(\cob{k+2,\alpha}(Z),C^{k,\alpha}(Z)\right)$ be open such that, for $\widetilde{L}\in\mathcal{V},$ the restricted operator $\widetilde{L}_0:=\widetilde{L}|_{C^{k+2,\alpha}(Z;\partial Z)}$ is an isomorphism. The assignment $\widetilde{L}\mapsto\widetilde{L}_0^{-1}$ is smooth in $\mathcal{V}.$ For $\widetilde{L}\in\mathcal{V},$ the unique $\widetilde{u}$ with the desired properties is given by
		\[
		\widetilde{u}_{\widetilde{L}}=u-\widetilde{L}_0^{-1}\widetilde{L}u.
		\]
		The lemma follows.
	\end{proof}

	Next, we define interior regularity and regular convergence for families of special Lagrangian cylinders. As a preliminary, we recall the notion of ends (see~\cite{freudenthal}). Let $A$ be a topological space. The set of ends $\mathcal{E}(A)$ is given by
\[
\mathcal{E}(A) := \varprojlim_{K \subset\subset A} \pi_0(A\setminus K).
\]
Thus, an end $E \in \mathcal{E}(A)$ determines for every compact $K\subset A$ a connected component $E(K)$ of the complement $A\setminus K$, such that for two compact subsets $K\subset K'$ we have $E(K')\subset E(K).$ For our purposes the space $A$ will always be a connected component of $\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right),$ which is 1-dimensional. We thus use the following terminology, which is somewhat more elementary and equivalent in the case at hand. Let $C$ be a connected non-compact $1$-dimensional manifold. That is, $C$ is a curve diffeomorphic to the real line. A \emph{ray} in $C$ is a connected open proper subset $U\subsetneq C$ with non-compact closure. Two rays, $U,V\subset C$ are said to be equivalent if $U\subset V$ or $V\subset U.$ Finally, an \emph{end} is an equivalence class of rays. Every curve $C$ as above has exactly two ends.

	\begin{dfn}
		\label{definition: interior regularity}
		$\;$
		\begin{enumerate}
			\item\label{interior regularity first part} Let $U\subset\mathcal{SLC}\left(N;\Lambda_0,\Lambda_1\right)$ be open and connected. An \emph{interior-regular parameterization} of $U$ is a smooth immersion $\Phi:N\times[0,1]\times(a,b)\to X$ satisfying the following conditions:
			\begin{enumerate}[ref=(\alph*)]
				\item\label{item: boundary embedding} The restriction of $\Phi$ to a boundary component $\Phi|_{N\times\{i\}\times(a,b)}$ is an embedding for $i = 0,1$.
				\item\label{item: rep} For $s\in(a,b),$ the restricted immersion $\Phi_s:=\Phi|_{N\times[0,1]\times\{s\}}$ represents an element of $U$.
				\item\label{item: diffeo s} The map $\chi:(a,b)\to U,\quad s\mapsto[\Phi_s],$ is a diffeomorphism.
			\end{enumerate}
			The subset $U$ is said to be \emph{interior-regular} if it admits an interior-regular parameterization.
			\item \label{compatible with harmonics}
Let $U$ be as in~\eqref{interior regularity first part} and let $\Phi$ be an interior-regular parameterization of $U$. For $s\in(a,b),$ write $Z_s:=[\Phi_s]$ and let $\sigma_s$ denote the fundamental harmonic on $Z_s.$ The parameterization $\Phi$ is \emph{compatible with the harmonics of $U$} if the following conditions hold.
			\begin{enumerate}[ref=(\alph*)]
				\item\label{t compatible}
For $(p,t,s)\in N\times[0,1]\times(a,b)$ we have
				\[
				\sigma_s(\Phi(p,t,s))=t.
				\]
				\item\label{orthogonal} For $(p,t,s)\in N\times[0,1]\times(a,b)$, the derivative $\pderiv{t}\Phi(p,t,s)$ is orthogonal to the $t$-level set of $\sigma_s.$
			\end{enumerate}
			\item\label{regular convergence to intersection point} Let $\mathcal{Z}\subset\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right)$ be a connected component, let $E$ be an end of $\mathcal{Z},$ and let $q\in\Lambda_0\cap\Lambda_1.$ A \emph{regular parameterization} of $E$ about $q$ is a smooth map $\Phi:S^{n-1}\times[0,1]\times[0,\epsilon)\to X$ satisfying the following conditions.
			\begin{enumerate}[ref=(\alph*)]
				\item\label{critical point} For $(p,t)\in S^{n-1}\times[0,1]$ we have $\Phi(p,t,0)=q.$
				\item\label{item:interior regular} The restricted map $\Phi|_{S^{n-1}\times[0,1]\times(0,\epsilon)}$ is an interior-regular parameterization of $U,$ for some ray $U\subset\mathcal{Z}$ representing $E.$
				\item \label{item:derivative Phi immersion} The derivative
				\[
				\left.\pderiv{s}\right|_{s=0}\Phi(\cdot,\cdot,s):S^{n-1}\times[0,1]\to T_qX
				\]
				is an immersion and the restriction $\left.\left.\pderiv{s}\right|_{s=0}\Phi(\cdot,\cdot,s)\right|_{S^{n-1}\times\{i\}}$ is an embedding for $i = 0,1.$
				\item \label{item:nowhere tangent Euler} The Euler vector field on $T_qX$ is nowhere tangent to the immersion $\left.\pderiv{s}\right|_{s=0}\Phi(\cdot,\cdot,s).$
			\end{enumerate}
In this case, we also say that $\Phi$ is a regular parameterization of $U$ about~$q.$ A regular parameterization $\Phi$ as above is said to be compatible with the harmonics of $E$ if the interior-regular restricted map $\Phi|_{S^{n-1}\times[0,1]\times(0,\epsilon)}$ is. We say the end $E$ \emph{converges regularly} to the intersection point $q$ if it admits a regular parameterization about $q.$ We may use a half-open interval with arbitrary endpoints, open either from below or above, in place of the half-open interval $[0,\epsilon).$
		\end{enumerate}
	\end{dfn}
The following lemmas make it easier to check regularity.
\begin{lemma}\label{lemma: immersion nowhere tangent to Euler}
Let $N$ be a compact manifold possibly with boundary. Let $G : N \times [0,\epsilon) \to \R^m$ be a smooth map satisfying the following.
\begin{enumerate}
  \item
The restriction
$G|_{N \times \{0\}}$
is an immersion.
  \item
The Euler vector field on $\R^m$ is nowhere tangent to the immersion $G|_{N \times \{0\}}.$
\end{enumerate}
Let $F : N \times [0,\epsilon) \to \R^m$ be given by
\[
F(p,s) = sG(p,s).
\]
Then, possibly after shrinking $\epsilon,$ the restriction $F|_{N \times (0,\epsilon)}$ is an immersion. Moreover, if $N = S^{m-1},$ and $G|_{N \times \{0\}}$ is an embedding, then $F|_{N \times (0,\epsilon)}$ is an embedding.

Furthermore, there exists a $C^1$ neighborhood $G \in U$ in the space of smooth maps $C^\infty(N \times [0,\epsilon),\R^m)$ such that the conclusions of the lemma continue to hold with a uniform choice of $\epsilon$ if we replace $G$ with any map in $U.$
\end{lemma}
\begin{proof}
Possibly after shrinking $\epsilon,$ the restriction $G|_{N \times \{s\}}$ is an immersion to which the Euler vector field is nowhere tangent for $s \in [0,\epsilon).$ Letting $\varepsilon$ denote the Euler vector field, we have
\[
\pderiv[F]{s}(p,s) = \varepsilon(G(p,s)) + s \pderiv[G]{s}(p,s).
\]
So, possibly shrinking $\epsilon$ again, we may assume that
\[
\pderiv[F]{s}(p,s) \notin \mathrm{Im}\left(d\left(G|_{N \times \{s\}}\right)_{(p,s)}\right), \qquad (p,s) \in N \times [0,\epsilon).
\]
On the other hand, for $v \in T_pN$ and $w \in T_s [0,\epsilon)\simeq \R,$ we have
\[
dF_{(p,s)}(v,w) =  sd(G|_{N \times \{s\}})_{(p,s)}(v) + w\pderiv[F]{s}(p,s).
\]
It follows that $dF_{(p,s)}$ is injective for $(p,s) \in N \times (0,\epsilon),$ so $F|_{N \times (0,\epsilon)}$ is an immersion.

We turn to the case that $N = S^{m-1}$ and $G|_{N \times \{0\}}$ is an embedding. Let $\pi : \R^m \to \orientedprojective(\R^m)$ denote the quotient map. We first show that $\pi \circ G|_{N \times \{s\}}$ is a diffeomorphism for $s \in [0,\epsilon).$ Indeed, since the Euler vector field is nowhere tangent to $G|_{N \times \{s\}},$ it follows that $\pi \circ G|_{N \times \{s\}}$ is a local diffeomorphism. The compactness of $N$ implies that $\pi \circ G|_{N \times \{s\}}$ is a covering map. When $m \geq 3,$ since $S^{m-1}$ is simply connected, $\pi \circ G|_{N \times \{s\}}$ must be a diffeomorphism. On the other hand, when $m = 2,$ since $G|_{N \times \{0\}}$ is an embedding nowhere tangent to the Euler vector field, the winding number about zero is $\pm 1.$ The winding number is preserved under isotopy. So, again, $\pi \circ G|_{N \times \{s\}}$ must be a diffeomorphism.

Next, we show that $F|_{N \times (0,\epsilon)}$ is injective. Define a diffeomorphism $\varphi : N \times [0,\epsilon) \to N \times [0,\epsilon)$ by
\[
\varphi(p,s) = (\pi \circ G|_{N \times \{s\}})^{-1}(\pi \circ G|_{N \times \{0\}}(p,0)).
\]
It suffices to show $F \circ \varphi$ is injective, so we may replace $F$ with $F \circ \varphi$ and $G$ with $G \circ \varphi,$ and thus we may assume $\pi \circ F(p,s)$ is independent of $s.$
Since $\pi \circ G|_{N \times \{s\}}$ is injective, so is $G|_{N \times \{s\}}$ for $s \in [0,\epsilon).$
Thus, we need only show that for $p \in N$ and $s < s' \in [0,\epsilon)$ we have $|F(p,s)| < |F(p,s')|.$ But
\[
\left.\pderiv[|F(p,s)|]{s}\right|_{s = 0^+} = |G(p,0)|.
\]
By compactness of $N,$ after possibly shrinking $\epsilon,$ we have $\pderiv[|F(p,s)|]{s} > 0$ for $(p,s) \in N \times [0,\epsilon).$ So, $F|_{N \times (0,\epsilon)}$ is injective.

To complete the proof, we show that $F|_{N \times (0,\epsilon)}$ is a homeomorphism onto its image. Let $B_\epsilon(0)$ denote the ball of radius $\epsilon$ in $\R^m$ and identify $S^{m-1}$ with the unit sphere in $\R^m.$ Define $\widetilde F : B_{\epsilon}(0) \to \R^m$ by $\widetilde F(sp) = F(p,s).$ Then, $\widetilde F$ is continuous and injective, so for $0 < \epsilon' < \epsilon,$ the restriction of $\widetilde F$ to the compact space $\overline{B_{\epsilon'}(0)}$ is a homeomorphism onto its image. So, we conclude by replacing $\epsilon$ with $\epsilon'.$

The choice of $\epsilon$ throughout the proof depends only on $G$ and its first derivatives, so the conclusions of the lemma continue to hold with uniform $\epsilon$ for any map that is $C^1$ close to~$G.$
\end{proof}

\begin{lemma}\label{lemma:easy regularity}
Let $M$ and $N$ be manifolds and let $q \in M.$ Let $F : N \times [0,\epsilon) \to M$ be a smooth map satisfying the following.
\begin{enumerate}
  \item
$F(p,0) = q$ for all $p \in N.$
  \item
The derivative
\[
\left.\pderiv[F]{s}\right|_{s = 0} : N \to T_qM
\]
is an immersion.
  \item
The Euler vector field on $T_qX$ is nowhere tangent to the immersion $\left.\pderiv[F]{s}\right|_{s = 0}.$
\end{enumerate}
Then, possibly after shrinking $\epsilon,$ the restriction $F|_{N \times (0,\epsilon)}$ is an immersion. Moreover, writing $m = \dim M,$ if $N = S^{m-1},$ and $\left.\pderiv[F]{s}\right|_{s = 0}$ is an embedding, then $F|_{N \times (0,\epsilon)}$ is an embedding.
\end{lemma}
\begin{proof}
This follows by choosing coordinates on $M$ about $q$ and applying Lemmas~\ref{lemma: milnor lemma} and~\ref{lemma: immersion nowhere tangent to Euler}.
\end{proof}
The following is an immediate consequence of Lemma~\ref{lemma:easy regularity}.
\begin{cor}\label{rem: easy regularity}
If $\Phi:S^{n-1}\times[0,1]\times[0,\epsilon)\to X$ satisfies conditions~\eqref{regular convergence to intersection point}\ref{critical point}, \eqref{regular convergence to intersection point}\ref{item:derivative Phi immersion}, \eqref{regular convergence to intersection point}\ref{item:nowhere tangent Euler} and~\eqref{interior regularity first part}\ref{item: rep} of Definition~\ref{definition: interior regularity}, then possibly after diminishing $\epsilon,$ it also satisfies condition~\eqref{regular convergence to intersection point}\ref{item:interior regular} and thus it is a regular parameterization.
\end{cor}

	\begin{rem}
		\label{remark: every cylinder in interior-regular family has regular harmonics}
		It follows from Lemma~\ref{lemma:immersive cylinder has regular harmonics} that if a connected open subset $U\subset\mathcal{SLC}\left(N;\Lambda_0,\Lambda_1\right)$ is interior-regular, every element of $U$ has regular harmonics.
	\end{rem}

	\begin{rem}
		\label{remark: regular parameterization gives rise to family of cone-immersions}
		Let $\mathcal{Z},E$ and $q,$ be as in Definition~\ref{definition: interior regularity}~\eqref{regular convergence to intersection point}, and suppose $\Phi$ is a regular parameterization of $E$ about $q.$ Recalling Definition~\ref{definition:polar coordinates}, let
\[
\kappa: S^{n-1}\times[0,\epsilon) \to \R^n
\]
be a polar coordinate map centered at zero with image $U.$ For a fixed $t\in[0,1],$ denote by $\Psi_t: U \to X$ the unique map such that
\[
\Phi|_{S^{n-1}\times\{t\}\times[0,\epsilon)} = \Psi_t \circ \kappa.
\]
Then, Lemma~\ref{lemma:immersion criterion} together with properties~\ref{item:derivative Phi immersion} and~\ref{item:nowhere tangent Euler} of a regular parameterization shows that $\Psi_t$ is cone immersive at zero. It follows from property~\ref{item:interior regular} that $(\Psi_t,0)$ is a cone immersion.
	\end{rem}

	\begin{lemma}
		\label{lemma: interior and end regular parameterization compatible with harmonics}
		$\:$
		\begin{enumerate}[label=(\alph*)]
			\item \label{interior regular parameterization compatible with harmonics}
			Let $U\subset\mathcal{SLC}\left(N;\Lambda_0,\Lambda_1\right)$ be interior-regular and let \[
\Phi: N \times [0,1] \times (0,1) \to X
\]
be an interior regular parameterization. Then $U$ admits a unique interior-regular parameterization $\widehat \Phi : N \times [0,1]\times (0,1) \to X$ compatible with its harmonics such that $\widehat \Phi|_{N \times \{0\} \times (0,1)} = \Phi|_{N \times \{0\} \times (0,1)}.$
			\item\label{end regular parameterization compatible with harmonics} Let $\mathcal{Z}\subset\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right)$ be a connected component and let $E$ be an end of $\mathcal{Z}.$ Let $q\in\Lambda_0\cap\Lambda_1$ be a transverse intersection point, suppose $E$ converges regularly to $q$ and suppose $\Phi : S^{n-1} \times [0,1] \times [0,\epsilon) \to X$ is a regular parameterization of $E$ about $q.$ Then $E$ admits a unique regular parameterization $\widehat \Phi : S^{n-1} \times [0,1] \times [0,\epsilon) \to X$ about $q$ which is compatible with the harmonics of $E$ such that $\widehat \Phi|_{N \times \{0\} \times [0,\epsilon)} = \Phi|_{N \times \{0\} \times [0,\epsilon)}.$
  		\end{enumerate}
	\end{lemma}

	\begin{proof}
		We prove~\ref{end regular parameterization compatible with harmonics}. The proof of~\ref{interior regular parameterization compatible with harmonics} is similar and less involved.

		We identify a neighborhood of $q$ in $X$ with a ball $V\subset\mathbb{C}^n$ via Darboux coordinates carrying $q$ to the origin and $\Lambda_i,\;i=0,1,$ to linear subspaces. For $s \in [0,1),$ write
\[
V_s:= \{z \in \C^n| s z \in V\}.
\]
For $s\in(0,1)$ let $M_s:V_s\to V$ denote rescaling by $s.$ Define a family of complex structures and $n$-forms on $V_s$ by
		\begin{equation}
		\label{equation:rescaled J and Omega}
		J_s:=M_s^*J,\quad\Omega_s:=s^{-n}M_s^*\Omega,\quad s\in(0,1),
		\end{equation}
		where $J$ and $\Omega$ are the complex structure and Calabi-Yau form of $X,$ respectively. Then as $s\searrow0,$ $J_s$ and $\Omega_s$ converge in the $C^\infty$-topology on compact subsets to a constant complex structure and a constant $n$-form on $V_0 = \C^n$ denoted by $J_0$ and $\Omega_0$ respectively. For $s\in[0,1),$ the quadruple $(V_s,\omega,J_s,\Omega_s)$ is a Calabi-Yau manifold. We let $g_s$ denote the associated K\"ahler metric. We define $\rho_s$ as in~\eqref{equation: rho}.

		Let $\Phi:S^{n-1}\times[0,1]\times[0,1)\to V$ be a regular parameterization of $E$ about the origin $q.$ By Lemma~\ref{lemma: milnor lemma}, we have
		\[
		\Phi(p,t,s)=s\cdot\chi(p,t,s),\quad(p,t,s)\in S^{n-1}\times[0,1]\times[0,1),
		\]
		where $\chi:S^{n-1}\times[0,1]\times[0,1)\to \C^n$ is smooth with
		\[
		\chi(p,t,0)=\pderiv[\Phi]{s}(p,t,0).
		\]
		For $s\in[0,1),$ write $\chi_s:=\chi|_{S^{n-1}\times[0,1]\times\{s\}}.$ Then $\chi_s$ is an immersion representing an $\Omega_s$-imaginary special Lagrangian cylinder with boundary components in $\Lambda_i,\;i=0,1.$ Define a smooth family of elliptic linear differential operators
		\[
		\Delta_s:C^\infty\left(S^{n-1}\times[0,1]\right)\to C^\infty\left(S^{n-1}\times[0,1]\right),\quad u\mapsto*_sd((\rho_s\circ\chi_s)*_sdu),
		\]
		where $*_s$ denotes the Hodge star operator of the metric $\chi_s^*g_s.$ Let $\widetilde{\sigma}_s$ denote the fundamental harmonic on $S^{n-1}\times[0,1]$ with respect to $\Delta_s.$ Namely, $\widetilde{\sigma}_s:S^{n-1}\times[0,1]\to\mathbb{R}$ is the unique function satisfying
		\[
		\Delta_s(\widetilde{\sigma}_s)=0,\quad\widetilde{\sigma}_s|_{S^{n-1}\times\{i\}}=i,\quad i=0,1.
		\]
		Note that all the cylinders $[\chi_s]$ have regular harmonics. Indeed, for $s\in(0,1),$ the cylinder $[\chi_s]$ is merely a rescaling of $[\Phi_s],$ which has regular harmonics by Remark~\ref{remark: every cylinder in interior-regular family has regular harmonics}, whereas $[\chi_0]$ has regular harmonics by Lemma~\ref{lemma: cylinder with regular harmonics in Euclidean space} and property~\ref{item:nowhere tangent Euler} of Definition~\ref{definition: interior regularity}~(\ref{regular convergence to intersection point}). Hence, all the functions $\widetilde{\sigma}_s$ have no critical points. By Lemma~\ref{lemma: regular harmonics is open property}, the function $\widetilde{\sigma}_s$ depends smoothly on $s.$ Define a smooth family of vector fields on $S^{n-1}\times[0,1]$ by
		\[
		Y_s:=\frac{\nabla\widetilde{\sigma}_s}{|\nabla\widetilde{\sigma}_s|^2},\quad s\in[0,1),
		\]
		where the gradient and modulus are taken with respect to $\chi_s^*g_s.$ Then $Y_s$ is $\chi_s^*g_s$-orthogonal to the level sets of $\widetilde{\sigma}_s$ and satisfies $Y_s(\widetilde{\sigma}_s)\equiv1.$ Let
\[
\varphi_s : W_s \subset \R \times \left(S^{n-1}\times [0,1]\right) \to S^{n-1}\times [0,1]
\]
denote the flow of $Y_s.$ Note that for $(p,t,s)\in S^{n-1}\times[0,1]\times[0,1)$ we have
		\[
(t,(p,0)) \in W_s, \qquad
		\widetilde{\sigma}_s(\varphi_s(t,(p,0)))=t.
		\]
Finally, we set
		\[
		\widehat{\chi}:S^{n-1}\times[0,1]\times[0,1)\to V,\quad(p,t,s)\mapsto\chi_s(\varphi_s(t,(p,0)))
		\]
		and
		\[
		\widehat{\Phi}:S^{n-1}\times[0,1]\times[0,1)\to V,\quad(p,t,s)\mapsto s\cdot\widehat{\chi}(p,t,s).
		\]
		By construction, $\widehat{\Phi}$ is a regular parameterization of $E$ about $q$ compatible with the harmonics and $\widehat \Phi|_{N \times \{0\} \times [0,\epsilon)} = \Phi|_{N \times \{0\} \times [0,\epsilon)}.$

Suppose $\widehat \Phi'$ is another such regular parameterization. Since
\begin{equation}\label{equation:restriction to t=0}
\widehat \Phi|_{N \times \{0\} \times [0,\epsilon)} = \Phi|_{N \times \{0\} \times [0,\epsilon)} = \widehat \Phi'|_{N \times \{0\} \times [0,\epsilon)},
\end{equation}
it follows from the definition of regular parameterization that $\left[\widehat \Phi'_s\right] = \left[\widehat \Phi_s\right]$ for $s \in (0,\epsilon).$ So, there exists a diffeomorphism $\psi_s : N \times [0,1] \to N \times [0,1]$ such that $\widehat \Phi'_s = \widehat \Phi_s \circ \psi_s.$ By~\eqref{equation:restriction to t=0}, we have also $\psi_s|_{N\times\{0\}} = \id_N.$ By the chain rule,
\[
d\widehat\Phi_s\left(\pderiv[\psi_s]{t}\right) = \pderiv[\widehat\Phi_s']{t}.
\]
On the other hand, by compatibility with harmonics,
\[
\pderiv[\widehat\Phi_s]{t}\circ \psi_s = \pderiv[\widehat\Phi_s']{t}.
\]
Since $d\widehat\Phi_s$ is injective, it follows that
\[
\pderiv[\psi_s]{t} = \pderiv{t}\circ \psi_s.
\]
Consequently, $\psi_s = \id_{N \times [0,1]}.$
\end{proof}
Let $\diff(N \times (0,1) \to (0,1))\subset \diff(N \times (0,1))$ denote the subgroup consisting of diffeomorphisms that carry fibers of the projection $N \times (0,1) \to (0,1)$ to other such fibers.	For $\Phi : N \times [0,1] \times (0,1) \to X$ an interior regular parameterization of $U$ and $\phi \in \diff(N \times (0,1) \to (0,1)),$ write
\[
\Phi_\phi = \Phi \circ \left(\phi \times \id_{[0,1]}\right): N \times [0,1]\times (0,1) \to X.
\]
\begin{cor}\label{corollary:action}
Let $U\subset\mathcal{SLC}\left(N;\Lambda_0,\Lambda_1\right)$ be interior-regular. Then, the map $(\Phi,\phi) \mapsto \Phi_\phi$ gives a free transitive action of the group $\diff(N \times (0,1) \to (0,1))$ on the set of interior regular parameterizations of $U$ compatible with the harmonics.
\end{cor}
\begin{proof}
It follows from the definitions that the action preserves regular parameterizations compatible with the harmonics. Freeness is immediate. Transitivity follows from the uniqueness claim of Lemma~\ref{lemma: interior and end regular parameterization compatible with harmonics}~\ref{interior regular parameterization compatible with harmonics}.
\end{proof}

	\section{The relation between cylinders and geodesics}
	\label{section: the relation between cylinders and geodesics}

	In this section we establish the relation between geodesics of positive Lagrangians and families of imaginary special Lagrangian cylinders. We fix an ambient Calabi-Yau manifold $(X,\omega,J,\Omega),$ a smooth manifold $L,$ a finite subset $S \subset L$ and a finite subset $C_0 \subset X.$ In the following, all geodesics are geodesics of cone-immersed Lagrangians of type $(L,S)$ with cone locus image $C_0$ unless otherwise mentioned.

	\begin{lemma}
		\label{geodesic yields unique cylinder}
		Let $(\Lambda_t)_{t\in[0,1]}$ be a geodesic of positive Lagrangians of type $(L,S)$, and let $\Psi_t:L\to\Lambda_t,\;t\in[0,1],$ be a horizontal lifting of $(\Lambda_t)$. Let $(h_t)_t$ be the Hamiltonian of $(\Lambda_t)_t$ and let $h = h_t\circ \Psi_t:L\to\mathbb{R}$ be the Hamiltonian of $(\Lambda_t)_t$ with respect to the lifting $(\Psi_t)_t$.
		For $c$ in the image of $h,$ define
		\[
\Phi_c:\left(h^{-1}(c)\setminus\crit(h)\right)\times[0,1]\to X,\qquad (p,t)\mapsto\Psi_t(p).
		\]
		Then $\Phi_c$ is an imaginary special Lagrangian immersion.
	\end{lemma}

	\begin{proof}
		Fix $t_0\in[0,1]$ and $p\in h^{-1}(c)\setminus\crit(h)$. Let $u_1,\ldots,u_{n-1}\in T_ph^{-1}(c)$ be a basis and write
		\[
		w_i:=d(\Psi_{t_0})_p(u_i)\in T_{\Psi_{t_0}(p)}\Lambda_{t_0},\qquad i=1,\ldots,n-1.
		\]
		The tangent vectors $w_1,\ldots,w_{n-1}$ are linearly independent as $\Psi_{t_0}$ is a smooth immersion away from critical points of $h$. Let $v\in T_{\Psi_{t_0}(p)}X$ be the $t$-derivative of $\Phi_c$. That is,
		\[
		v:=\pderiv[\Phi_c]{t}(p,t_0)=\deriv{t}\Psi_t(p).
		\]
		Since $p$ is a regular point of $h$, we have $v\not\in T_{\Psi_{t_0}(p)}\Lambda_{t_0}$. It follows that the tangent vectors $v,w_1,\ldots,w_{n-1},$ are linearly independent and $\Phi_c$ is indeed an immersion.

		As the immersion $\Psi_{t_0}$ is Lagrangian, we have
		\[
		\omega(w_i,w_j)=0,\qquad i,j=1,\ldots,n-1.
		\]
		By definition of the derivative of a Lagrangian path (see Remark~\ref{remark: tangent space hands on}), we have
		\[
		\omega(v,w_i)=dh_p(u_i)=0,\qquad i=1,\ldots,n-1.
		\]
		The immersion $\Phi_c$ is thus Lagrangian. Finally, by horizontality of the lifting $(\Psi_t)$, we have
		\[
		\real\Omega(v,w_1,\ldots,w_{n-1})=0,
		\]
		and $\Phi_c$ is indeed an imaginary special Lagrangian immersion.
	\end{proof}

	\begin{dfn}
		\label{definition: associated cylinders}
		In the setting of Lemma~\ref{geodesic yields unique cylinder}, for $c$ in the image of the Hamiltonian $h,$ we call the immersed special Lagrangian cylinder $[\Phi_c]$ the \emph{associated cylinder of $c$ level sets.} Note that $[\Phi_c]$ is independent of the horizontal lifting $(\Psi_t)_t.$ It depends only on the choice of additive constant in the intrinsic Hamiltonian $(h_t)_t.$
	\end{dfn}

	\begin{lemma}
		\label{lemma: time function}
		Let $(\Lambda_t)_t,(\Psi_t)_t,h$ and $(\Phi_c)_c,$ be as in Lemma~\ref{geodesic yields unique cylinder}. For $c$ in the image of $h,$ let $Z_c$ denote the associated cylinder $[\Phi_c]$ and define a function
		\[
		\sigma_c:Z_c\to[0,1],
\]
by
\[
\sigma_c(\Phi_c(q,t))=t.
		\]
		\begin{enumerate}[label=(\alph*)]
			\item\label{time function is the derivative} Let $N$ be an $(n-1)$-dimensional smooth manifold, let $c_0<c_1\in\mathbb{R}$ and let $\beta:N\times(c_0,c_1)\to L$ be a smooth embedding with
			\begin{equation}
			\label{local level slicing}
			h(\beta(q,c))=c,\quad(q,c)\in N\times(c_0,c_1).
			\end{equation}
			For $c\in(c_0,c_1)$ define a smooth immersion
			\[
			\phi_c:N\times[0,1]\to X,\quad(q,t)\mapsto\Phi_c(\beta(q,c),t)=\Psi_t(\beta(q,c)),
			\]
and let
\[
v = \deriv{c}\phi_c.
\]
Then,
\[
i_v \omega = -d(\sigma_c\circ \phi_c).
\]
\item\label{time function is harmonic} For $c$ in the image of $h$ we have $\Delta_\rho(\sigma_c)=0.$
		\end{enumerate}
	\end{lemma}

	\begin{proof}
		Let $N$ and $\beta$ be as in~\ref{time function is the derivative}.
		Let $t$ denote the $[0,1]$-coordinate on $N\times[0,1].$ By Remark~\ref{remark: tangent space hands on} and the definition of a geodesic, we have
		\begin{align}
		\label{same time derivative}
i_v\omega\left(\pderiv{t}\right) &=-\omega\left(\deriv{t}\Psi_t,\deriv{c}(\Psi_t\circ\beta)\right)\\
		&=-\deriv{c}(h\circ\beta)\nonumber\\
        &=-1 \nonumber \\
		&=-\pderiv[(\sigma_c\circ\phi_c)]{t}.\nonumber
		\end{align}
Let $w$ be a vector field on $N \times [0,1]$ tangent to $N.$ Then, since $\Psi_t$ is a Lagrangian immersion,
\begin{align}\label{same tangent derivative}
i_v\omega\left(w\right) &=-\omega\left(d\Psi_t(w),\deriv{c}(\Psi_t\circ\beta)\right)\\
		&=0\nonumber\\
		&= -w(\sigma_c\circ\phi_c).\nonumber
\end{align}
		Part~\ref{time function is the derivative} now follows from~\eqref{same time derivative} and~\eqref{same tangent derivative}.

		Let $p\in L$ be a regular point of $h$ and let $c_0<c_1$ with $h(p)\in(c_0,c_1).$ Let $B$ denote the standard open ball of dimension $n-1.$ Let $p\in W\subset L$ be a ball containing only regular points of $h,$ and let $\beta:B\times(c_0,c_1)\to W$ be a diffeomorphism with
		\[
		h\circ\beta(q,c)=c,\quad(q,c)\in B\times(c_0,c_1).
		\]
		For $c\in(c_0,c_1),$ define $\phi_c$ as in~\ref{time function is the derivative}. Part~\ref{time function is harmonic} now follows from~\ref{time function is the derivative}, Lemma~\ref{lemma: linearized operator}~\ref{more general lemma} and Lemma~\ref{geodesic yields unique cylinder}.
	\end{proof}

	\begin{rem}
		\label{remark: time function is the fundamental harmonic}
		Let $(\Lambda_t)_{t\in[0,1]},\,(h_t)_t,\,(Z_c)_c,$ and $(\sigma_c)_c,$ be as in Lemma~\ref{lemma: time function}. Let $c$ be such that $Z_c$ is compact. In particular, $c$ must be a regular value of $(h_t)_t.$  It follows from Lemma~\ref{lemma: time function}~\ref{time function is harmonic} that $\sigma_c$ is the fundamental harmonic on $Z_c.$ Also, by its definition, the function $\sigma_c$ has no critical points. The cylinder $Z_c$ thus has regular harmonics.
	\end{rem}

	\begin{proof}[Proof of Theorem~\ref{theorem: family}]
		Follows from Lemmas~\ref{geodesic yields unique cylinder} and~\ref{lemma: time function}.
	\end{proof}

	Let $(\Lambda_t)_t$ be a geodesic of positive Lagrangians with Hamiltonian $(h_t:\Lambda_t\to\mathbb{R})_t.$ For $c$ in the image of $h_t,$ let $Z_c$ denote the associated cylinder of $c$ level sets. Recall from Definition~\ref{definition: cylindrical transform} that the family of cylinders $(Z_c)_c$ is called the \emph{cylindrical transform} of the geodesic $(\Lambda_t)_t$. The subset $\{Z_c\;|\;c\;\mathrm{is\;a\;regular\;value\;of}\; h\}$ is the \emph{non-singular cylindrical transform}. We show that every component in the non-singular cylindrical transform is interior-regular with relative Lagrangian flux given by the Hamiltonian of the geodesic.

	\begin{lemma}
		\label{lemma: non-singular cylindrical transform}
		Let $(\Lambda_t)_t$ be a geodesic with Hamiltonian $(h_t)_t$ and assume the functions $h_t$ are proper. For $c$ in the image of $h_t,$ let $Z_c$ denote the associated cylinder of $c$ level sets. Since the functions $h_t$ are proper, the cylinder $Z_c$ is compact when $c$ is a regular value of $(h_t)_t.$ Let $c_0<c_1\in\mathbb{R}$ be such that the interval $(c_0,c_1)$ consists of regular values of $(h_t)_t.$
		\begin{enumerate}[label=(\alph*)]
			\item\label{cylindrical transform is interior-regular} The family of cylinders $U:=\{Z_c\;|\;c\in(c_0,c_1)\}$ is interior-regular.
			\item\label{flux equals Hamiltonian} For $b_0<b_1\in(c_0,c_1)$ we have
			\[
			\mathrm{RelFlux}\left((Z_c)_{c\in[b_0,b_1]}\right)=b_1-b_0.
			\]
		\end{enumerate}
	\end{lemma}

	\begin{proof}
		Let $(\Psi_t:L\to X)_t$ be a horizontal lifting of $(\Lambda_t)_t,$ and let $h:L\to\mathbb{R}$ be the corresponding Hamiltonian,
		\[
		h_t=[(\Psi_t,h)].
		\]
		By assumption there exist an $(n-1)$-dimensional smooth manifold $N$ and a diffeomorphism
		\[
		\beta:N\times(c_0,c_1)\to h^{-1}(c_0,c_1)
		\]
		with
		\[
		h\circ\beta(q,c)=c,\quad (q,c)\in N\times(c_0,c_1).
		\]
		Define
		\[
		\Phi:N\times[0,1]\times(c_0,c_1)\to X,\quad(q,t,c)\mapsto\Psi_t(\beta(q,c)).
		\]
		Then $\Phi$ is an interior-regular parameterization of $U$ verifying~\ref{cylindrical transform is interior-regular}. Part~\ref{flux equals Hamiltonian} follows from Lemma~\ref{lemma: time function}~\ref{time function is the derivative} and Definition~\ref{definition:relative Lagrangian flux}.
	\end{proof}

	Proposition~\ref{proposition: interior-regular family gives rise to geodesic} below is a converse to Lemma~\ref{lemma: non-singular cylindrical transform}~\ref{cylindrical transform is interior-regular} showing that an interior-regular family of special Lagrangian cylinders gives rise to a geodesic of positive Lagrangians. We use the following notion.

	\begin{dfn}
		\label{definition: swept submanifold}
		Let $\Lambda_0,\Lambda_1\subset X$ be smooth Lagrangians and let $N$ be a closed connected smooth manifold of dimension $n-1.$ Let $U\subset\mathcal{SLC}(N;\Lambda_0,\Lambda_1)$ be open, connected and interior-regular. Let $\Phi:N\times[0,1]\times(0,1)\to X$ be an interior regular parameterization of $U.$ For $i=0,1,$ the submanifold of $\Lambda_i$ \emph{swept by $U$} is the image of the embedding $\Phi|_{N\times\{i\}\times(0,1)}.$ This is independent of $\Phi.$ Similarly, suppose $U\subset\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right)$ is a ray and $\Phi : S^{n-1}\times [0,1]\times [0,\epsilon) \to X$ is a regular parameterization of $U$ about an intersection point $q \in \Lambda_0 \cap \Lambda_1.$ For $i = 0,1,$ the \emph{unpunctured} submanifold of $\Lambda_i$ swept by $U$ is the image of the restricted map~$\Phi|_{S^{n-1}\times\{i\}\times[0,\epsilon)}.$
	\end{dfn}

	\begin{prop}
		\label{proposition: interior-regular family gives rise to geodesic}
		Let $\Lambda_0,\Lambda_1\subset X$ be smoothly embedded positive Lagrangians, let $N$ be a closed connected smooth manifold of dimension $n-1$ and let
\[
U\subset\mathcal{SLC}(N;\Lambda_0,\Lambda_1)
\]
be open, connected and interior regular. For $i=0,1,$ let $\Lambda_i^U$ denote the submanifold of $\Lambda_i$ swept by $U.$ Then, there exists a geodesic of positive Lagrangians between $\Lambda_0^U$ and $\Lambda_1^U$ with cylindrical transform $U.$ This geodesic is unique up to reparameterization and has empty critical locus.
	\end{prop}

	\begin{proof}
Let $\Phi:N\times[0,1]\times(0,1)\to X$ be an interior-regular parameterization of~$U.$ Since the points of $U$ are imaginary special Lagrangians, $N$ must be orientable. Orient $N$ so that the open embedding
\[
\Phi|_{N \times \{0\}\times(0,1)} : N \times \{0\} \times (0,1) \to \Lambda_0
\]
is orientation preserving. By Lemma~\ref{lemma: interior and end regular parameterization compatible with harmonics}, we may assume that $\Phi$ is compatible with the harmonics of $U.$ For $s\in(0,1)$ write $\Phi_s:=\Phi|_{N\times[0,1]\times\{s\}}$ and $Z_s:=[\Phi_s].$ Let $\sigma_s$ denote the fundamental harmonic on $Z_s.$ Let $\xi$ denote the $t$-derivative of $\Phi,$
		\[
		\xi=\xi(p,t,s)=\pderiv{t}\Phi(p,t,s).
		\]
		For $t\in[0,1]$ write
		\[
		\Psi_t:=\Phi|_{N\times\{t\}\times(0,1)}.
		\]
		Then $\Psi_t$ is a smooth immersion by the definition of an interior regular parameterization. Let $\Lambda_t^U:=[\Psi_t]$ with orientation given by the orientation on $N \times (0,1).$ We show in three steps that $\left(\Lambda_t^U\right)_{t\in[0,1]}$ is a geodesic of positive Lagrangians.

		\textbf{Step 1:} \emph{For $t\in[0,1],$ the immersed submanifold $\Lambda_t^U$ is positive Lagrangian.} To show this, pick $(p,t,s)\in N\times[0,1]\times(0,1).$ As $\Phi$ is compatible with the harmonics of $U,$ the immersed submanifold
		\[
		K_{t,s}:=\left[\Phi|_{N\times\{t\}\times\{s\}}\right]
		\]
		is the $t$-level set of $\sigma_{s}$ in the Lagrangian cylinder $Z_s.$ In particular, $K_{t,s}$ is $\omega$-isotropic. By Proposition~\ref{proposition: space of special lag cylinders is one dimensional}, there exists $a(s)\in\R$ such that
		\begin{equation}
		\label{specific harmonic}
		\deriv{s}Z_s=a(s)\sigma_{s}.
		\end{equation}
		Hence, by Remark~\ref{remark: tangent space hands on}, for $v\in T_{\Psi_t(p,s)}K_{t,s}$ we have
		\begin{align*}
		\omega\left(\pderiv{s}\Psi_t(p,s),v\right)
		&=a(s)d(\sigma_s)_{\Psi_t(p,s)}(v)\\
		&=0,
		\end{align*}
		showing that $\Lambda_t^U$ is Lagrangian.

It remains to establish positivity. Fix a basis $v_1,\ldots,v_{n-1}\in T_{\Psi_t(p,s)}K_{t,s}.$ Let $0\ne w\in T_{\Psi_t(p,s)}\Lambda_t^U$ be orthogonal to $K_{t,s}.$ Define $E_{p,t,s}\subset T_{\Psi_t(p,s)}X$ by
		\begin{align*}
		E_{p,t,s}:
		&=(T_{\Psi_t(p,s)}K_{t,s})^g\cap(T_{\Psi_t(p,s)}K_{t,s})^\omega\\
		&=\{u\in T_{\Psi_t(p,s)}X\;|\;g(u,v_i)=0 = \omega(u,v_i),\;i=1,\ldots,n-1\}.
		\end{align*}
		Then, $E_{p,t,s}$ is a $J$-invariant linear subspace of real dimension $2.$ It follows from Definition~\ref{definition: interior regularity}\eqref{compatible with harmonics}\ref{orthogonal} and the fact that $Z_s$ is Lagrangian that $\xi(p,t,s) \in E_{p,t,s}.$ Since $\Lambda_t^U$ is Lagrangian, it follows that $w\in E_{p,t,s}.$ Since $\Phi$ is an immersion, $\xi\ne0$, so
		\begin{equation}
		\label{complex dependence}
		w=\alpha\xi+\beta J\xi
		\end{equation}
		for some $\alpha,\beta\in\R.$ Also, as $\Phi$ is an immersion, we have $\beta\ne0.$ By Lemma~\ref{lemma: Omega does not vanish on Lags} and since $Z_s$ is imaginary special Lagrangian, we have
		\[
		0\ne\Omega(v_1,\ldots,v_{n-1},\xi)\in\ii\R.
		\]
		Finally, as $\Omega$ is of type $(n,0),$ we have
		\begin{align*}
		\real\Omega(v_1,\ldots,v_{n-1},w)
		&=\real\left(\left(\alpha+\ii\beta\right)\Omega(v_1,\ldots,v_{n-1},\xi)\right)\\
		&=\ii\beta\Omega(v_1,\ldots,v_{n-1},\xi)\\
		&\ne0.
		\end{align*}
Since $\Lambda_0^U$ is positive by assumption, the positivity of $\Lambda_t^U$ follows by continuity.

		\textbf{Step 2:} \emph{The family of immersions $(\Psi_t)_{t\in[0,1]}$ is horizontal.} To prove this, we need to verify
		\begin{equation}
		\label{horizontal 1}
		\real\Omega(\xi,v_1,\ldots,v_{n-1})=0
		\end{equation}
		and
		\begin{equation}
		\label{horizontal 2}
		\real\Omega\left(\xi,v_1,\ldots\widehat{v}_i\ldots,v_{n-1},w\right)=0,\quad i=1,\ldots,n-1,
		\end{equation}
		where $v_1,\ldots,v_{n-1}$ and $w$ are as above. Equality~\eqref{horizontal 1} holds true as all the cylinders in $U$ are imaginary special Lagrangian. Equality~\eqref{horizontal 2} holds true by virtue of equality~\eqref{complex dependence}, as $\Omega$ is of type $(n,0).$ This completes the proof of Step 2.

		\textbf{Step 3:} \emph{Pick $s_0\in(0,1)$. Define a family of functions $h_t^U \in C^\infty(\Lambda_t)$ by
			\[
			h_t^U\circ\Psi_t(p,s_1)=\mathrm{RelFlux}\left((Z_s)_{s\in[s_0,s_1]}\right),\quad s_1\in(0,1).
			\]
			In particular, $h_t^U\circ\Psi_t(p,s_1)$ is independent of $t$ and the point $p\in N.$ Then,
\[
\deriv{t}\Lambda_t^U = dh_t^U.
\]
}

To show this, recall the function $a:(0,1)\to\R$ defined in equality~\eqref{specific harmonic}. By Definition~\ref{definition:relative Lagrangian flux} we have
		\begin{equation*}
		\label{derivative of RelFlux}
		\deriv{s_1}\mathrm{RelFlux}\left((Z_s)_{s\in[s_0,s_1]}\right)=-a(s_1).
		\end{equation*}
So, recalling from Definition~\ref{definition: interior regularity}\eqref{compatible with harmonics}\ref{t compatible} the meaning of $\Phi$ being compatible with the harmonics of $U,$
		\begin{align*}
		\pderiv{s}h_t^U\circ\Psi_t(p,s)
		&=-a(s)		\\
		&=-a(s)d(\sigma_s)_{\Psi_t(p,s)}\left(\xi(p,t,s)\right)\\
&=-\omega\left(\pderiv{s}\Psi_t(p,s),\xi(p,t,s)\right)\\
&=\omega\left(\xi(p,t,s),\pderiv{s}\Psi_t(p,s)\right),
		\end{align*}
On the other hand, for $v$ a vector field on $N \times [0,1]$ tangent to $N,$ since $h_t^U\circ\Psi_t(p,s)$ is independent of $p,$ we have
\begin{equation*}
v\left(h_t^U\circ\Psi_t\right) = 0 = \omega(\xi, d\Psi_t(v)).
\end{equation*}
By Remark~\ref{remark: tangent space hands on}, it follows that $\deriv{t}\Lambda^U_t = dh_t^U.$ This completes the proof of Step 3.

Since $h_t^U\circ \Psi_t$ is independent of $t,$ it follows that $\left(\Lambda_t^U\right)_{t\in[0,1]}$ is a geodesic. By construction, the cylindrical transform of this geodesic is $U.$ Finally, since by Definition~\ref{definition: interior regularity}~\eqref{interior regularity first part} the parameterization $\Phi$ is an immersion, the time derivative $\deriv{t}\Psi_t=\pderiv{t}\Phi = \xi$ is nowhere vanishing, implying that $\left(\Lambda_t^U\right)_t$ has empty critical locus.

It remains to prove uniqueness. Let $\left(\Lambda_t^{U'}\right)_{t\in[0,1]}$ be another geodesic with cylindrical transform $U.$ Let $(\Psi_t' : N \times (0,1) \to X)_t$ be a horizontal lifting with $\Psi_0' = \Psi_0,$ and let $h' : N \times (0,1) \to \R$ denote the associated Hamiltonian. Let $h$ denote the Hamiltonian of $\left(\Lambda_t^U\right)_t$ with respect to the horizontal lifting $(\Psi_t)_t.$ By Lemma~\ref{lemma: non-singular cylindrical transform}~\ref{flux equals Hamiltonian}, after possibly adding a constant to $h'$, we have $h' = h.$ Let $t_0 \in [0,1],$ let $c$ be a value of $h,$ and let $s \in (0,1)$ be such that $h^{-1}(c) = N \times \{s\}.$ Since the cylindrical transforms of $\left(\Lambda_t^U\right)_t$ and $\left(\Lambda_t^{U'}\right)_t$ coincide, the cylinder of $c$ level sets $Z_c$ associated with $\left(\Lambda_t^U\right)_t$ coincides with the cylinder of $c$ level sets $Z_c'$ associated with $\left(\Lambda_t^{U'}\right)_t.$ By Remark~\ref{remark: time function is the fundamental harmonic}, the immersed submanifolds $\left[\Psi_{t_0}|_{N \times \{s\}}\right]$ and $\left[\Psi_{t_0}'|_{N \times \{s\}}\right]$ both coincide with the $t_0$ level set of the fundamental harmonic of the cylinder $Z_c = Z_c'.$ Since $c$ was arbitrary, it follows that $\Lambda_{t_0}^U = \Lambda_{t_0}^{U'}.$ Since $t_0$ was arbitrary, the claim follows.
	\end{proof}

	\begin{lemma}
		\label{lemma: geodesic of small Lagrangians}
		Let $\Lambda_0,\Lambda_1\subset X$ be smoothly embedded positive Lagrangians intersecting transversally at a point $q.$ Suppose there exists a connected component $\mathcal{Z}\subset\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right)$ with an end $E$ converging regularly to $q.$ Let $U \subset \mathcal{Z}$ be a ray representing $E$ admitting a regular parameterization about $q.$ For $i = 0,1,$ let $\Lambda_i^U$ denote the unpunctured submanifold of $\Lambda_i$ swept by $U$. Then, there exists a geodesic of positive Lagrangians between $\Lambda_0^U$ and $\Lambda_1^U$ with cylindrical transform~$U.$ This geodesic is unique up to reparameterization and has critical locus $\{q\}.$
	\end{lemma}

	\begin{proof}
		Let $\Phi:S^{n-1}\times[0,1]\times[0,\epsilon)\to X$ be a regular parameterization of $U$ about $q$ compatible with the harmonics. For $t\in[0,1],$ define
		\[
		\widetilde{\Psi}_t:S^{n-1}\times[0,\epsilon)\to X,\qquad(c,s)\mapsto\Phi(c,t,s).
		\]
Recalling Definition~\ref{definition:polar coordinates}, let
\[
\kappa: S^{n-1}\times[0,\epsilon) \to \R^n
\]
be a polar coordinate map centered at zero with image $W.$
		For $t\in[0,1],$ let
\[
\Psi_t:W\to X
\]
be the unique map such that
\[
		\widetilde{\Psi}_t=\Psi_t\circ\kappa.
\]
		By Remark~\ref{remark: regular parameterization gives rise to family of cone-immersions}, the pair $(\Psi_t,0)$ is a cone-immersion for $t\in[0,1].$ Write
\[
\Psi_t^\circ:=\Psi_t|_{W\setminus\{0\}},\qquad \Lambda_t^U:=[(\Psi_t,0)], \qquad \Lambda_t^{U^\circ}:=\left[\Psi_t^\circ\right].
\]
By Proposition~\ref{proposition: interior-regular family gives rise to geodesic}, the family $\left(\Lambda_t^{U^\circ}\right)_{t\in[0,1]}$ is a geodesic with empty critical locus and cylindrical transform $U.$ Recalling the meaning of a regular parameterization about $q$ from Definition~\ref{definition: interior regularity}\eqref{regular convergence to intersection point}\ref{critical point}, it follows that $\Psi_t(0) = q$ for $t \in [0,1].$ Thus $\Lambda_t^U \in \mathcal{L}(X,W;\{0\},\{q\})$ as in Definition~\ref{definition: geodesic}. It follows from Lemma~\ref{lemma:extension of geodesic to cone point} that $\left(\Lambda_t^U\right)_t$ is a geodesic with critical locus $\{q\}.$ The uniqueness of $\left(\Lambda_t^U\right)_t$ follows from the uniqueness of $\left(\Lambda_t^{U^\circ}\right)_{t\in[0,1]}$ given by Proposition~\ref{proposition: interior-regular family gives rise to geodesic}.
	\end{proof}

	The following lemmas are preliminary to the proof of Theorem~\ref{theorem: geodesic-cylinder correspondence}. We first establish relevant notation. Let $(\Lambda_t)_{t\in[0,1]} \in \mathcal{L}(X,L;S,C_0)$ be a geodesic. Let $(h_t:\Lambda_t\to\mathbb{R})_{t\in[0,1]}$ denote the Hamiltonian of $(\Lambda_t)_t.$ Let $\pi:\widetilde{L}_S\to L$ denote the blowup projection. Let $((\Psi_t:L\to X,S))_{t\in[0,1]}$ be a horizontal lifting of $(\Lambda_t)_t,$ and let $h:L\to\mathbb{R}$ be the Hamiltonian of $(\Lambda_t)_t$ with respect to $((\Psi_t,S))_t.$ That is,
		\[
		h_t=[((\Psi_t,S),h)].
		\]
		Then $(h,S)$ is cone-smooth and $S$ is contained in its critical locus by Lemma~\ref{lemma:derivative 1-form of cone smooth path}. Recalling Definition~\ref{definition: cone-smooth differential form etc}, Remark~\ref{remark:differential of cone-smooth function}, Lemma~\ref{lemma:cone-smooth pullback} and notation~\eqref{eq:pullbackabbrev}, let $\nabla^t h$ denote the gradient of $h$ with respect to the pullback metric $\Psi_t^*g.$ So, $\nabla^t h$ is a cone-smooth vector field on~$(L,S).$  Composing with the differential $d\Psi_t,$ we consider $\nabla^th$ as a cone-smooth vector field along $\Psi_t.$ Recall that we denote by $\widetilde{\nabla^t h}$ the blowup vector field, which is a section of $\pi^*\Psi_t^*TX.$ Let $\theta_t = \theta_{\Psi_t}:\widetilde{L}_S \to S^1$ denote the phase function.

\begin{lemma}\label{lemma: time derivative of cone derivative}
Let $p_0 \in S,$ let $0\neq v \in T_{p_0}L$ and let $\widetilde p: = [v] \in E_{p_0}.$ Then,
\[
\deriv{t}d\Psi_t(v) =-J\nabla_v\widetilde{\nabla^th}-\tan(\theta_t(\widetilde p))\nabla_v\widetilde{\nabla^th},
\]
where $\nabla$ is any connection on $TX.$
\end{lemma}
\begin{proof}
  By~\cite[Remark~5.6]{solomon}, we have
		\begin{equation}
		\label{horizontal vector formula}
		\deriv{t}\Psi_t\circ\pi(p)=-J\widetilde{\nabla^th}(p)-\tan(\theta_t(p))\widetilde{\nabla^th}(p),\quad p\in\widetilde{L}_S,t\in[0,1].
		\end{equation}
Let $s,x^1,\ldots,x^{n-1},$ be local cone-coordinates on $L$ around $\widetilde{p}$ with $\pderiv[\pi]{s}|_{\widetilde p}=v.$ As $p_0$ is a critical point of $h$ and by Remark~\ref{remark: critical cone point}, we have
		\[
		\widetilde{\nabla^th}\left(\widetilde{p}\right)=0,\quad t\in[0,1].
		\]
		Hence, covariantly differentiating~\eqref{horizontal vector formula} at $\widetilde{p}$ in the direction of $v$, we obtain
		\begin{align*}
\deriv{t}d\Psi_t(v) &=
		\frac{D}{\partial t}\frac{\partial\Psi_t}{\partial s}\left(\widetilde{p}\right) \\
		&=\frac{D}{\partial s}\frac{\partial\Psi_t}{\partial t}\left(\widetilde{p}\right)\\
		&=\nabla_v\left(-J\widetilde{\nabla^th}-\tan(\theta_t(\widetilde p))\widetilde{\nabla^th}\right)\\
		&=-J\nabla_v\widetilde{\nabla^th}-\tan(\theta_t(\widetilde p))\nabla_v\widetilde{\nabla^th}.
		\end{align*}
\end{proof}

	\begin{lemma}
		\label{lemma: critical points are non-degenerate}
		Suppose $\Lambda_0$ and $\Lambda_1$ intersect transversally at $q \in C_0.$  Then, $q$ is a non-degenerate critical point of $h_t,\, t \in [0,1].$
	\end{lemma}

	\begin{proof}
Let $p \in S$ with $\Psi_t(p) = q$ for $t \in [0,1].$ By way of contradiction, suppose that $p$ is a degenerate critical point of $h$ and let $0\ne v\in T_{p}L$ with $\nabla_vdh=0.$ It follows from Lemma~\ref{lemma: time derivative of cone derivative} that
		\[
		d\Psi_0\left(v\right)=d\Psi_1\left(v\right)
		\]
		contradicting the transversality of $\Lambda_0$ and $\Lambda_1.$
	\end{proof}
Suppose $\Lambda_0$ and $\Lambda_1$ intersect transversally at a single point, so $C_0 = \{q\}.$ Moreover, for $t \in [0,1]$ assume $q$ is an absolute minimum or maximum of $h_t$. Write $S = \{p\},$ so $\Psi_t(p) = q.$ By Lemma~\ref{lemma: critical points are non-degenerate}, $p$ is a non-degenerate critical point of $h.$ By Lemma~\ref{lemma: nice level sets},
choose a positive $\epsilon$ and a polar coordinate map $\kappa:S^{n-1}\times[0,\epsilon)\to L$ centered at $p$ such that
for each $s\in(0,\epsilon)$ the restricted map $\kappa|_{S^{n-1}\times\{s\}}$ parameterizes a level set of $h.$
\begin{lemma}\label{lemma:rocking lemma}
Let $\mathcal{Z}$ denote the cylindrical transform of $(\Lambda_t)_t.$ Then, one end of $\mathcal{Z}$ converges regularly to $q.$
In fact, the map
		\[
		\Phi:S^{n-1}\times[0,1]\times[0,\epsilon)\to X,\qquad(c,t,s)\mapsto\Psi_t(\kappa(c,s))
		\]
is a regular parameterization of $\mathcal{Z}$ about $q.$
\end{lemma}
\begin{proof}
We verify the conditions of Definition~\ref{definition: interior regularity}~\eqref{regular convergence to intersection point} one by one. Condition~\ref{critical point} is immediate. By Lemma~\ref{lemma: non-singular cylindrical transform}~\ref{cylindrical transform is interior-regular}, $\Phi|_{S^{n-1}\times [0,1]\times (0,\epsilon)}$ is interior regular verifying condition~\ref{item:interior regular}. To verify conditions~\ref{item:derivative Phi immersion} and~\ref{item:nowhere tangent Euler}, we show that
\[
\left.\pderiv[\Phi]{s}\right|_{S^{n-1} \times [0,1]\times \{0\}} : S^{n-1} \times [0,1]\times \{0\} \to T_qX
\]
is an immersion nowhere tangent to the Euler vector field. Indeed, by Lemma~\ref{lemma:immersion criterion}, for $t \in [0,1]$ the restriction
\[
\left.\pderiv[\Phi]{s}\right|_{S^{n-1} \times \{t\}\times \{0\}} : S^{n-1} \times \{t\}\times \{0\} \to T_qX
\]
is an immersion nowhere tangent to the Euler vector field. Let $(c,t) \in S^{n-1} \times [0,1],$ and write
\[
v = \pderiv[\kappa]{s}(c,0) \in T_pL, \qquad \widetilde p = [v] \in E_p \simeq \orientedprojective(T_pL).
\]
Let
\[
\widetilde q = [(\orientedprojective((d\Psi_t)_p),\widetilde p)] \in \orientedprojective(TC_q\Lambda_t).
\]
By Lemma~\ref{lemma: time derivative of cone derivative} we have
\begin{equation}\label{equation:Jw}
\pderiv{t}\pderiv[\Phi]{s}(c,t,0) = \deriv{t} d\Psi_t(v) = -J\nabla_v\widetilde{\nabla^th}-\tan(\theta_t(\widetilde p))\nabla_v\widetilde{\nabla^th}.
\end{equation}
Write $w = \nabla_v\widetilde{\nabla^th}.$ As $p$ is a critical point of $h,$ Remark~\ref{remark: critical cone point} gives $\widetilde{\nabla^th}(\widetilde p) = 0,$ and it follows that $w \in T_{\widetilde q}\Lambda_t \subset T_qX.$ Moreover, $w \neq 0$ since $p$ is a non-degenerate critical point of $h.$ Write
\[
\Upsilon := d\left(\left.\pderiv[\Psi_t\circ\kappa]{s}\right|_{s=0}\right)\left(T_cS^{n-1}\right) = d \left(\left.\pderiv[\Phi]{s}\right|_{S^{n-1} \times \{t\}\times \{0\}} \right)\left(T_cS^{n-1}\right)
\subset T_q X.
\]
Recalling Definition~\ref{definition:polar coordinates}, let $\sigma = \left.\pderiv[\kappa]{s}\right|_{s=0}: \orientedprojective(T_pL) \to T_pL \setminus \{0\}$ denote the section associated to $\kappa.$ Let $\widehat p \subset T_pL$ and $\widehat q \subset T_q X$ denote the $1$-dimensional subspaces generated by the rays $\widetilde p, \widetilde q,$ respectively.
Then,
\[
\widetilde{TL}_S|_{\widetilde p} = T_pL = \widehat p \oplus d\sigma\left(T_{\widetilde p}\orientedprojective(T_pL)\right).
\]
So, we obtain
\[
T_{\widetilde q}\Lambda_t = \widetilde{d\Psi}_t\left(\left(\widetilde{TL}_S\right)_{\widetilde p}\right) = \widehat q \oplus \Upsilon.
\]
Since $T_{\widetilde q}\Lambda_t$ is Lagrangian, we have $T_{\widetilde q}\Lambda_t \cap JT_{\widetilde q}\Lambda_t = \{0\}.$ It follows from equation~\eqref{equation:Jw} that
\[
\pderiv{t}\pderiv[\Phi]{s}(c,t,0) \notin T_{\widetilde q}\Lambda_t.
\]
So, the linear subspace
\begin{multline*}
\R\left\langle\pderiv{t}\pderiv[\Phi]{s}(c,t,0)\right\rangle \oplus T_{\widetilde q}\Lambda_t =\\
 =\R\left\langle\pderiv{t}\pderiv[\Phi]{s}(c,t,0)\right\rangle \oplus \widehat q\oplus d \left(\left.\pderiv[\Phi]{s}\right|_{S^{n-1} \times \{t\}\times \{0\}} \right)\left(T_cS^{n-1}\right) \subset T_qX
\end{multline*}
is $n+1$ dimensional. Since the Euler vector field $\varepsilon$ on $T_qX$ satisfies
\[
0 \neq \varepsilon\left(\pderiv[\Phi]{s}(c,t,0)\right) \in \widehat q,
\]
the lemma follows.
\end{proof}

	We henceforth restrict the discussion to the setting of Theorem~\ref{theorem: geodesic-cylinder correspondence}. Namely, we assume $\Lambda_0,\Lambda_1\subset X$ are smoothly embedded positive Lagrangian spheres intersecting transversally at the points $q_0$ and $q_1$ and nowhere else.

	\begin{rem}
		\label{rem:simple topology}
		Note that, under the above assumption, if there exists a geodesic connecting $\Lambda_0$ and $\Lambda_1$, its critical locus necessarily consists of exactly two points. In particular, letting $h$ denote the Hamiltonian of the geodesic and $[m,M]$ the image of $h,$ every $c\in(m,M)$ is a regular value.
	\end{rem}

	We define regularity for a connected component in $\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right).$

	\begin{dfn}
		\label{definition: regularity}
		Let $\mathcal{Z}\subset\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right)$ be a connected component. A \emph{regular parameterization} of $\mathcal{Z}$ is a smooth map $\Phi:S^{n-1}\times[0,1]\times[0,1]\to X$ satisfying the following conditions:
		\begin{enumerate}
			\item The restricted map $\Phi|_{S^{n-1}\times[0,1]\times(0,1)}$ is an interior-regular parameterization of $Z.$
			\item The restricted maps $\Phi|_{S^{n-1}\times[0,1]\times[0,1/2)}$ and $\Phi|_{S^{n-1}\times[0,1]\times(1/2,1]}$ are regular parameterizations of the two ends of $\mathcal{Z}$ about the intersection points $q_0$ and $q_1,$ respectively.
		\end{enumerate}
		We say $\mathcal{Z}$ is \emph{regular} if it admits a regular parameterization.
	\end{dfn}

	\begin{rem}
		\label{remark: regular connected component}
		$\;$
		\begin{enumerate}[label=(\alph*)]
			\item \label{equivalent definition of regularity}	A connected component $\mathcal{Z}\subset\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right)$ is regular if and only if each $Z \in \mathcal{Z}$ has an interior-regular neighborhood and the ends of $\mathcal{Z}$ converge regularly to the intersection points~$q_0$ and $q_1$ respectively. This follows from Lemma~\ref{lemma: interior and end regular parameterization compatible with harmonics} and Corollary~\ref{corollary:action}. Indeed, we use the compactness of $[0,1]$ and the uniqueness claim to glue together parameterizations of the ends of~$\mathcal{Z}$ with finitely many interior regular parameterizations of open intervals in~$\mathcal{Z}.$
			\item Suppose $\mathcal{Z}\subset\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right)$ is a regular connected component. Then $\mathcal{Z}$ admits a regular parameterization compatible with the harmonics. This again follows from Lemma~\ref{lemma: interior and end regular parameterization compatible with harmonics}.
		\end{enumerate}
	\end{rem}

	\begin{proof}[Proof of Theorem~\ref{theorem: geodesic-cylinder correspondence}]
		Suppose there exists a geodesic $(\Lambda_t)_{t\in[0,1]}$ between the positive Lagrangian spheres $\Lambda_0$ and~$\Lambda_1.$ Let $\mathcal{Z}\subset\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right)$ denote the cylindrical transform. Then Lemma~\ref{lemma: non-singular cylindrical transform}, Lemma~\ref{lemma:rocking lemma} and Remark~\ref{remark: regular connected component}\ref{equivalent definition of regularity}, imply that $\mathcal{Z}$ is regular.

		Conversely, let $\mathcal{Z}\subset\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right)$ be a regular connected component. Let $\Phi : S^{n-1} \times [0,1]\times [0,1]$ be a regular parameterization of $\mathcal{Z}.$ It follows from Remark~\ref{remark: regular parameterization gives rise to family of cone-immersions} and Corollary~\ref{cor:cone-immersion open mapping} that the image of $\Phi|_{S^{n-1}\times \{i\} \times [0,1]} \subset \Lambda_i$ is open. Since $S^{n-1} \times [0,1]$ is compact, the image is also closed. Thus, the submanifolds swept by $\mathcal{Z}$ are $\Lambda_i\setminus\{q_0,q_1\}=:\Lambda_i^\circ,\;i=0,1.$ By Proposition~\ref{proposition: interior-regular family gives rise to geodesic}, there exists a geodesic $\left(\Lambda_t^\circ\right)_{t\in[0,1]}$ between $\Lambda_0^\circ$ and $\Lambda_1^\circ$ with cylindrical transform $\mathcal{Z}.$ For $j = 0,1,$ let $U_j$ be a ray representing the end of $\mathcal{Z}$ that converges regularly to $q_j.$ Let $\Lambda_i^{U_j}$ denote the unpunctured submanifold of $\Lambda_i$ swept by $U_j.$ By Lemma~\ref{lemma: geodesic of small Lagrangians}, there exists a geodesic $\left(\Lambda^{U_j}_t\right)_{t\in[0,1]}$ between $\Lambda_0^{U_j}$ and $\Lambda_1^{U_j}$ with cylindrical transform $U_j.$  By the uniqueness claim of Proposition~\ref{proposition: interior-regular family gives rise to geodesic}, the three above geodesics glue together to the desired geodesic between $\Lambda_0$ and $\Lambda_1.$
	\end{proof}

	\section{Proof of the perturbation theorem}
	\label{section: proof of perturbation theorem}

	In this section we prove Theorem~\ref{theorem: perturbation of geodesic}. We assume the setting of the theorem. That is, $\mathcal{O}$ is a Hamiltonian isotopy class of positive Lagrangian spheres in $X,$ the spheres $\Lambda_0, \Lambda_1 \in \mathcal{O}$ intersect transversally at $\Lambda_0 \cap \Lambda_1 = \{q_0,q_1\},$ and $(\Lambda_t)_{t\in[0,1]}$ is a geodesic between $\Lambda_0$ and $\Lambda_1.$ Let $\mathcal{Z}$ denote the cylindrical transform of $(\Lambda_t)_t.$ By Theorem~\ref{theorem: geodesic-cylinder correspondence} the cylindrical transform $\mathcal{Z}$ is a regular connected component of the 1-dimensional manifold $\mathcal{SLC}\left(S^{n-1};\Lambda_0,\Lambda_1\right).$ Let $\Phi : S^{n-1} \times [0,1] \times [0,1] \to X$ be a regular parameterization of $\mathcal{Z}.$ For $s \in (0,1),$ we write
	\[
	\Phi_s := \Phi|_{S^{n-1} \times [0,1] \times \{s\}},\quad Z_s := [\Phi_s] \in \mathcal{Z}.
	\]
Fix a Weinstein neighborhood $\Lambda_1 \subset W \subset X$ with projection $\pi : W \to \Lambda_1$ satisfying
	\[
	W \cap \Lambda_0 = \pi^{-1}(\{q_0, q_1\}).
	\]
	In particular, spheres in $\mathcal{O}$ that are close to $\Lambda_1$ are identified with exact 1-forms on $\Lambda_1.$ For a function $h \in C^\infty(\Lambda_1)$ small enough in the $C^1$-sense, we let $\Lambda_{1, h}$ denote the element of $\mathcal{O}$ obtained by this identification as the graph of $dh.$ We write $\Lambda_0 \cap \Lambda_{1, h} = \{q_{0, h}, q_{1, h}\}.$ Thus a sufficiently small $C^{k+1,\alpha}$ open set $0 \in \mathcal{W} \subset C^\infty(\Lambda_1)$ corresponds to a $C^{k,\alpha}$ open neighborhood of $\Lambda_1$ in $\mathcal{O}.$

	\begin{ntn}
		\label{notation: Hamiltonian flow with cutoff}
		Let $\chi : X \to [0,1]$ be smooth with compact support contained in $W,$ and let $h \in C^{k, \alpha}(\Lambda_1)$ for some $k \ge 2$ and $\alpha \in [0,1].$ Then the function $\chi \cdot h \circ \pi$ is well-defined and of regularity $C^{k, \alpha}$ on $X$ with compact support. We let $\varphi_{h, \chi}$ denote the time-1 Hamiltonian flow of $\chi \cdot h \circ \pi.$
	\end{ntn}

	\begin{rem}
		\label{remark: cutoff}
		Let $\chi$ and $h$ be as in Notation~\ref{notation: Hamiltonian flow with cutoff}. Then the map $\varphi_{h, \chi} : X \to X$ is a $C^{k - 1, \alpha}$ symplectomorphism of $X$ restricting to the identity on $\Lambda_0.$
	\end{rem}
In the proofs below, we will use the following well-known technical lemma. Let $\Omega^*_{C^{k,\alpha}}$ denote differential forms of regularity $C^{k,\alpha}.$
\begin{lemma}\label{lemma: Holder pullback}
Let $P,Q,$ be smooth manifolds. For $r,s,l \geq 1,$ the map
\[
C^{r,\alpha}(P,Q) \times \Omega^*_{C^{s,\alpha}}(Q) \to \Omega^*_{C^{\min\{r-1,s-l\},\alpha}}(P), \qquad (g,\eta) \mapsto g^*\eta,
\]
is of regularity $C^l.$
\end{lemma}

Abbreviate $L = S^{n-1}\times [0,1].$

	\begin{prop}
		\label{proposition: interior big family of cylinders}
		Fix $\alpha \in (0,1).$ Let $s_0  \in (0, 1).$ Then there exist a positive $\epsilon,$ a $C^{3, \alpha}$-open $0 \in \mathcal{W} \subset C^\infty(\Lambda_1)$ and a family of smooth immersions
\[
f_{s,h} : L \to X, \qquad (s, h) \in (s_0 - \epsilon, s_0 + \epsilon) \times \mathcal{W},
\]
smooth in $s$ and continuous with respect to the $C^{3,\alpha}$ topology on $\mathcal{W}$ and the $C^{1,\alpha}$ topology on $C^{\infty}(L,X)$			with the following properties:
			\begin{enumerate}[label=(\arabic*)]
				\item \label{item: special} For $s$ and $h$ as above the immersion $f_{s,h}$ represents an imaginary special Lagrangian cylinder $Z_{s, h} \in \mathcal{SLC}(S^{n-1};\Lambda_0,\Lambda_{1,h}).$
\item \label{item:h = 0}
We have $f_{s,0} = \Phi_s$ for $s \in (s_0 - \epsilon, s_0 + \epsilon).$
				\item \label{item: interior regular} For $h \in \mathcal{W}$ the family of cylinders $(Z_{s, h})_s$ is interior regular.
			\end{enumerate}
Moreover, there exists a $C^{1,\alpha}$ open set $\mathcal V \subset C^\infty(L,X)$ with
\begin{equation}\label{equation: fshinW}
f_{s,h} \in \mathcal V, \qquad (s,h) \in (s_0-\epsilon,s_0 + \epsilon)\times \mathcal{W},
\end{equation}
such that if $f \in \mathcal V$ represents an imaginary special Lagrangian cylinder
\[
[f] \in \mathcal{SLC}(S^{n-1};\Lambda_0,\Lambda_{1,h})
\]
with $h \in \mathcal W,$ then $[f] = [f_{s,h}]$ for some $s \in (s_0-\epsilon,s_0 + \epsilon).$
	\end{prop}

	\begin{proof}
By Lemma~\ref{lemma: Weinstein with boundary}, choose an immersed Weinstein neighborhood $(Y,\psi)$ of $Z_{s_0}$ compatible with $\Lambda_0$ and $\Lambda_1,$ where $Y \subset T^*L$ and $\psi : Y \to X$ with $\psi|_L = \Phi_{s_0}.$ Let $\pi_L : T^*L \to L$ denote the projection. For $u\in\cob{2,\alpha}(L),$ let $\mathrm{Graph}(du) \subset T^*L$ denote the graph. For $u$ small enough that $\mathrm{Graph}(du) \subset Y,$ let $j_u : L \to X$ be given by
\[
j_u = \psi \circ \left( \pi_L|_{\mathrm{Graph}(du)}\right)^{-1}.
\]
Let $U \subset X$ be an open set such that
\[
\overline U \subset W \setminus \Lambda_0, \qquad \Phi_{s_0}({S^{n-1}\times \{1\}}) \subset U.
\]
Let $\chi : X \to [0,1]$ be a smooth function such that $\chi|_U \equiv 1$ and $\mathrm{supp}(\chi) \subset W \setminus \Lambda_0.$
Choose an open set $\Phi_{s_0}(S^{n-1}\times\{1\}) \subset U' \subset\subset U$ and an open set $0 \in \mathcal{A} \subset C^{3, \alpha}(\Lambda_1)$ such that for $h \in \mathcal{A}$ we have
		\[
		\varphi_{h, \chi}(U' \cap \Lambda_1) \subset \Lambda_{1, h}.
		\]
		Let $0 \in \mathcal{U} \subset \cob{2,\alpha}(L)$ be open such that for $u\in\mathcal{U},$ we have $j_u(S^{n-1}\times \{1\}) \subset U'.$ Consider the differential operator
		\[
		\mathcal{F} : \mathcal{U} \times \mathcal{A} \to C^\alpha\left(L\right),\quad(u,h) \mapsto *j_u^* \varphi_{h, \chi}^* \real \Omega.
		\]
For $(u, h) \in \mathcal{U} \times \mathcal{A},$ the immersion $\varphi_{h,\chi} \circ j_u$ represents a Lagrangian cylinder in $\mathcal{LC}\left(S^{n-1}; \Lambda_0, \Lambda_{1, h}\right)$, and this cylinder is imaginary special Lagrangian if and only if $\mathcal{F}(u, h) = 0.$

We claim $\mathcal{F}$ is continuously differentiable, and for a fixed $h \in \mathcal{A} \cap C^\infty(\Lambda_1),$ the map $u \mapsto \mathcal{F}(u,h)$ is smooth. Indeed, recalling Remark~\ref{remark: cutoff}, since the map
\[
\mathcal{A} \to C^{2,\alpha}(X,X), \qquad h \mapsto \varphi_{h,\chi},
\]
is continuously differentiable, it follows from Lemma~\ref{lemma: Holder pullback} that the map
\[
\mathcal{A} \to \Omega^*_{C^{1,\alpha}}(X), \qquad h \mapsto \varphi^*_{h,\chi}\real \Omega,
\]
is continuously differentiable. Similarly, since the map
\[
\mathcal{U} \to C^{1,\alpha}(L,X), \qquad u \mapsto j_u,
\]
is smooth, it follows from Lemma~\ref{lemma: Holder pullback} that the map
\[
\mathcal{U} \times \Omega^*_{C^{s,\alpha}}(X) \to \Omega^*_{C^{\alpha}}(L), \qquad (u,\eta) \mapsto j_u^* \eta,
\]
is of regularity $C^{s}.$
So, the map $\mathcal{F}$ is continuously differentiable as the composition of two continuously differentiable  maps. Moreover, for fixed $h \in \mathcal{A}\cap C^\infty(\Lambda_1),$ we have $\varphi^*_{h,\chi}\real \Omega \in \Omega^*_{C^{\infty}}(X),$ so the map $u \mapsto \mathcal{F}(u,h)$ is smooth.

		Consider the linearization of $\mathcal{F},$
		\[
		d \mathcal{F}_{(0, 0)} : \cob{2, \alpha}(L) \times C^{3, \alpha}(\Lambda_1) \to C^\alpha(L).
		\]
		By Lemma~\ref{lemma: linearized operator} and Lemma~\ref{lemma: Delta rho is a cool operator},  the restriction of $d \mathcal{F}_{(0, 0)}$ to the subspace
		\[
		C^{2, \alpha}(L; \partial L) \times \{0\} \subset \cob{2, \alpha}(L) \times C^{3, \alpha}(\Lambda_1)
		\]
		is an isomorphism onto $C^\alpha(L).$ Abbreviate $\mathbb{V} := C^{2, \alpha}(L; \partial L).$ Recall that
		\[
		\codim\left(\mathbb{V} \subset \cob{2, \alpha}(L)\right) = 1.
		\]
		Let $\ell \subset \cob{2, \alpha}(L)$ be a line consisting of smooth functions such that
		\[
		\cob{2, \alpha}(L) = \mathbb{V} \oplus \ell.
		\]
		By the implicit function theorem, there exist open neighborhoods
		\[
		0 \in \mathbb{V}_0 \subset \mathbb{V}, \quad 0 \in \ell_0 \subset \ell, \quad 0 \in \mathcal{A}_0 \subset \mathcal{A},
		\]
		such that for $(l, h) \in \ell_0 \times \mathcal{A}_0$ there exists a unique $v = v(l, h) \in \mathbb{V}_0$ with
		\[
		\mathcal{F}(v + l, h) = 0.
		\]
		By elliptic regularity (e.g.\ \cite[Chapter 17]{gilbarg-trudinger}), the function $v(l, h)$ is smooth if $h$ is. Write
		\[
		\mathcal{W} := C^\infty(\Lambda_1) \cap \mathcal{A}_0.
		\]
		Define a family of immersions by
\begin{equation}
		\label{big family of cylinders}
		\widetilde f_{l,h} := \varphi_{h, \chi} \circ j_{v(l, h) + l}, \qquad (l, h) \in \ell_0 \times \mathcal{W}.
		\end{equation}
By construction, the immersions $\widetilde f_{l,h}$ represent special Lagrangian cylinders in  $\mathcal{SLC}\left(S^{n-1}; \Lambda_0, \Lambda_{1, h}\right)$ for $(l, h) \in \ell_0 \times \mathcal{W}.$ Moreover, the map
\[
\ell_0 \times \mathcal{W} \to C^{\infty}(L,X), \qquad (l,h) \mapsto \widetilde f_{l,h},
\]
is continuous with respect to the $C^{3,\alpha}$ topology on $\mathcal{W}$ and the $C^{1,\alpha}$ topology on $C^\infty(L,X).$ Also, for fixed $h \in \mathcal{W},$ the map $l \mapsto \widetilde f_{l,h}$ is smooth.

		For a small $\epsilon > 0,$ there is a unique open embedding $i : (s_0 - \epsilon, s_0 + \epsilon) \hookrightarrow \ell_0$ such that
		\[
		\left[\widetilde f_{i(s),0}\right] = Z_s, \qquad s \in (s_0 - \epsilon, s_0 + \epsilon).
		\]
For $s \in (s_0 - \epsilon, s_0 + \epsilon),$ let $\zeta_s \in \diff(L)$ be the diffeomorphism such that
\[
\widetilde f_{i(s),0}\circ \zeta_s = \Phi_s.
\]
Observe that the map
\[
(s_0 - \epsilon, s_0 + \epsilon) \to C^\infty(L,L), \qquad s \mapsto \zeta_s
\]
is smooth.
Take
\[
f_{s,h} := \widetilde f_{i(s),h} \circ \zeta_s.
\]
Since the maps $(l,h) \mapsto \widetilde f_{l,h}$ and $s \mapsto \zeta_s$ are continuous in the topologies specified above, it follows that the map $(s,h) \mapsto f_{s,h}$ is continuous as desired. Moreover, for fixed $h \in \mathcal{W},$ the map $s \mapsto f_{s,h}$ is smooth.

Write $Z_{s,h} := [f_{s,h}].$ Properties~\ref{item: special} and~\ref{item:h = 0} claimed in the proposition are immediate from the construction. To establish property~\ref{item: interior regular}, we claim that after possibly shrinking $\mathcal{W},$ for $h \in \mathcal{W},$ the map
\[
\Phi^h : L \times (s_0 - \epsilon, s_0 + \epsilon) \to X,
\]
given by
\[
\Phi^h(p,s) = f_{s,h}(p)
\]
is a regular parameterization of the family $(Z_{s,h})_{s \in (s_0 - \epsilon, s_0 + \epsilon)}.$ Indeed, the map $\Phi^h$ is smooth because the map $s \mapsto f_{s,h}$ is smooth. Conditions~\ref{item: rep} and~\ref{item: diffeo s} of Definition~\ref{definition: interior regularity}~\eqref{interior regularity first part} hold by construction. The remaining conditions in Definition~\ref{definition: interior regularity}~\eqref{interior regularity first part} are open, and $\Phi^0 = \Phi|_{S^{n-1}\times [0,1]\times (s_0 - \epsilon, s_0 + \epsilon)}$, which is interior regular. So, possibly after shrinking $\mathcal W$ and $\epsilon,$ the map $\Phi^h$ is also an interior regular parameterization for $h \in \mathcal W.$

Finally, we construct the open set $\mathcal{V}.$ Given $\mathcal{Q} \subset \mathcal{W}$ and $\mathcal{P}\subset C^\infty(L,X),$ let
\[
\mathcal{B}(\mathcal{Q},\mathcal{P}) := \{(h,f) \in \mathcal{Q}\times\mathcal{P}\,|\, f \text{ represents a cylinder } [f]\in \mathcal{LC}(S^{n-1};\Lambda_0,\Lambda_{1,h})\}.
\]
First, we claim that perhaps after shrinking~$\mathcal{W},$ there exists a $C^{1,\alpha}$ open set $\mathcal{V}_1 \subset C^{\infty}(L,X)$ such that
\[
\Phi_s \in \mathcal{V}_1, \qquad s \in (s_0-\epsilon,s_0 + \epsilon),
\]
and if $(h,f) \in \mathcal{B}(\mathcal{W},\mathcal{V}_1),$ then
\begin{equation*}
\varphi_{h,\chi}^{-1}\circ f(L) \subset \psi(Y).
\end{equation*}
Here, the set $\psi(Y)$ is not open in $X$ because $Y$ is a manifold with boundary. Indeed, $Y$ is an open neighborhood of the zero section of $T^*L,$ and $L$ is a manifold with boundary.
Nonetheless, for $i = 0,1,$ we have $\varphi_{h,\chi}^{-1}\circ f(S^{n-1} \times \{i\}) \subset \Lambda_i.$  Moreover, $\varphi_{h,\chi}^{-1}\circ f$ is close in the $C^{1,\alpha}$ topology to $\Phi_s$ for some $s \in (s_0-\epsilon,s_0 +\epsilon),$ and $\Phi_s(L) \subset \psi(Y).$
So, the claim follows.

For $(h,f) \in \mathcal{B}(\mathcal{W},\mathcal{V}_1),$  let
\[
\kappa_{h,f}: = \pi_L \circ \psi^{-1} \circ \varphi_{h,\chi}^{-1}\circ f : L \to L.
\]
The map
\[
\mathcal{B}(\mathcal{W},\mathcal{V}_1) \to C^\infty(L,L), \qquad (h,f) \mapsto \kappa_{h,f},
\]
is continuous with respect to the $C^{3,\alpha}$ topology on~$\mathcal{W}$ and the $C^{1,\alpha}$ topologies on~$\mathcal{V}_1$ and $C^\infty(L,L).$ Moreover, diffeomorphisms are open in $C^\infty(L,L)$ in the $C^{1,\alpha}$ topology and $\kappa_{0,\Phi_s} = \zeta_s$ is a diffeomorphism for $s \in (s_0-\epsilon,s_0+\epsilon).$
So, possibly shrinking $\mathcal{W},$ we choose an open $\mathcal{V}_2 \subset \mathcal{V}_1$ such that $\Phi_s \in \mathcal{V}_2$ for $s \in (s_0 - \epsilon, s_0 + \epsilon)$ and if $(h,f) \in \mathcal{B}(\mathcal{W},\mathcal{V}_2),$ then $\kappa_{h,f}$ is a diffeomorphism.

For $(h,f) \in \mathcal{B}(\mathcal{W},\mathcal{V}_2),$ by Lemma~\ref{lemma: local description of space of cylinders} there exists $u_{h,f} \in \cob{\infty}(L)$ such that
\[
j_{u_{h,f}} = \varphi_{h,\chi}^{-1}\circ f \circ \kappa_{h, f}^{-1}.
\]
The map
\[
\mathcal{B}(\mathcal{W},\mathcal{V}_2) \to \cob{\infty}(L), \qquad (h,f) \mapsto u_{h,f},
\]
is continuous with respect to the $C^{3,\alpha}$ topology on $\mathcal{W},$ the $C^{1,\alpha}$ topology on $\mathcal{V}_2$ and the $C^{2,\alpha}$ topology on $\cob{\infty}(L).$ Hence, possibly shrinking $\mathcal{W},$ we choose an open $\mathcal{V} \subset \mathcal{V}_2$ such that $\Phi_s \in \mathcal{V}$ for $s \in (s_0 - \epsilon, s_0 + \epsilon)$ and if $(h,f) \in \mathcal{B}(\mathcal{W},\mathcal{V}),$ then
\[
u_{h,f} \in \mathbb{V}_0 + \ell_0.
\]
For $(h,f) \in \mathcal{B}(\mathcal{W},\mathcal{V}),$ let $l_{h,f}$ be the projection of $u_{h,f}$ along $\mathbb{V}$ to $\ell.$ If $[f]$ is imaginary special Lagrangian, then $\mathcal{F}(u_{h,f},h) = 0.$ So, by uniqueness of $v_{l,h},$ it follows that $u_{h,f} = l_{h,f} + v_{l_{h,f},h}.$ Thus, $[f] = [f_{s,h}]$ for $s = i^{-1}(l_{h,f}).$ By continuity of $f_{s,h}$ and property~\ref{item:h = 0} of the proposition, after possibly shrinking $\mathcal{W}$ again, we obtain condition~\eqref{equation: fshinW}.
\end{proof}

	\begin{prop}
		\label{proposition: end big family of cylinders}
		Fix $\alpha \in (0,1)$. Then, there exist a positive $\epsilon,$ a $C^{3, \alpha}$-open $0 \in \mathcal{W} \subset C^\infty(\Lambda_1)$ and a family of smooth maps
\[
f_{s,h} : L \to X, \qquad (s, h) \in [0,\epsilon) \times \mathcal{W},
\]
smooth in $s$ and continuous with respect to the $C^{3,\alpha}$ topology on $\mathcal{W}$ and the $C^{1,\alpha}$ topology on $C^{\infty}(L,X)$			with the following properties:
\begin{enumerate}[label=(\arabic*)]
\item \label{item: special end} For $(s,h) \in (0,\epsilon) \times \mathcal{W}$ the map $f_{s,h}$ is an immersion  representing an imaginary special Lagrangian cylinder $Z_{s, h} \in \mathcal{SLC}(S^{n-1};\Lambda_0,\Lambda_{1,h}).$
\item \label{item:h = 0 end}
We have $f_{s,0} = \Phi_s$ for $s \in [0, \epsilon).$
\item \label{item: regularly converges}
For $h \in \mathcal{W},$ the map
\[
\Phi^h : L \times [0,\epsilon) \to X,
\]
given by
\[
\Phi^h(p,s) = f_{s,h}(p)
\]
is a regular parameterization about $q_{0,h}$ of the family of imaginary special Lagrangian cylinders $(Z_{s,h})_{s \in (0,\epsilon)}.$
\item \label{item: end continuity}
The map $\mathcal{W} \to C^\infty(L,TX)$ given by
\begin{equation}\label{equation:pdsfsh0}
h \mapsto \left.\pderiv[f_{s,h}]{s}\right|_{s = 0}
\end{equation}
is continuous with respect to the $C^{3,\alpha}$ topology on $\mathcal W$ and the $C^{1,\alpha}$ topology on $C^{\infty}(L,TX).$
			\end{enumerate}
A similar family of smooth immersions $f_{s,h}$ exists for $s \in (1-\epsilon,1].$
	\end{prop}

	\begin{proof}
We prove the proposition for $s \in (0,\epsilon).$
	Let $\chi : X \to [0,1]$ be smooth with compact support in $W$ and equal to 1 in a neighborhood of the intersection point~$q_0.$ Then there exist open sets $q_0 \in U \subset X$ and $0 \in \mathcal{A}_1 \subset C^1(\Lambda_1)$ such that for $h \in \mathcal{A}_1$ we have
	\[
	\varphi_{h, \chi}(U \cap \Lambda_1) \subset \Lambda_{1, h}, \qquad \varphi_{h, \chi}(U \cap \Lambda_0) \subset \Lambda_0.
	\]
	Let $\delta > 0$ such that for $s \in (0, \delta)$ we have $Z_s \subset U.$ Thus, for $s \in (0,\delta)$ and $h \in \mathcal{A}_1$,
	\[
	[\varphi_{h, \chi}\circ \Phi_s] \in \mathcal{LC}\left(S^{n-1}; \Lambda_0, \Lambda_{1, h}\right).
	\]

After possibly shrinking $U,$ identify $U$ with a ball $V \subset \C^n$ via a Darboux parameterization
		\[
		\mathbf{X} : V \to U
		\]
		such that $\mathbf{X}^{-1}(\Lambda_0)$ and $\mathbf{X}^{-1}(\Lambda_1)$ are contained in real linear subspaces. For $s \geq 0,$ let $M_s : \C^n \to \C^n$ denote multiplication by $s,$ and write
		\[
		V_s := M_s^{-1}(V).
		\]

		Let $\mathcal{A} := \mathcal{A}_1 \cap C^{3,\alpha}(\Lambda_1).$
		For $h\in\mathcal{A},$ define a complex structure and an $n$-form on $V$ by
		\[
		J_h = J_{h,1} := \mathbf{X}^* \varphi_{h, \chi}^* J, \quad \Omega_h = \Omega_{h,1} := \mathbf{X}^* \varphi_{h, \chi}^*\Omega.
		\]
		For $h \in \mathcal{A}$ and $s \in (0, \delta),$ define a complex structure and an $n$-form on $V_s$ by
		\[
		J_{h,s}:=M_s^*J_h,\quad\Omega_{h,s}:=s^{-n}M_s^*\Omega_h.
		\]
		The complex structures and $n$-forms defined in this manner are of regularity $C^{1,\alpha}.$
For $h\in\mathcal{A}$ we have
		\[
		J_{h,s}\underset{s\searrow0}{\longrightarrow}J_{h,0},\quad\Omega_{h,s}\underset{s\searrow0}{\longrightarrow}\Omega_{h,0},
		\]
		where $J_{h, 0}$ and $\Omega_{h, 0}$ are a constant complex structure and a constant $n$-form on $V_0 = \C^n,$ and the convergence is with respect to the $C^{1, \alpha}$ topology on compact subsets. Moreover, writing $\Omega_{C^\alpha}$ for differential forms of regularity $C^\alpha,$ for $s_0 \in (0,\delta)$ the map
\[
[0,s_0) \to \Omega_{C^{\alpha}}(V_{s_0}), \qquad s \mapsto \Omega_{h,s}|_{V_{s_0}},
\]
is continuously differentiable.

		Recall the regular parameterization $\Phi : S^{n-1} \times [0, 1] \times [0, 1] \to X.$ By the choice of $\delta$, we have $\Phi \left(S^{n-1} \times [0,1] \times [0,\delta) \right) \subset U.$ By Lemma~\ref{lemma: milnor lemma}, we have
		\[
		\mathbf{X}^{-1}\circ\Phi(p,t,s) = s \cdot \Psi(p,t,s), \quad(p,t,s) \in S^{n-1} \times[0,1] \times [0,\delta),
		\]
		where $\Psi : S^{n-1} \times [0,1] \times [0,\delta)\to \C^n$ is smooth with
		\[
		\Psi(p,t,0) = \pderiv[(\mathbf{X}^{-1}\circ\Phi)]{s}(p,t,0), \quad (p,t) \in S^{n-1} \times [0,1].
		\]
For $s \in [0,\delta),$ write
\[
\Psi_s := \Psi|_{S^{n-1}\times[0,1]\times\{s\}}.
\]
For $s \in (0,\delta),$ the map $\Psi_s$ is an immersion representing an $\Omega_{0,s}$-imaginary special Lagrangian cylinder. As $\Phi$ is regular, it follows from Definition~\ref{definition: regularity} and Definition~\ref{definition: interior regularity} \eqref{regular convergence to intersection point} that the map
$
\Psi_0
$
is an immersion nowhere tangent to the Euler vector field. Thus, by continuity
\[
Z_{0,0}':=\left[\Psi_0\right]
\]
is an $\Omega_{0,0}$-imaginary special Lagrangian cylinder.

By Lemma~\ref{lemma: Weinstein with boundary}, choose an immersed Weinstein neighborhood $(Y,\psi)$ of $Z_{0,0}'$ compatible with $\mathbf{X}^{-1}(\Lambda_0)$ and $\mathbf{X}^{-1}(\Lambda_1),$ where $Y \subset T^*L$ and $\psi : Y \to \C^n$ with $\psi|_L = \Psi_{0}.$ Let $\pi_L : T^*L \to L$ denote the projection. For $u\in\cob{2,\alpha}(L),$ let $\mathrm{Graph}(du) \subset T^*L$ denote the graph.
Let $0\in\mathcal{U}\subset \cob{2,\alpha}(L)$ be open such that for $u\in\mathcal{U}$ we have $\mathrm{Graph}(du) \subset Y.$ For $u \in \mathcal{U},$ let $j_u : L \to X$ be given by
\[
j_u = \psi \circ \left( \pi_L|_{\mathrm{Graph}(du)}\right)^{-1}.
\]
If necessary, diminish $\delta$ so that $\psi(Y) \subset V_s$ for $s \in [0,\delta).$
Define a differential operator
		\[
		\mathcal{F} : \mathcal{U} \times \mathcal{A} \times[0,\delta) \to C^\alpha(L), \quad (u, h, s) \mapsto *j_u^* \real \Omega_{h,s}.
		\]
For $(u, h, s) \in \mathcal{U} \times \mathcal{A} \times (0,\delta),$ the immersion $\varphi_{h,\chi}\circ \mathbf{X} \circ M_s \circ j_u$ represents a Lagrangian cylinder in $\mathcal{LC}\left(S^{n-1}; \Lambda_0, \Lambda_{1, h}\right)$, and this cylinder is imaginary special Lagrangian if and only if $\mathcal{F}(u, h,s) = 0.$

We claim $\mathcal{F}$ is continuously differentiable, and for a fixed $h \in \mathcal{A} \cap C^\infty(\Lambda_1),$ the map $(u,s) \mapsto \mathcal{F}(u,h,s)$ is smooth. Indeed, recalling Remark~\ref{remark: cutoff}, since the map
\begin{equation}\label{equation: calA}
\mathcal{A} \to C^{2,\alpha}(V,X), \qquad h \mapsto \varphi_{h,\chi}\circ\mathbf{X},
\end{equation}
is continuously differentiable, it follows from Lemma~\ref{lemma: Holder pullback} that the map
\begin{equation}\label{equation: vhc*o}
\mathcal{A} \to \Omega^n_{C^{1,\alpha}}(V), \qquad h \mapsto \Omega_h = \mathbf{X}^*\varphi^*_{h,\chi}\real \Omega,
\end{equation}
is continuously differentiable. Similarly, since the map
\[
\mathcal{U}\times [0,\delta) \to C^{1,\alpha}(L,V), \qquad (u,s) \mapsto M_s \circ j_u,
\]
is smooth, it follows from Lemma~\ref{lemma: Holder pullback} that the map
\[
\Upsilon : \mathcal{U} \times [0,\delta)\times\Omega^*_{C^{l,\alpha}}(V) \to \Omega^*_{C^{\alpha}}(L), \qquad (u,s,\eta) \mapsto j_u^*M_s^* \eta,
\]
is of regularity $C^l.$ We claim that the map
\[
\widetilde{\Upsilon}: \mathcal{U} \times [0,\delta)\times\Omega^n_{C^{l,\alpha}}(V) \to \Omega^n_{C^{\alpha}}(L), \qquad (u,s,\eta) \mapsto s^{-n} j_u^*M_s^* \eta,
\]
is also of regularity $C^{l}.$ Indeed, letting $I$ run over multi-indices of length $n,$ write
\[
\eta = \sum_I \eta_I dx^I.
\]
Then,
\[
s^{-n} M_s^* \eta = \sum_I (\eta_I\circ M_s) dx^I.
\]
So,
\[
s^{-n}j_u^* M_s^* \eta = \sum_I (\eta_I \circ M_s \circ j_u) j_u^* dx^I =  \sum_I \Upsilon(u,s,\eta_I) \Upsilon(u,1,dx^I).
\]
It follows that $\widetilde \Upsilon$ is $C^l$ regular as the sum of products of $C^l$ regular maps.
So, the map $\mathcal{F}$ is continuously differentiable as the composition of the two continuously differentiable maps~\eqref{equation: vhc*o} and $\widetilde \Upsilon.$  Moreover, for fixed $h \in \mathcal{A}\cap C^\infty(\Lambda_1),$ we have $\Omega_h = \mathbf{X}^*\varphi^*_{h,\chi}\real \Omega \in \Omega^*_{C^{\infty}}(V),$ so the map $(u,s) \mapsto \mathcal{F}(u,h,s)$ is smooth.

		Consider the linearization of $\mathcal{F},$
		\[
		d \mathcal{F}_{(0, 0,0)} : \cob{2, \alpha}(L) \times C^{3, \alpha}(\Lambda_1)\times \R \to C^\alpha(L).
		\]
		By Lemma~\ref{lemma: linearized operator} and Lemma~\ref{lemma: Delta rho is a cool operator},  the restriction of $d \mathcal{F}_{(0, 0,0)}$ to the subspace
		\[
		C^{2, \alpha}(L; \partial L) \times \{0\}\times\{0\} \subset \cob{2, \alpha}(L) \times C^{3, \alpha}(\Lambda_1) \times \R
		\]
		is an isomorphism onto $C^\alpha(L).$ Abbreviate $\mathbb{V} := C^{2, \alpha}(L; \partial L).$
		Let $\ell \subset \cob{2, \alpha}(L)$ be a one dimensional subspace consisting of smooth functions such that
		\[
		\cob{2, \alpha}(L) = \mathbb{V} \oplus \ell.
		\]
		By the implicit function theorem, there exist open neighborhoods
		\[
		0 \in \mathbb{V}_0 \subset \mathbb{V}, \quad 0 \in \ell_0 \subset \ell, \quad 0 \in \mathcal{A}_0 \subset \mathcal{A},
		\]
and $\epsilon \leq \delta$ such that for $(l, h,s) \in \ell_0 \times \mathcal{A}_0 \times [0,\epsilon)$ there exists a unique
\[
v = v(l, h,s) \in \mathbb{V}_0
\]
with
		\[
		\mathcal{F}(v + l, h,s) = 0.
		\]
		By elliptic regularity (e.g.\ \cite[Chapter 17]{gilbarg-trudinger}), the function $v(l, h,s)$ is smooth if $h$ is. Write
		\[
		\mathcal{W} := C^\infty(\Lambda_1) \cap \mathcal{A}_0.
		\]
Since $\mathcal{F}$ is continuously differentiable, it follows that
\begin{equation}\label{equation: v}
v : \ell_0 \times \mathcal{A}_0 \times [0,\epsilon) \to \mathbb{V}_0
\end{equation}
is continuously differentiable. Moreover, since for fixed $h \in C^{\infty}(\Lambda_1) \cap \mathcal{A}$ the map $(u,s) \mapsto \mathcal{F}(u,h,s)$ is smooth, it follows that for fixed $h \in \mathcal{W}$ the map
\[
\ell_0 \times [0,\epsilon) \to \mathbb{V}_0, \qquad (l,s) \mapsto v(l,h,s),
\]
is smooth.

Since $\Psi_0 = j_0$ and $(\Psi_s)_{s \in [0,\delta)}$ is a smooth family of immersions, after possibly shrinking $\epsilon,$ for each $s \in [0,\epsilon)$ there exists a unique $u_s \in \mathbb{V}_0 + \ell_0$ such that $j_{u_s}$ and $\Psi_s$ represent the same immersed $\Omega_{0,s}$-imaginary special Lagrangian cylinder. In particular, $\mathcal{F}(u_s,0,s) = 0.$
Decompose
\[
u_s = v_s + l_s, \qquad v_s \in \mathbb{V}_0, \quad l_s \in \ell_0.
\]
Since $v = v(l_s,0,s)$ is the unique solution to $\mathcal{F}(v + l_s ,0,s) = 0,$ we conclude that
\begin{equation}\label{equation:v_s}
v(l_s,0,s) = v_s.
\end{equation}

Define a family of smooth maps
		\begin{equation}
		\label{another big family of cylinders}
		\widetilde f_{s, h} := \varphi_{h, \chi}\circ \mathbf{X}\circ M_s\circ j_{l_s + v(l_s,h, s)}, \qquad (s, h) \in [0, \epsilon) \times \mathcal{W}.
		\end{equation}
For $(s, h) \in (0,\epsilon) \times \mathcal{W},$ the maps $\widetilde f_{s,h}$ are immersions representing imaginary special Lagrangian cylinders in  $\mathcal{SLC}\left(S^{n-1}; \Lambda_0, \Lambda_{1, h}\right).$ Moreover, the map
\[
[0,\epsilon) \times \mathcal{W} \to C^{\infty}(L,X), \qquad (s,h) \mapsto \widetilde f_{s,h},
\]
is continuous with respect to the $C^{3,\alpha}$ topology on $\mathcal{W}$ and the $C^{1,\alpha}$ topology on $C^\infty(L,X).$ Also, for fixed $h \in \mathcal{W},$ the map $s \mapsto \widetilde f_{s,h}$ is smooth.

Recall that for $s \in [0,\epsilon),$ the immersion $j_{u_s}$ represents the same immersed cylinder as the immersion $\Psi_s.$
Let $\zeta_s \in \diff(L)$ be the diffeomorphism such that
\[
j_{u_s}\circ \zeta_s = \Psi_s.
\]
Observe that the map
\[
[0, \epsilon) \to C^\infty(L,L), \qquad s \mapsto \zeta_s,
\]
is smooth. Moreover, since $\Psi_0 = j_0,$ we have
\begin{equation}\label{equation:zeta0}
\zeta_0 = \id_L.
\end{equation}
By equation~\eqref{equation:v_s} we have $u_s = l_s + v(l_s,0,s),$ so it follows from equation~\eqref{another big family of cylinders} that
\[
\widetilde f_{s,h}\circ \zeta_s = \Phi_s.
\]
Take
\[
f_{s,h} := \widetilde f_{s,h} \circ \zeta_s.
\]
Since the maps $(s,h) \mapsto \widetilde f_{s,h}$ and $s \mapsto \zeta_s$ are continuous in the topologies specified above, it follows that the map $(s,h) \mapsto f_{s,h}$ is continuous as desired. Moreover, for fixed $h \in \mathcal{W},$ the map $s \mapsto f_{s,h}$ is smooth.

Write $Z_{s,h} : = [f_{s,h}]$ for $(s,h) \in (0,\epsilon)\times \mathcal{W}.$ Properties~\ref{item: special end} and~\ref{item:h = 0 end} claimed in the proposition are immediate from the construction. We proceed with the proof of property~\ref{item: end continuity}. Indeed, by equation~\eqref{equation:zeta0}, we have
\[
\left.\pderiv[f_{s,h}]{s}\right|_{s=0} = d(\varphi_{h,\chi}\circ \mathbf{X})\circ j_{l_0 + v(l_0,h,0)}.
\]
So, the continuity of the map~\eqref{equation:pdsfsh0} follows from the continuity of the maps~\eqref{equation: calA} and~\eqref{equation: v}.

To establish property~\ref{item: regularly converges} claimed in the proposition, we argue as follows. The map $\Phi^h$ is smooth because the map $s \mapsto f_{s,h}$ is smooth. Condition~\ref{critical point} of Definition~\ref{definition: interior regularity}~\eqref{regular convergence to intersection point} is a consequence of the fact that $\varphi_{h,\chi} \circ \mathbf{X} \circ M_0$ is the constant map with image $q_{0,h}.$
Conditions~\ref{item:derivative Phi immersion} and~\ref{item:nowhere tangent Euler} of Definition~\ref{definition: interior regularity}~\eqref{regular convergence to intersection point} hold after possibly shrinking $\mathcal W$ by the following argument. Observe that
\[
\left.\pderiv[\Phi^0]{s}\right|_{s=0} = \left.\pderiv[\Phi]{s}\right|_{s=0},
\]
which satisfies Conditions~\ref{item:derivative Phi immersion} and~\ref{item:nowhere tangent Euler} of Definition~\ref{definition: interior regularity}~\eqref{regular convergence to intersection point} by assumption.
Since immersions, embeddings and transverse maps, are open in the $C^1$ topology, it suffices to show that the map
\begin{equation}\label{equation: calW}
\mathcal{W} \to C^{1,\alpha}(L,TX), \qquad h \mapsto \left.\pderiv[\Phi^h]{s}\right|_{s=0},
\end{equation}
is continuous. Since $\Phi(p,s) = f_{s,h}(p),$ this is equivalent to property~\ref{item: end continuity} of the proposition.
Condition~\ref{item:interior regular} of Definition~\ref{definition: interior regularity}~\eqref{regular convergence to intersection point} requires that $\Phi^h|_{L \times (0,\epsilon)}$ be an interior regular parameterization, which we prove as follows.  Conditions~\ref{item: rep} and~\ref{item: diffeo s} of Definition~\ref{definition: interior regularity}~\eqref{interior regularity first part} hold by construction. It remains to show that $\Phi^h|_{L \times (0,\epsilon)}$ is an immersion and $\Phi^h|_{\partial L \times (0,\epsilon)}$ is an embedding. Possibly after shrinking $\epsilon,$ this follows from Corollary~\ref{rem: easy regularity} for fixed $h.$ By the construction of $f_{s,h}$ and Lemma~\ref{lemma: immersion nowhere tangent to Euler}, we can choose $\epsilon$ uniformly in $h.$
	\end{proof}

\begin{dfn}\label{dfn:topology on geodesics}
Let $\mathcal{O}$ be a Hamiltonian isotopy class of positive Lagrangian spheres. For $\Lambda_0,\Lambda_1 \in \mathcal{O},$ we write $\Lambda_0 \pitchfork_2 \Lambda_1$ if $\Lambda_0$ and $\Lambda_1$ intersect transversally at exactly two points.  Let
\[
\mathfrak{Z}_\mathcal{O} := \left\{(\Lambda_0,\Lambda_1,\mathcal{Z})\left |
\begin{matrix}
\Lambda_i \in \mathcal{O},\; i = 0,1,\quad \Lambda_0 \pitchfork_2 \Lambda_1, \; \\
\mathcal{Z} \subset \mathcal{SLC}(\Lambda_0,\Lambda_1) \text{ a regular component}
\end{matrix}
\right.\right\}.
\]
We define the strong and weak $C^{k,\alpha}$ topologies on $\mathfrak{Z}_\mathcal{O}$ as follows.
For
\[
\mathcal{V} \subset C^\infty(S^{n-1}\times [0,1],X), \qquad \mathcal{U} \subset C^\infty(S^{n-1}\times [0,1],TX),
\]
open subsets in the $C^{k,\alpha}$ topology, write
\[
\mathcal{T}_{\mathcal{U},\mathcal{V}} := \left \{(\Lambda_0,\Lambda_1,\mathcal{Z}) \in \mathfrak{Z}_\mathcal{O} \left|
\begin{matrix}
\forall Z \in \mathcal{Z}, \; \exists f : S^{n-1}\times [0,1] \to X \text{ representing } Z \\
\text{such that } f \in \mathcal{V},\\
\forall E \text{ an end of }\mathcal{Z},\; \exists \Phi : [0,\epsilon) \to X \text{ a regular}\\
\text{parameterization of $E$ such that }\left.\pderiv[\Phi]{s}\right|_{s = 0} \in \mathcal{U}
\end{matrix}
\right.\right\}
\]
and
\[
\mathcal{X}_{\mathcal{V}} = \left \{(\Lambda_0,\Lambda_1,\mathcal{Z}) \in \mathfrak{Z}_\mathcal{O} \left|
\begin{matrix}
\exists Z \in \mathcal{Z}, \; \exists f : S^{n-1}\times [0,1] \to X \text{ representing } Z \\
\text{such that } f \in \mathcal{V}
\end{matrix}
\right.\right\}.
\]
Then, a basis for the strong $C^{k,\alpha}$ topology on $\mathfrak{Z}_\mathcal{O}$ is given by sets of the form $\mathcal{T}_{\mathcal{U},\mathcal{V}}$ and a sub-basis for the weak $C^{k,\alpha}$ topology on $\mathfrak{Z}_\mathcal{O}$ is given by sets of the form $\mathcal{X}_\mathcal{V}.$  Let
\[
\mathfrak{G}_\mathcal{O} : = \{(\Lambda_t)_{t \in [0,1]} | (\Lambda_t)_{t \in [0,1]} \text{ is a geodesic with } \Lambda_0,\Lambda_1 \in \mathcal{O}, \quad \Lambda_0 \pitchfork_2 \Lambda_1\}
\]
denote the space of geodesics with endpoints in $\mathcal{O}$ intersecting transversally at two points.
By Theorem~\ref{theorem: geodesic-cylinder correspondence}, the cylindrical transform gives a bijection
\[
\mathfrak{G}_\mathcal{O} \simeq \mathfrak{Z}_\mathcal{O}.
\]
So, the strong and weak $C^{k,\alpha}$ topologies on $\mathfrak{Z}_\mathcal{O}$ give rise to topologies on $\mathfrak{G}_\mathcal{O},$ which we also call the strong and weak $C^{k,\alpha}$ topologies respectively.
\end{dfn}

	\begin{proof}[Proof of Theorem~\ref{theorem: perturbation of geodesic}]
By Propositions~\ref{proposition: interior big family of cylinders} and~\ref{proposition: end big family of cylinders} and the compactness of $[0,1],$ we find a finite cover of $[0,1]$ by relatively open intervals $I_j,\, j = 0,\ldots,N,$ subsets $0 \in \mathcal{W}^j \subset C^\infty(\Lambda_1)$ open in the $C^{3,\alpha}$ topology, and families of smooth immersions
\[
f_{s,h}^j : L \to X, \qquad (s,h) \in I_j \times \mathcal{W}^j,
\]
continuous with respect to the $C^{3,\alpha}$ topology on $\mathcal{W}^j$ and the $C^{1,\alpha}$ topology on $C^\infty(L,X)$ that satisfy properties~\ref{item: special}-\ref{item: interior regular} of Proposition~\ref{proposition: interior big family of cylinders} if $0,1 \notin I_j$ and properties~\ref{item: special end}-\ref{item: end continuity} of Proposition~\ref{proposition: end big family of cylinders} otherwise. Moreover, if $0,1 \notin I_j,$ we have a $C^{1,\alpha}$ open set $\mathcal{V}^j \in C^\infty(L,X)$ with
\[
f_{s,h}^j \in \mathcal V^j, \qquad (s,h) \in I_j \times \mathcal{W}^j,
\]
such that for $h \in \mathcal{W}^j,$
\begin{equation}\label{equation: implication}
f \in \mathcal V^j, \quad
[f] \in \mathcal{SLC}(S^{n-1};\Lambda_0,\Lambda_{1,h}) \quad \Rightarrow \quad \exists s \in I_j, \quad [f] = [f_{s,h}^j].
\end{equation}
After possibly shrinking and relabeling the intervals $I_j,$ we can assume that $0 \in I_0,\, 1 \in I_N,$ and $I_j \cap I_k = \emptyset$ unless $k = j \pm 1.$ Moreover, we can assume that $N \geq 2.$

Let $\mathcal{W} = \cap_{j = 0}^N \mathcal{W}^j.$ For $h \in \mathcal{W},$ let
\[
U^h_j \subset \mathcal{SLC}(S^{n-1};\Lambda_0,\Lambda_{1,h})
\]
be the interval consisting of the cylinders $Z_{s,h}^j = [f_{s,h}^j]$ for $s \in I_j.$
Choose
\[
s_j \in I_j \cap I_{j+1}, \qquad j = 0,\ldots, N-1.
\]
Possibly shrinking~$\mathcal{W},$ we may assume by continuity that
\[
f^j_{s_j,h} \in \mathcal{V}^{j+1}, \qquad h \in \mathcal{W}, \quad j = 0,\ldots, N-2,
\]
and
\[
f^N_{s_{N-1},h} \in \mathcal{V}^{N-1}, \qquad h \in \mathcal{W}.
\]
It follows from implication~\eqref{equation: implication} that for $h \in \mathcal{W},$ we have
\[
U^h_j \cap U^h_{j+1} \neq \emptyset, \qquad j = 0,\ldots,N-1.
\]
Thus, the sets $\{U^h_j\}_{j=0}^N$ cover an open interval $\mathcal{Z}^h \subset \mathcal{SLC}(S^{n-1};\Lambda_0,\Lambda_{1,h}).$ For $j = 1,\ldots,N-1,$ the interval $U^h_j$ is interior regular by property~\ref{item: interior regular} of Proposition~\ref{proposition: interior big family of cylinders}. For $j = 0$ (resp. $N$) the interval $U^h_j$ converges regularly to $q_{0,h}$ (resp.~$q_{1,h}$) by property~\ref{item: regularly converges} of Proposition~\ref{proposition: end big family of cylinders}. So, the interval $\mathcal{Z}^h$ is a regular connected component by Remark~\ref{remark: regular connected component}\ref{equivalent definition of regularity}.  Take $\mathcal Y$ the $C^{2,\alpha}$ open neighborhood of $\Lambda_1$ in $\mathcal{O}$ corresponding to~$\mathcal{W}$ and take $\mathcal{X} := \mathcal{X}_{\mathcal{V}^1}.$ Let $(\Lambda_t^h)_{t \in [0,1]}$ be the geodesic corresponding to $\mathcal{Z}^h$ by Theorem~\ref{theorem: geodesic-cylinder correspondence}. By construction, $(\Lambda_t^h)_{t \in [0,1]} \in \mathcal{X}.$ For $h \in \mathcal{W}$ suppose $(\Lambda_t')_{t \in [0,1]}$ is a geodesic in $\mathcal{X}$ with $\Lambda_0' = \Lambda_0$ and $\Lambda_1' = \Lambda_{1,h}.$  Let $\mathcal{Z}' \subset \mathcal{SLC}(S^{n-1};\Lambda_0,\Lambda_{1,h})$ denote its cylindrical transform. It follows from implication~\eqref{equation: implication} that $\mathcal{Z}' \cap \mathcal{Z}^h \neq \emptyset$ and thus $\mathcal{Z}' = \mathcal{Z}^h.$ So, Theorem~\ref{theorem: geodesic-cylinder correspondence} gives $(\Lambda_t')_{t} = (\Lambda_t^h)_t.$ We have proven the existence and uniqueness part of Theorem~\ref{theorem: perturbation of geodesic}. The continuity claim follows from the continuity of the families $f_{s,h}^j$ and property~\ref{item: end continuity} of Proposition~\ref{proposition: end big family of cylinders}.
	\end{proof}

	\bibliographystyle{amsabbrvcnobysame}
	\bibliography{bibli}

\end{document}