text:
\section{Background and Geometric Framework}\label{SEC_background}
Here we will recall some standard background material, and then we
  provide a geometric framework very specific to the problem of study.
We begin with the notions of an almost complex manifold and
  pseudoholomorphic maps.

\begin{definition}[almost complex structures; compatible and tame]
  \label{DEF_almost_complex_structures}
  \hfill\\
Let \(W\) be a smooth manifold, not necessarily closed, possibly with
  boundary, and let \(J\in \Gamma(\rm{End}(TW))\) be a smooth
  section for which \(J\circ J=-\mathds{1}\).
We call \(J\) an almost complex structure for \(W\), and the pair
  \((W, J)\) an almost complex manifold.
In the case that \(\Omega\) is a symplectic form on \(W\), we say \(J\) is
  \(\Omega\)-compatible provided that \(g(v,w):=\Omega(v, Jw)\) is a
  Riemannian metric.
Under the weaker assumption that \(\Omega(v, Jv) \geq 0\) for all \(v\in
  TW\), and with equality if and only if \(v=0\), we say instead that \(J\)
  is only \(\Omega\)-tame.

\end{definition}

\begin{definition}[pseudoholomorphic map]
  \label{DEF_pseudoholomorphic_map}
  \hfill\\
Let \((S, j)\) and \((W, J)\) be smooth almost complex manifolds with
  \({\rm dim}(S)=2\), each possibly with boundary.
A \(C^\infty\)-smooth map \(u:S\to W\) is said to be
  \emph{pseudoholomorphic} provided \(J\cdot Tu = Tu\cdot j\).
That is, the tangent map of \(u\) intertwines the almost complex
  structures on domain and target.
Unless otherwise specified, we allow \(S\) to be disconnected.
We say such a map is \emph{proper} provided the pre-image of any compact
  set is compact.
\end{definition}

It is worth noting that constant maps are always pseudoholomorphic.
We also note that we will allow the domains of our pseudoholomorphic maps
  to be disconnected, which in conjunction with the fact that constant
  maps are always pseudoholomorphic allows for the possibility that any
  given map may have many constant components -- perhaps infinitely many.
At present we allow this, while noting that second countability of the
  domain Riemann surfaces forces any given map to have at most countably
  many constant components.

Because the domains of our pseudoholomorphic maps are Riemann surfaces
  which need not be compact, we make the notion of \emph{genus} precise with
  the following definition.

\begin{definition}[Genus]
  \label{DEF_genus}
Let \(S\) be a two-dimensional oriented manifold, possibly with boundary,
  with at most countably many connected components, and with the property
  that each connected component of \(\partial S\) is compact.
Then
\begin{enumerate}
    \item If \(S\) is closed and connected, then define \({\rm
      Genus}(S):=g\) where \(\chi(S)=2-2g\) is the Euler characteristic
      of \(S\).
    \item If \(S\) is compact and connected with \(n\) boundary
      components, define \(\widetilde{S}=\big(S \sqcup (\sqcup_{k=1}^n
      D^2)\big)/\sim\) to be the closed surface capped off by \(n\) disks,
      and define \({\rm Genus}(S):={\rm Genus}(\widetilde{S})\).
    \item If \(S\) is compact (possibly with boundary), then \({\rm
      Genus}(S)\) is defined to be the sum of the genera of each connected
      component.
    \item If \(S\) is not compact, then \({\rm Genus}(S)\)
      is defined by taking any nested sequence \(S_1\subset S_2\subset
      S_3\subset\cdots\) of compact surfaces (possibly with boundary) such
      that \(S_k\subset S\) for all \(k\in \mathbb{N}\) and such that \(S
      =\cup_k S_k\); then we define \({\rm Genus}(S):=\lim_{k\to \infty}
      {\rm Genus}(S_k)\).
  \end{enumerate}
\end{definition}

We now turn our attention to geometric structures more specific to the
  proof of Theorem \ref{THM_main_maybe}.
The first key idea is what we call a \emph{framed Hamiltonian energy
  pile} (see Definition \ref{DEF_framed_hamiltonian_energy_pile}) which is
  essentially the neighborhood of a compact energy level with enough
  structure to regard it as something like a family of framed Hamiltonian
  manifolds.
This relationship is made precise with Lemma
  \ref{LEM_energy_levels_are_framed_Hamiltonian_manifolds}.
Also important is Lemma
  \ref{LEM_localization_to_framed_Hamiltonian_energy_pile}, which is a means
  to identify the neighborhood of a compact energy level in a general
  symplectic manifold, with the structure of a framed Hamiltonian energy
  pile, thereby localizing the almost existence problem.

\begin{definition}[framed Hamiltonian energy pile]
  \label{DEF_framed_hamiltonian_energy_pile}
  \hfill\\
Let \(\mathcal{I}_\epsilon\) be the interval \((-\epsilon, \epsilon)\)
  equipped with the coordinate \(s\), and let \(M\) be a closed
  odd-dimensional manifold.
Assume \(\mathcal{I}_\epsilon \times M \) is equipped with the symplectic
  form \(\Omega\).
Let \(H\colon \mathcal{I}_\epsilon \times M \to \mathbb{R}\) be the smooth
  Hamiltonian \(H(s,p)=s\), and let \(X_H\) be the associated Hamiltonian
  vector field determined by \(i_{X_H}\Omega = - d H\)
and let  \(\hat{\lambda}\) be a one-form on \(\mathcal{I}_\epsilon \times M\)
  which satisfies the following three conditions:
\begin{enumerate}[(H1)]

  \item
  \(\hat{\lambda}(\partial_s) = 0\),

  \item
  \(\mathcal{L}_{\partial_s} \hat{\lambda}=0\) where \(\mathcal{L}\) is
  the Lie derivative,

  \item \(\hat{\lambda}(X_H) >0\).

  \end{enumerate}
We call the triple \((\mathcal{I}_\epsilon \times M,\Omega, \hat{\lambda})
  \) a framed Hamiltonian energy pile.
\end{definition}
If we consider local coordinates $x_1,...,x_N$ on $M$, $N=2n+1$, and the
  coordinate $s$ on $\mathcal{I}_{\varepsilon}$ then $\hat{\lambda}$ can be
  written at the point $(s,x)$ as
\begin{eqnarray}\label{Hope}
\hat{\lambda}(s,x)= \sum_{i=1}^N a_i(x)dx_i.
\end{eqnarray}
due to the imposed conditions (H1) and (H2).

We shall show in Lemma
  \ref{LEM_localization_to_framed_Hamiltonian_energy_pile} how framed
  Hamiltonian energy piles arise near a compact regular energy surface.
At first, however, we begin by deriving a few geometric structures which
  arise as an immediate consequence of having a framed Hamiltonian energy
  pile.
We illuminate them at present.
Define the two-plane distribution \(\hat{\rho}\) on
  $\mathcal{I}_{\varepsilon}\times M$ by
  \begin{align} \label{EQQQ_xi}
    \hat{\rho} = {\rm Span}(\partial_s , X_H),
    \end{align}
  and define the codimension-two plane distribution \(\xi\) on
  $\mathcal{I}_{\varepsilon}\times M$ by
  \begin{align}\label{EQ_xi}
    \xi = {\rm ker}\; (ds \wedge \hat{\lambda})= ({\rm ker}\; ds) \cap
    ({\rm ker}\; \hat{\lambda}).
    \end{align}
We observe that the vector bundle \(\xi\rightarrow
  \mathcal{I}_{\varepsilon}\times M\) is ${\mathbb R}$-invariant in the
  following sense.
Given $s_0,s_1\in\mathcal{I}_{\varepsilon}$,  the map
  $(h+s_0,m)\rightarrow (h+s_1,m)$, which is defined for small \(\vert
  h\vert \), pushes forward (the obvious restrictions of)  $\xi$ to $\xi$.
Moreover, we have  the splitting
\[T(\mathcal{I}_\epsilon \times M) =
  \hat{\rho}\oplus \xi,
  \]
   and the associated projections:
  \begin{align*}
    \pi^{\hat{\rho}}\colon \hat{\rho}\oplus \xi \to \hat{\rho}
    \qquad\text{and}\qquad
    \pi^{\xi}\colon \hat{\rho}\oplus \xi \to \xi.
    \end{align*}
Define the two-form \(\hat{\omega}\) by
\begin{align}   \label{XX_decomp}
  \hat{\omega} = \Omega\circ (\pi^\xi \times \pi^\xi).
  \end{align}
Here and throughout, we will also employ the following notation,
  \begin{align*}
    \hat{X}:= \frac{X_H}{\hat{\lambda}(X_H)}.
    \end{align*}

\begin{lemma}[localization to a framed Hamiltonian energy pile]
  \label{LEM_localization_to_framed_Hamiltonian_energy_pile}
  \hfill\\
Let \((\widetilde{W},\widetilde{\Omega})\) be a symplectic manifold
  without boundary, and let \(\widetilde{H}\colon \widetilde{W} \to
  \mathbb{R}\) be a \(C^\infty\)-smooth proper Hamiltonian.
Suppose further that zero is a regular value of
  \(\widetilde{H}\).
Then there exists an \(\epsilon>0\), a framed Hamiltonian energy pile
  \((\mathcal{I}_\epsilon\times M, \Omega, \hat{\lambda})\), and a
  \(C^\infty\)-smooth diffeomorphism
  \begin{align*}
    \Phi\colon \mathcal{I}_\epsilon \times M \to \{|\widetilde{H}| <
    \epsilon\} \subset \widetilde{W}
    \end{align*}
  for which
  \begin{align*}
    (\widetilde{H}\circ \Phi)(s,p) = s\qquad\text{and}\qquad
    \Phi^*\widetilde{\Omega} = \Omega.
    \end{align*}
Additionally, the framed Hamiltonian energy pile can be found such that
  along the energy level \(\{0\}\times M\), the following are true
\begin{enumerate}
  \item
  \(\hat{\lambda}(X_H)=1.\)
  \item
  The vector bundles \(\hat{\rho}\) and \(\xi\) are symplectic complements
    over $\{0\}\times M$.
  That is, for each \(q\in \{0\}\times M\), each \(v_q \in \hat{\rho}_q\),
    and each \(w_q\in \xi_q\), we have \(\Omega(v_q, w_q)=0\).
  \end{enumerate}
\end{lemma}

\begin{proof}
We begin by fixing an auxiliary \(\widetilde{\Omega}\)-compatible almost
  complex structure on \(\widetilde{W}\); denote it \(\widetilde{J}\).
This gives rise to the Riemannian metric \(g_{\widetilde{J}} =
  \widetilde{\Omega}\circ ({\rm Id}\times \widetilde{J})\).
Using this metric to compute the gradient of \(\widetilde{H}\), we let
  \(\varphi^t\) be the time \(t\) flow associated to the vector field
  \(\|\nabla \widetilde{H}\|_{g_{\widetilde{J}}}^{-2}\nabla
  \widetilde{H}\) in a neighborhood \(\{|\widetilde{H}|<\epsilon\}\) for
  some small \(\epsilon>0\) yet to be determined.
Defininig \(M:=\widetilde{H}^{-1}(0)\), it follows that for all
  sufficiently small \(\epsilon>0\)
  \begin{align*}
    &\Phi\colon \mathcal{I}_\epsilon\times M \to \{|\widetilde{H}|
    <\epsilon\} \subset \widetilde{W}\\
    &\Phi(s,p) = \varphi^s(p)
    \end{align*}
  is a diffeomorphism, and \(H:= \widetilde{H}\circ \Phi\)
  satisfies \(H(s,p) = s\).

By construction, the vector field \(X_{\widetilde{H}}\) is tangent to level
  sets of \(\widetilde{H}\), and thus along \(\{\widetilde{H}=0\}\) we can
  define \(\tilde{\lambda}\) to be the one form uniquely determined by the
  conditions
  \begin{align*}
  \tilde{\lambda}(X_{\widetilde{H}})=1 \qquad\text{and}\qquad
    {\rm ker}\; \tilde{\lambda} = \mathbb{R} \nabla \widetilde{H} \, \oplus
    \big({\rm Span}(X_{\widetilde{H}}, \nabla\widetilde{H})\big)^\bot
    \end{align*}
  where \(\bot\) denotes the \(\widetilde{\Omega}\)-symplectic complement.
We then define
  \begin{align*}
    \hat{\lambda} = {\rm pr}_2^* \tilde{\lambda}
    \end{align*}
  where \({\rm pr}_2\colon \mathcal{I}_\epsilon \times M\to M\) is the
  canonical projection.
It is straightforward to verify that \(\hat{\lambda}(\partial_s)=0\), and
  that \(\mathcal{L}_{\partial_s} \hat{\lambda}=0\).
Because \(\hat{\lambda}(X_H)\big|_{\{0\}\times M} = 1\), it then follows
  that \(\hat{\lambda}(X_H)>0\) for all sufficiently small \(\epsilon>0\).
By construction then, \((W, \Omega, \hat{\lambda})\) is a framed
  Hamiltonian energy pile provided that \(\epsilon>0\) is sufficiently
  small.
Moreover by construction, along \(\{0\}\times M\) it is both the case that
  \(\hat{\lambda}(X_H)=1\) and the case \(\hat{\rho}\) and \(\xi\) are
  \(\Omega\)-symplectic complements.
This completes the proof of Lemma
  \ref{LEM_localization_to_framed_Hamiltonian_energy_pile}.
\end{proof}

We make the following important observation.

\begin{lemma}[energy levels are framed Hamiltonian manifolds]
  \label{LEM_energy_levels_are_framed_Hamiltonian_manifolds}
  \hfill\\
Let \(\big(\mathcal{I}_\epsilon\times M, \Omega, \hat{\lambda}\big)\) be a
  framed Hamiltonian energy pile, and let \(\hat{\rho}\), \(\xi\), and
  \(\hat{\omega}\) be the associated structures defined above, see
  (\ref{EQQQ_xi}), (\ref{EQ_xi}), and (\ref{XX_decomp}).
Then for each \(s_0\in \mathcal{I}_\epsilon\), the restriction of
  \(\hat{\lambda}\) and \(\hat{\omega}\) to the energy level \(\{s_0\}\times
  M\) is a framed Hamiltonian structure for this energy level.
\end{lemma}

\begin{proof}
We begin by recalling that a framed Hamiltonian structure \(\eta=(\lambda,
  \omega)\) for an odd-dimensional manifold \(M\) is a one-form
  \(\lambda\) and a closed two-form \(\omega\) for which  \(\lambda
  \wedge \omega \wedge \cdots \wedge \omega =: {\rm vol}_M\) is a volume form.
To show this latter non-degeneracy condition is satisfied on our energy
  levels, first note that on \(\mathcal{I}_\epsilon \times M\) we have that
  \(T(\mathcal{I}_\epsilon\times M) = \hat{\rho}\oplus \xi\)  is a
  splitting and \(\Omega\) is non-degenerate on each of \(\hat{\rho}\) and
  \(\xi\).
Moreover, by construction
  \begin{align*}
    \xi = {\rm ker}\; (ds\wedge \hat{\lambda})\qquad\text{and}\qquad
    \hat{\rho} = {\rm ker}\; (\hat{\omega}) = { \rm Span}(\partial_s,
    X_H),
    \end{align*}
  and \(\hat{\omega}=\Omega\circ(\pi^\xi \times \pi^\xi)\) so that
  \begin{align*}
    ds\wedge \hat{\lambda} \wedge \hat{\omega}^n >0 \qquad\text{on
    }\mathcal{I}_\epsilon\times M;
    \end{align*}
    here \({\rm dim} (M) = 2n +1\).
It immediately follows that
  \begin{align*}
    \hat{\lambda} \wedge \hat{\omega}^n >0 \qquad\text{on } \{s_0\}\times
    M \text{ for each }s_0\in \mathcal{I}_\epsilon.
    \end{align*}
This establishes the non-degeneracy condition.

Next, we establish that the restriction of \(\hat{\omega}\) to each energy
  level \(\{s_0\}\times M\) is closed.
To that end, let \(v,w\in T(\{s_0\}\times M)\).
Then there exist \(a,b\in \mathbb{R}\) and \(v^\xi, w^\xi \in \xi\) such
  that
  \begin{align*}
    v = a X_H + v^\xi\qquad\text{and}\qquad w = b X_H + v^\xi,
    \end{align*}
  and then
  \begin{align*}
    \Omega(v, w) &= \Omega\big(a X_H + v^\xi,b X_H + w^\xi\big)
    \\
    &=
    \Omega(v^\xi,w^\xi)
    \\
    &=
    \Omega\big(\pi^\xi(v),\pi^\xi(w)\big)
    \\
    &=
    \hat{\omega}(v,w)
    \end{align*}
  which shows that for the inclusion \(\iota\colon \{s_0\}\times M
  \hookrightarrow \mathcal{I}_\epsilon \times M \) we have
  \(\hat{\omega}\big|_{\{s_0\}\times M}
  = \iota^*\Omega\), and thus the restriction of \(\hat{\omega}\) to
  \(\{s_0\}\times M\) is closed.
This completes the proof of Lemma
  \ref{LEM_energy_levels_are_framed_Hamiltonian_manifolds}.
\end{proof}

For the following considerations of adapted almost complex structures we
  summarize the salient points of the previous discussion.
By definition a framed Hamiltonian manifold \((\check{M}, \check{\lambda},
  \check{\omega})\) has an associated Reeb vector field
  \(\check{X}\) uniquely determined by the equations
  \begin{align*}
    \check{\lambda}(\check{X}) = 1\qquad\text{and}\qquad
    i_{\check{X}}\check{\omega} =0.
    \end{align*}
Given a framed Hamiltonian energy pile $(\mathcal{I}_{\varepsilon}\times
  M,\Omega,\hat{\lambda})$ we can consider the Hamiltonian $H$ given by
  $H(s,m)=s$, which has Hamiltonian vector field $X_H$ is defined by
  $i_X\Omega=-dH$.
We normalize it by setting
  \[
  \widehat{X}=X_H/\hat{\lambda}(X_H).
  \]
Then we observe that $\widehat{X}$ satisfies
  \begin{align*}
    \hat{\lambda}(\widehat{X}) = 1, \qquad ds(\widehat{X})=0,
    \qquad\text{and}\qquad i_{\widehat{X}}\hat{\omega}=0.
    \end{align*}
Consequently, on each energy level \(\{s_0\}\times M\) the Reeb
  vector field associated to the framed Hamiltonian structure
  \((\hat{\lambda}, \hat{\omega})\big|_{\{s_0\}\times M}\) is the
  restriction of \(\widehat{X}\).

We now turn to equipping framed
  Hamiltonian energy piles with a certain class of almost complex
  structures, which we now define.

\begin{definition}[weakly adapted almost complex structure]
  \label{DEF_weakly_adapted_J}
  \hfill\\
Let \(\big(\mathcal{I}_\epsilon\times M, \Omega, \hat{\lambda}\big)\) be a
  framed Hamiltonian energy pile, and let \(\hat{\rho}\), \(\xi\), and
  \(\hat{\omega}\) be the associated structures defined above.
Let \(J\) be an almost complex structure on \(\mathcal{I}_\epsilon\times
  M\) which satisfies the following conditions.
\begin{enumerate}[(J1)]

  \item
  It preserves the splitting \(\hat{\rho}\oplus \xi\); that is,
  \begin{align*}
    J: \hat{\rho}\to \hat{\rho} ,
    \qquad\text{and}\qquad
    J\colon \xi \to \xi.
    \end{align*}

  \item There exists a smooth function of the form
  \begin{align*}
    &\phi\colon \mathcal{I}_\epsilon \times M\to (0, 1]
    \\
    &\phi(s,p) = \phi(s)
    \end{align*}
  with the property that
  \begin{align*}
    \phi \cdot J \partial_s =  \widehat{X}\qquad\text{where}\qquad
    \widehat{X}=\frac{X_H}{\hat{\lambda}(X_H)}.
    \end{align*}

  \item
  The following is a Riemannian metric
  \begin{align}\label{EQ_Riemannian_metric}
    g_J(v,w) =(ds \wedge \hat{\lambda} + \hat{\omega})(v, J w).
    \end{align}

  \end{enumerate}
In this case we say \(J\) is an almost complex structure \textit{weakly adapted} to
  the framed Hamiltonian energy pile \((\mathcal{I}_\epsilon \times M,
  \Omega, \hat{\lambda})\).
\end{definition}

We pause for a moment to discuss the manner in which these almost complex
  structures are weakly adapted.
Consider a fixed framed Hamiltonian energy pile
  $(\mathcal{I}_{\varepsilon}\times M,\Omega,\hat{\lambda})$.
Associated to this data we have the plane field bundles
\begin{align*}
  \xi = {\rm ker}\; (ds\wedge \hat{\lambda})\qquad\text{and}\qquad
  \hat{\rho} = {\rm ker}\; (\hat{\omega}) = { \rm Span}(\partial_s,
  X_H)={ \rm Span}(\partial_s, \widehat{X}),
  \end{align*}
   with the two-form \(\hat{\omega}=\Omega\circ(\pi^\xi \times \pi^\xi)\).
 Here we recall that
  \[
  T(\mathcal{I}_{\varepsilon}\times M)= \hat{\rho}\oplus \xi.
  \]
Following motivation from Symplectic Field Theory (SFT) as proposed
  in \cite{EGH}, there is a somewhat natural choice of class of almost
  complex structures here, namely those of the form
  \begin{align*}
    J\partial_s = \widehat{X} \qquad\text{and}\qquad J\colon \xi \to \xi
    \end{align*}
  where \(\hat{\omega}\circ ({\rm Id}\times J)\big|_\xi\) is symmetric and
  positive definite.
There are two issues of note, which make such a choice somewhat
  different from the weakly adapted almost complex structures defined above.
The first such difference is that in general \(\hat{\omega}\) is not
  translation invariant,
Put another way, this means that \(\widehat{X}\) has \(s\) dependency, or
  rather that \(\widehat{X}(s,p) \neq \widehat{X}(s',p)\) in general for
  \(s\neq s'\).
Because \(\widehat{X}\) is not translation invariant, and in fact in
  general the line bundle \(\mathbb{R} \widehat{X}\subset \hat{\rho}\)
  will fail to be translation invariant, it follows that any almost complex
  structure which preserves \(\hat{\rho}\) will also fail to be translation
  invariant.

It is worth recalling that the framework of SFT
  typically requires a translation invariant almost complex structure (in
  cylindrical homogeneous regions of symplectizations) or else is only
  required to be symplectically tame (in the inhomogenous or cobordant
  regions).
In contrast, Definition \ref{DEF_weakly_adapted_J}  requires the almost
  complex structure to be carefully ``adapted'' in an inhomogeneous region.
Perhaps a more important feature of Definition \ref{DEF_weakly_adapted_J} is the
  ``weakness'' condition, which allows the almost complex structure to
  have the form \(\phi\cdot J\partial_s = \widehat{X}\).
This is best understood as a partial degeneration of the almost complex
  structure which can be undone by a finite amount of
  ``stretching the neck'' along the first factor of
  \(\mathcal{I}_\epsilon\times M\).
More precisely, we define the embedding.
  \begin{align}
    &\Psi \colon\mathcal{I}_\epsilon \times M  \to \mathbb{R}\times M
    \label{EQ_Psi_new}
    \\
    &\Psi(s, p) = \big(\psi(s), p\big)\notag
    \end{align}
  where
  \begin{align}\label{EQ_psi_new}
    \psi(s) = \int_0^s \frac{1}{\phi(t)} dt.
    \end{align}
To see the utility of the map \(\Psi\), we consider an example, which
  already explains its key features with respect to the other relevant data.
Specifically, we consider the case that \(\phi(s) = \delta\) where \(\delta\)
  is some very small positive number, and we assume that \(J_\xi :=
  J\big|_\xi \colon \xi \to \xi\) is translation invariant in the \(s\)
  direction; that is, \(\mathcal{L}_{\partial_s} J_\xi = 0\).
For \(v^\xi\in \xi\) we then have
  \begin{align*}
    J\big(a \partial_s + b \widehat{X} + v^\xi \big)
    =
    -\delta b \partial_s + \frac{a}{\delta} \widehat{X} +
    J_\xi v^\xi.
    \end{align*}
We also see that in this case
  \begin{align*}
    &\Psi\colon (-\epsilon, \epsilon) \times M \to
    \Big(-\frac{\epsilon}{\delta}, \frac{\epsilon}{\delta}\Big)\times M\ \
    (\textrm{diffeomorphism})
    \\
    &\Psi(s, p)=(s\delta^{-1} , p)
    \end{align*}
  and thus
  \begin{align*}
    (\Psi_*J)\big(a \partial_{\check{s}} + b (\Psi_*\widehat{X}) + v^\xi
    \big)
    =
    - b \partial_{\check{s}} + a (\Psi_*\widehat{X}) +
    J_\xi v^\xi,
    \end{align*}
  where \(\check{s}\) is the coordinate on \((-\epsilon \delta^{-1},
  \epsilon \delta^{-1})\).
Note that if we abuse notation by identifying \(\{s_0\}\times M\) with
  \(M\), then
  \begin{align*}
    (\Psi_* \widehat{X})(\check{s},p) = \widehat{X}(\delta \check{s}, p).
    \end{align*}
Similarly, if \(J_\xi = J\big|_\xi\) has \(s\) dependence, then by abusing
  notation again, we have
  \begin{align*}
    (\Psi_* J_\xi)(\check{s} , p) = J_\xi(\delta \check{s}, p).
    \end{align*}
Put another way, because the tangent bundle splits as \(\hat{\rho}\oplus
  \xi\), and because the almost complex structures preserve this
  splitting, and the stretching direction is contained in \(\hat{\rho}\),
  it follows that this ``neck stretching'' does not degenerate the
  restriction \(J_\xi\), and it is chosen to stretch the compressed \(J\)
  into something more well behaved.
For example, \((\Psi_*J)\partial_{\check{s}}= \Psi_*\widehat{X}\).
It is left to the reader to observe that this qualitative behavior is
  preserved when one changes from the example case of \(\phi(s)=\delta\)
  to the more general case that \(\phi\colon \mathcal{I}_\epsilon\times
  M\to (0, 1]\) with \(\phi(s,p)=\phi(s)\).

We will elaborate further on this stretching construction below, however
  it will be helpful to put it in the context of ``adiabatic degeneration'',
  which amounts to fully neck-stretching along a continuum of energy
  levels simultaneously.
We make this precise with the following.

\begin{definition}[adiabatically degenerating almost complex structures]
  \label{DEF_adiabatically_degenerating_almost_complex_structures}
  \hfill\\
Let \(\big(\mathcal{I}_\epsilon\times M, \Omega, \hat{\lambda}\big)\) be a
  framed Hamiltonian energy pile, and let \(\hat{\rho}\), \(\xi\), and
  \(\hat{\omega}\) be the associated structures defined above, and let
  \(\widehat{X}=X_H/\hat{\lambda}(X_H)\) as above.
Let \(\mathcal{J}=\{J_k\}_{k\in \mathbb{N}}\) be a sequence of weakly
  adapted almost complex structures in the sense of Definition
  \ref{DEF_weakly_adapted_J}, with \(\phi_k \cdot J_k \partial_s =
  \widehat{X}\), which also satisfies the following conditions.
\begin{enumerate}[($\mathcal{J}$1)]

  \item
  \emph{({Adiabatic Degeneration})}
  We require that \(\phi_k\to 0\) in \(C^\infty\) as \(k\to
  \infty\).

  \item
  \emph{({Symplectic Area Controls Metric Area})}
  For each \(k\in \{1, 2, 3, \ldots\}\), and each  \(v\in
  T(\mathcal{I}_\epsilon\times M)\), we require that
  \begin{align*}
    (ds \wedge \hat{\lambda} + \hat{\omega})(v, J_kv )\leq 2 \Omega(v ,
    J_k v)
    \end{align*}

  \item
  \emph{({Geometrically Bounded})}
  There exists a sequence \(\{C_n\}_{n=0}^\infty\) of positive
  constants and an auxiliary translation invariant metric \(\tilde{g}\) on
  \(\mathbb{R}\times M\) for which
  \begin{align*}
    \sup_{k\in \mathbb{N}} \| (\Psi_k)_* J_k \|_{C^n} \leq C_n
    \end{align*}
  for each \(n\in \mathbb{N}\); here \(\|\cdot \|_{C^n}\) is the
  \(C^n\)-norm on \(\mathbb{R}\times M\) with respect to the
  auxiliary metric \(\tilde{g}\), and the \(\Psi_k\) are the embeddings as
  in equation (\ref{EQ_Psi_new}).
  \end{enumerate}
We then say that \(\mathcal{J}=\{J_1, J_2, \ldots\}\) is a sequence of
  adiabatically degenerating almost complex structures adapted to the
  framed Hamiltonian energy pile given by \((\mathcal{I}_\epsilon\times M,
  \Omega, \hat{\lambda})\).
\end{definition}

We note that given a framed Hamiltonian energy pile, it is easy to
  construct a candidate sequence of adiabatically degenerating almost
  complex structures.
For example, one might fix a translation invariant \(J_\xi = J\big|_\xi\),
  and then define
  \begin{align*}
    J_k \partial_s = k \widehat{X}\qquad\text{and}\qquad J_k\big|_\xi  =
    J_\xi.
    \end{align*}
In such a candidate, one can easily see that \(\phi_k = k^{-1}\to 0\) as
  \(k\to \infty\), and in fact \((\Psi_k)_* J_k\) is independent of \(k\).
Consequently (\(\mathcal{J}1\)) and (\(\mathcal{J}3\)) are immediately
  satisfied.
However, (\(\mathcal{J}2\)) is less obvious.
That is, it is unclear if the following estimate holds for all \(J_k\):
  \begin{align*}
    (ds \wedge \hat{\lambda} + \hat{\omega})(v, J_kv )\leq 2 \Omega(v ,
    J_k v)
    \end{align*}
Geometrically, the concern is that while the almost complex structures
  \(J_k\) are all \(\Omega\)-tame, they are not \(\Omega\)-compatible.
Put another way, although the \(J_k\) preserve the splitting
  \(T(\mathcal{I}_\epsilon\times M) = \hat{\rho}\oplus \xi\), and
  \(\hat{\rho}\) and \(\xi\) are each symplectic subspaces, these
  subspaces  are not symplectic complements; that is there exist \(v\in
  \hat{\rho}\)
  and \(w\in \xi\) such that \(\Omega(v,w)\neq 0\).
That the cross terms might cause an issue is then compounded by
  the fact that we are degenerating adiabatically so that \(\phi_k\to 0\)
  as \(k\to \infty\).

We resolve this issue by making use of the fact that Lemma
  \ref{LEM_localization_to_framed_Hamiltonian_energy_pile} allows us to
  guarantee that along \(\{0\}\times M\) the sub-bundles \(\hat{\rho}\) and
  \(\xi\) \emph{are} \(\Omega\)-symplectic complements.
As it will turn out, condition (\(\mathcal{J}\)2) can then be achieved by
  first fixing \(\epsilon>0\) sufficiently small.
We will couple this with a convenient means of obtaining a sequence of
  adiabatically degenerating almost complex structures from some easily
  verified bounds and some \(C^\infty\) converging functions
  \(\phi_k\).
This is achieved via Proposition
  \ref{PROP_adiabatically_degenerating_constructions} below.

\setcounter{CounterSectionAdiabatic}{\value{section}}
\setcounter{CounterPropositionAdiabatic}{\value{lemma}}

\begin{proposition}[adiabatically degenerating constructions]
  \label{PROP_adiabatically_degenerating_constructions}
  \hfill\\
Let \(\big(\mathcal{I}_{\ell}\times M, \Omega, \hat{\lambda}\big)\) be a
  framed Hamiltonian energy pile, and let \(\hat{\rho}\), \(\xi\), and
  \(\hat{\omega}\) be the associated structures defined above, and assume,
  as in the conclusions of Lemma
  \ref{LEM_localization_to_framed_Hamiltonian_energy_pile}, that along
  \(\{0\}\times M\) the sub-bundles \(\hat{\rho}\) and \(\xi\) are
  symplectic complements and \(\hat{\lambda}(X_H)=1\).
Let \(\widehat{X}=X_H/\hat{\lambda}(X_H)\) as above.
Let \(\{\check{J}_k\}_{k\in \mathbb{N}}\) be a sequence of almost
  complex structures on \(\mathcal{I}_\ell \times M\) which satisfy the
  following conditions.
\begin{enumerate}[(D1)]
  \item
  \(\check{J}_k\colon \hat{\rho} \to \hat{\rho}\; \) and  \(\;
  \check{J}_k\colon \xi \to \xi\) for each \(k\in \mathbb{N}\),
  \item
  \((ds\wedge \hat{\lambda} +\hat{\omega})\circ ({\rm Id}\times
  \check{J}_k)\) is a Riemannian metric for each \(k\in \mathbb{N}\)
  \item
  there exist constants  \(\{C_n'\}_{n\in \mathbb{N}}\) such that
  \begin{align*}
    \sup_{k\in \mathbb{N}} \|\check{J}_k\|_{C^n} \leq C_n'
    \end{align*}
  \end{enumerate}
Then there exist an \(\epsilon>0\) with the following significance.
For any sequence of functions \(\phi_k : (-\epsilon, \epsilon)\to (0, 1]\)
  which converge in \(C^\infty\) (though not necessarily to
  zero), the weakly adapted almost complex structures defined by
  \begin{align}\label{EQ_def_Jk_in_body}
    \phi_k \cdot J_k \partial_s = \widehat{X}\qquad\text{and}\qquad
    J_k\big|_\xi := \check{J}_k\big|_\xi
    \end{align}
  satisfy the following properties
\begin{enumerate}[(E1)]
  \item For each \(v\in T(\mathcal{I}_{\epsilon}\times M)\) we have
  \begin{align*}
    (ds \wedge \hat{\lambda} + \hat{\omega})(v, J_kv )\leq 2 \Omega(v ,
    J_k v).
    \end{align*}
  \item
  There exists a sequence \(\{K_n\}_{n=0}^\infty\) of positive
  constants and an auxiliary translation invariant metric \(\tilde{g}\) on
  \(\mathbb{R}\times M\) for which
  \begin{align*}
    \sup_{k\in \mathbb{N}} \| (\Psi_k)_* J_k \|_{C^n} \leq K_n
    \end{align*}
  for each \(n\in \mathbb{N}\); here \(\|\cdot \|_{C^n}\) is the
  \(C^n\)-norm on \(\mathbb{R}\times M\) with respect to the
  auxiliary translation invariant metric \(\tilde{g}\), and the \(\Psi_k\)
  are the embeddings as in equation (\ref{EQ_Psi_new}).
  \end{enumerate}
In particular, if \(\epsilon' \in (0, \epsilon)\) and
  \(\phi_k\big|_{(-\epsilon', \epsilon')}\to 0\), then the almost complex
  structures \(J_k\) are adiabatically degenerating on
  \(\mathcal{I}_{\epsilon'}\times M\) in the sense of Definition
  \ref{DEF_adiabatically_degenerating_almost_complex_structures}, and
  on \(\mathcal{I}_{\epsilon}\times M\) these almost complex structures are
  all tame.
\end{proposition}

The proof of Proposition
  \ref{PROP_adiabatically_degenerating_constructions} is elementary but
  somewhat lengthy, so we relegate it to Appendix
  \ref{SEC_supporting_proofs}, however at present we indicate its utility.
We start by noting that a natural way to perform an adiabatic degeneration
  is to start with some fixed almost complex structure \(J\) which
  preserves the splitting \(\hat{\rho}\oplus \xi\) and for which \((ds
  \wedge \hat{\lambda} + \hat{\omega})\circ ({\rm Id} \times J)\) is a
  Riemannian metric.
This almost complex structure will then be \(\Omega\)-compatible along
  \(\{0\}\times M\), and hence \(\Omega\)-tame in some neighborhood of
  \(\{0\}\times M\).
Consequently, \(J\) can be adjusted away from \(\{0\}\times M\) so that it
  is \(\Omega\)-tame everywhere, and still satisfies our compatibility
  conditions in a neighborhood of \(\{0\}\times M\).
Proposition \ref{PROP_adiabatically_degenerating_constructions} then
  guarantees that we can find \(\phi_k\) which tend to zero in a
  neighborhood of zero so that the sequence of almost complex structures
  given by
  \begin{align*}
    \phi_k \cdot J_k\partial_s = \widehat{X}\qquad\text{and}\qquad
    J_k\big|_\xi = J\big|_\xi
    \end{align*}
  are everywhere tame, while also adiabatically degenerating a
  neighborhood  of \(\{0\}\times M\).
Among other things, this means
  \begin{align*}
    (ds \wedge \hat{\lambda} + \hat{\omega})(v, J_kv )\leq 2 \Omega(v ,
    J_k v)
    \end{align*}
  for all \(k\in \mathbb{N}\).
This latter condition is important, since by assumption the
  pseudoholomorphic curves we will study will have uniform bounds on
  \(\Omega\)-energy, which (as a consequence of the above inequality) will
  give us bounds on the area associated to the metric \((ds\wedge
  \hat{\lambda} +\hat{\omega})\circ ({\rm Id}\times J)\).

With Proposition \ref{PROP_adiabatically_degenerating_constructions}
  stated and its use outlined, we now turn our attention to proving our
  main result.
This is the topic of the next section.


TITLE: How to find an injective affine map mapping a square in $R^2$ to the triangle in $R^3$ with vertices $(0,0,0), (1,1,1), (1,1,2)$?
QUESTION [0 upvotes]: How to find an injective affine map mapping some square(sets of the form $[a,b] \times [c,d]$) in $R^2$ to the triangle in $R^3$ with vertices $(0,0,0), (1,1,1), (1,1,2)$?

REPLY [1 votes]: There is no such map.
An affine map will map lines to lines and preserve incidences between points and lines. An injective map has at most one preimage for every image. If your original square had corners $A,B,C,D$ and your triangle has corners $P,Q,R$, you could assume one of two cases: either all square corners get mapped to triangle corners. Then one triangle corner would have two preimages, in violation to the injective property. Or one square corner would get mapped to some point along the edge of the triangle. W.l.o.g. we may assume $A\mapsto P,B\mapsto Q,C\mapsto R,D\mapsto X$ with $X$ a point on the segment $PR$. The diagonal $AC$ in the original square must map to the line $PR$ as lines are preserved. But $D$ does not lie on $AC$, so if it still maps to $PR$ as well, that would mean some point on $PR$ would have two preimages. Again in violation of the injective property.


text:
\begin{document}

\begin{abstract}
This paper is concerned with the location of nodal sets
of eigenfunctions of the Dirichlet Laplacian
in thin tubular neighbourhoods of hypersurfaces
of the Euclidean space of arbitrary dimension.
In the limit when the radius of the neighbourhood tends to zero,
it is known that spectral properties of the Laplacian are
approximated well by an effective Schr\"odinger operator
on the hypersurface with a potential expressed solely
in terms of principal curvatures.
By applying techniques of elliptic partial differential equations,
we strengthen the known perturbation results
to get a convergence of eigenfunctions in H\"older spaces.
This enables us in particular to conclude that
every nodal set has a non-empty intersection with the boundary of
the tubular neighbourhood.
\end{abstract}

\maketitle

\section{Introduction}
The celebrated \emph{nodal-line conjecture} of L.~E.~Payne's
from 1967~\cite[Conj.~5]{Payne1}
states that any membrane with fixed boundary vibrating at
the second lowest resonant frequency cannot have
a closed nodal curve.
Mathematically, representing the membrane
by a bounded domain (i.e.\ open connected set) $\Omega \subset \R^2$,
the resonant frequencies~$\sqrt{\lambda}$
and corresponding vibrating modes~$u$
are obtained as solutions of the boundary-value problem

\begin{equation}\label{bv}
\left\{
\begin{aligned}
  -\Delta u &= \lambda u
  && \mbox{in} \quad \Omega \,,
  \\
  u &= 0
  && \mbox{on} \quad \partial\Omega \,.
\end{aligned}
\right.
\end{equation}

It is customary to arrange the eigenvalues
in a non-decreasing sequence (counting multiplicities)
$
  0 < \lambda_1 \leq \lambda_2 \leq \lambda_3 \leq \dots
$
and choose the corresponding eigenfunctions
$u_1, u_2, u_3, \dots$ real-valued.
It is well known that~$\lambda_1$ is non-degenerate
and that~$u_1$ does not change sign in~$\Omega$.
Since the eigenfunctions are mutually orthogonal in $L^2(\Omega)$,
the \emph{nodal set}

\begin{equation}\label{nodal.set}
  \mathcal{N}(u_n) := u_n^{-1}(\{0\})
\end{equation}

is non-trivial whenever $n \geq 2$
and forms peculiar shapes (also known as Chladni's patterns):
various crossing curves or closed loops.
The connected components of $\Omega\setminus\mathcal{N}(u_n)$
are called \emph{nodal domains} of~$u_n$.
It turns out that the shape of the nodal set
is related to acoustic properties of the membrane.
The conjecture then states that $\mathcal{N}(u_2)$
cannot form a closed curve, or more generally,
in any dimension and for an arbitrary~$\Omega$,
that the nodal set touches the boundary:

\begin{Conjecture}\label{Conj1}
$\overline{\mathcal{N}(u_2)} \cap \partial\Omega \not= \emptyset$
for any open domain $\Omega \subset \R^d$, $d \geq 2$.
\end{Conjecture}

If~$\Omega$ is not sufficiently regular or it is unbounded,
\eqref{bv}~should be interpreted as a spectral problem
for the Dirichlet Laplacian $-\Delta_D^{\Omega}$
in the Hilbert space $L^2(\Omega)$.
The conjecture still makes sense even for unbounded~$\Omega$,
provided that two eigenvalues below the essential spectrum exist.
We note that the spectral problem~\eqref{bv}
for unbounded tubular domains arises
in the context of stationary Schr\"odinger equation
in \emph{quantum waveguides}
(see \cite{DE,KKriz,Haag-Lampart-Teufel_2014}
and references therein),
where the eigenfunctions have the physical interpretation
of quantum bound states.

Since the solutions of~\eqref{bv} are analytic in~$\Omega$,
each $\mathcal{N}(u_n)$ decomposes into the disjoint union
of an analytic $(d-1)$-dimensional manifold
and a singular set contained in a countable number
of analytic $(d-2)$-dimensional manifolds
(cf.~\cite{Caffarelli-Friedman_1985}).
The \emph{Courant nodal domain theorem} then states that,
if the boundary~$\partial \Omega$ is sufficiently regular,
the $n$th~eigenfunction~$u_n$
has at most~$n$ nodal domains
(cf.~\cite{CH1,Chavel,Alessandrini_1998}).
In particular, $u_2$~has exactly two nodal domains.

The restriction to the second eigenfunction in Conjecture~\ref{Conj1}
is essential because higher eigenfunctions are known
to have nodal sets which possibly do not intersect the boundary.
As a matter of fact, there exist counterexamples even
to Conjecture~\ref{Conj1} as stated here,
given by domains~$\Omega$
which are multiply connected
\cite{H2ON,Fournais,Kennedy_2011}
or unbounded \cite{FK2}.
More significantly,
it has been shown recently by Kennedy~\cite{Kennedy_2013}
that the conjecture does not hold for $d \geq 3$,
even if the domain~$\Omega$ is bounded and simply connected.
The current status is that the validity of Conjecture~\ref{Conj1}
constitutes an open problem ``only'' for
simply connected bounded domains in $d=2$.

Although the conjecture is known to be violated in general,
it is important to identify domains for which it is satisfied,
and moreover, provide information on the location
and geometry of the nodal set.
The most general result obtained so far
was given by Melas~\cite{Melas},
who showed that Conjecture~\ref{Conj1} holds in the case
of convex planar domains (cf.~also~\cite{Alessandrini_1994}).
Independently of Melas, Jerison proved Conjecture~\ref{Conj1}
in the case of sufficiently \emph{long and thin convex}
planar domains~\cite{Jerison0}
and later extended the proof to higher dimensions~\cite{Jerison}.
Furthermore, Jerison's method enabled him to locate
the nodal set in the two-dimensional case
near the zero of an ordinary differential equation
associated to the convex domain in a natural way~\cite{Jerison1}
(cf.~also \cite{GJerison,Grieser-Jerison_2009}).

The only positive result for non-convex
and possibly multiply connected domains in any dimension
is that of Freitas and one of the present authors
obtained for \emph{thin curved tubes} of arbitrary cross-sec\-tion~\cite{FK4}.
The result resembles that of Jerison's in that
we also consider thin domains
and locate the nodal set near the zeros of an ordinary differential equation
associated to the curved tube, however,
the technical approach is actually very different.
Instead of a usage of a trial function
and refined applications of the maximum principle
developed by Jerison, we rather rely on a singular
perturbation theory in a Hilbert-space setting
(although the maximum principle, in addition to other techniques,
is also used eventually).
Moreover, our method works in all dimensions
and also for higher eigenfunctions.

The objective of the present paper is to prove Conjecture~\ref{Conj1}
for \emph{thin curved layers}~$\Omega_\varepsilon$,
i.e.\ $\varepsilon$-tubular neighbourhoods
of hypersurfaces~$\Sigma$ in~$\R^d$,
with $d \geq 2$ and $\varepsilon \to 0$.
The difference with respect to thin tubes considered in~\cite{FK4},
where the underlying manifold was just a curve
(of dimension~$1$ and codimension~$d-1$),
is that now the asymptotic geometry is that of a manifold
of dimension $d-1$ (and codimension~$1$).
It leads both to conceptual and technical complications.
Indeed, the location of the nodal sets is presently determined
by a \emph{partial} differential equation associated to the hypersurface,
which itself could have closed nodal sets
(even if $n=2$, cf.~\cite{Freitas1}).
More importantly, the method of~\cite{FK4}
to pass from $L^2$- to $C^0$-estimates
needed for the location of the nodal sets
does not work in this more complex situation.
Therefore new ideas that we explain below had to be developed.
Finally, contrary to~\cite{FK4} we do not exclude from the present paper
the possibility that the domain~$\Omega_\varepsilon$ is unbounded,
so that some of the results are new even in $d=2$,
when the geometrical setting coincides with that of~\cite{FK4}.

Let us state a consequence of the results established
in this paper.

\begin{theorem}\label{Thm.Conj}
Let~$\Omega_\varepsilon$ be an $\varepsilon$-tubular neighbourhood
of a smooth hypersurface~$\Sigma$ in~$\R^d$, $d \geq 2$.
Assume hypotheses \ref{eq:eps_bound}, \ref{eq:res_conv_ass},
and \ref{eq:reg_ass_final} stated below.
If~$\Sigma$ is unbounded, let us also assume that
the Dirichlet Laplacian $-\Delta_D^{\Omega_\varepsilon}$
possesses $m\geq 2$
eigenvalues below the essential spectrum
for all sufficiently small~$\varepsilon$.
Then for any $N\in\{2,3,\ldots\}$ or $N\in\{2,3,\ldots,m\}$
in the bounded or unbounded case, respectively, there exists a positive constant
$\varepsilon_0$
depending on the geometry of~$\Sigma$
such that
$$
  \overline{\mathcal{N}(u_n)} \cap \partial\Omega_\varepsilon \not= \emptyset
$$
holds for all $\varepsilon \leq \varepsilon_0$
and every $n \in \{2,\dots,N\}$.
In particular,
Conjecture~\ref{Conj1} holds for~$\Omega_\varepsilon$
whenever $\varepsilon \leq \varepsilon_0$.
\end{theorem}

As a matter of fact, Theorem~\ref{Thm.Conj} follows as a consequence
of a stronger result, Theorem~\ref{theo:sign_change},
in which we establish a \emph{local convergence}
of nodal sets for all eigenfunctions
of $-\Delta_D^{\Omega_\varepsilon}$
to nodal sets of eigenfunctions of a sum of
a $(d-1)$-dimensional Schr\"odinger operator~$h_\mathrm{eff}$
in~$L^2(\Sigma)$, cf.~\eqref{h.eff},
and the one-dimensional Dirichlet Laplacian in the cross-section.

The structure of the paper is as follows.

In the next Section~\ref{Sec.pre}
we collect basic facts and assumptions
about the geometry of the hypersurface~$\Sigma$
and its tubular neighbourhood~$\Omega_\varepsilon$.
We also introduce the effective
Schr\"odinger operator~$h_\mathrm{eff}$ in~$L^2(\Sigma)$,
which governs the location of nodal sets in~$\Omega_\varepsilon$
for small~$\varepsilon$, and recall from~\cite{KRT}
a result on a \emph{norm-resolvent convergence}
of $-\Delta_D^{\Omega_\varepsilon}$ to~$h_\mathrm{eff}$.
After an appropriate identification of Hilbert spaces,
the operator convergence implies the convergence
of the respective eigenfunctions in $L^2(\Omega_\varepsilon)$,
however, this convergence is not sufficient to deduce
the convergence of nodal sets.

Section~\ref{Sec.conv} consists of a number of subsections
devoted to a passage from the $L^2$-convergence
to a \emph{uniform}, $C^0$-convergence.
The main idea is to adapt techniques known
from elliptic regularity theory
for partial differential equations
in order to establish a convergence of eigenfunctions
in higher-order Sobolev spaces and apply the Sobolev embedding theorem.
We believe that our approach is of independent interest
because we deal with a highly singular problem
when some coefficients of the partial differential equation
depends on~$\varepsilon^{-2}$.
In the last subsection, we additionally use the maximum principle
to deduce a $C^{0,1}$-convergence needed for the convergence
of nodal sets.

The local convergence of nodal sets
is deduced from the previous convergence results
in Section~\ref{Sec.nodal}. The main result of this paper
is given by Theorem~\ref{theo:sign_change} there.
The last Section~\ref{Sec.special} deals with a special class
of underlying manifolds $\Sigma$ for which
one can extend the local results of
Section~\ref{Sec.nodal}
to a \emph{global convergence} along the whole nodal set.
For this class of manifolds,
we also demonstrate the convergence (in the Hausdorff sense) of nodal domains
employing the Courant nodal domain theorem.

When finishing this paper a related work of Lampart~\cite{Lampart} appeared.
On the one hand, it is concerned with
the much more general geometrical setting of the Laplacian
on a diminishing fibre bundle of arbitrary dimension.
On the other hand, the base space of the fibre
is assumed to be compact and of dimension at most~$3$ in~\cite{Lampart},
which would correspond to the restriction to compact $\Sigma$ and $d\leq 4$
in our setting. It is remarkable that we do not need these constraints
to prove Conjecture~\ref{Conj1}, but our fibre is always one-dimensional
(although there seem not to be any technical limitations
with extending our methods to higher-dimensional cross-sections,
cf.~\cite{FK4}).
The present work and~\cite{Lampart} are also independent
from the point of view of the technical handling of the limit $\varepsilon \to 0$.
While Lampart starts with an adiabatic perturbation theory
developed for fibre bundles in \cite{Lampart-Teufel}
and finally ends up with $\mathcal{H}^1$-estimates,
we rather rely on the norm-resolvent convergence established in~\cite{KRT}
and proceed to higher-order Sobolev spaces.

\section{Notation and preliminary results}\label{Sec.pre}
At the beginning let us point out that we will use both the ``prefix'' and
``postfix'' notation for partial derivatives. In the prefix notation a partial
derivative acts on the whole expression to the right, whereas in the postfix
notation a partial derivative acts only on the closest term to the left. This
convention shall save us from the over-bracketing.

\subsection{Geometry of the layer}
Mostly we adopt the notation of \cite{KRT}. In particular let $\Sigma$ be a
connected orientable $C^l$-hypersurface with $l\geq 4$  in $\R^d$ ($d\geq 2$)
with the Riemannian metric $g$ induced by the embedding. It may or may not have
a boundary, if it does, then we assume that the boundary is also $C^l$-smooth.
The orientation is given by a globally defined unit normal field $n:\Sigma\to
S^{d-1}$. For any $p\in\Sigma$, we introduce the Weingarten map
$$ L: T_p\Sigma \to T_p\Sigma: \  \big\{\xi \mapsto -\dd n(\xi)\big\}.$$
If $x^1,\ldots, x^{d-1}$ are some local coordinates on $\Sigma$,
then $L$ in the coordinate frame $(\partial_1,\ldots, \partial_{d-1})$ (where we
abbreviate $\partial_\mu=\partial_{x^{\mu}}$) has the matrix representation
$(L^{\mu}_{\ \nu})$ that reads $L^{\mu}_{\ \nu} = g^{\mu\rho} h_{\rho\nu}$,
where $(g^{\mu\rho})$ and $(h_{\rho\nu})$ is the matrix representation of
$g^{-1}$ and the second fundamental form of $\Sigma$, respectively. We set
$|g|:=\det{g}$.

The eigenvalues of $L$ are the principal curvatures,
$\kappa_1,\ldots,\kappa_{d-1}$, of $\Sigma$.
Re\-mark that they are in general given only locally. However, there are $d-1$
invariants of $L$ given by the formula
$$
  K_{\mu}:=\binom{d-1}{\mu}^{-1}
  \sum_{\alpha_{1}<\ldots< \alpha_{\mu}}
  \kappa_{\alpha_{1}}\ldots\kappa_{\alpha_{\mu}},\qquad\mu=1,\ldots,d-1,
$$
that are globally defined $C^{l-2}$-smooth functions.

Given $I:=(-1,1)$ and $\varepsilon>0$, we define a layer $\Omega_\varepsilon$ of
width $2\varepsilon$ along $\Sigma$ as the interior of the image of the mapping
\begin{equation}\label{eq:layer_def}
  \mathscr{L}: \
  \Sigma\times I\to\R^d: \ \big\{(x,u)\mapsto x+\varepsilon u n\big\},
\end{equation}
i.e.,
$$
  \Omega_\varepsilon
  :=\mathrm{int}\,\mathscr{L}(\Sigma\times I).
$$
We always assume that $\Omega_\varepsilon$ does not overlap itself, i.e.,
\begin{equation}\label{eq:eps_bound} \tag*{$\langle A1\rangle$}
  \varepsilon<\varrho_{m}
  :=\big(\max\big\{\|\kappa_{1}\|_{\infty},
  \ldots,\|\kappa_{d-1}\|_{\infty}\big\}\big)^{-1}
   \ \& \quad
  \mathscr{L} \mbox{ is injective},
\end{equation}
which implicitly involves the assumption
that the principal curvatures are bounded.

In the sequel we set $x^d:=u$, $\partial_{d}=\partial_{u}$, and
$x_{\shortparallel}=(x^1,\ldots,x^{d-1})$.
Also, unless otherwise stated,
we will always assume the range of Greek and Latin indices to be $1,\ldots,d-1$
and $1,\ldots,d$, respectively.

The metric induced by \eqref{eq:layer_def} is of the following block form, when
written in the coordinate frame $(\partial_1,\ldots, \partial_d)$,
\begin{equation}\label{eq:layer_metric}
  (G_{ij})=
  \begin{pmatrix}
         (G_{\mu\nu}) & 0\\0 & \varepsilon^2
  \end{pmatrix},
  \qquad
  G_{\mu\nu}=g_{\mu\rho}
  (\delta^{\rho}_{\sigma} -\varepsilon u L^{\rho}_{\ \sigma})
  (\delta^{\sigma}_{\nu} -\varepsilon u L^{\sigma}_{\ \nu}).
\end{equation}

In a straightforward manner one can derive the following matrix estimate
\begin{equation}\label{eq:metric_est}
 C_{-}g_{\mu\nu}\leq G_{\mu\nu}\leq C_{+}g_{\mu\nu}.
\end{equation}
with $C_{\pm}:=(1\pm\varepsilon\varrho_{m}^{-1})^2=1+\mathcal{O}(\varepsilon)$.

\subsection{Dirichlet Laplacian}
The Dirichlet Laplacian, $-\Delta_{D}^{\Omega_{\varepsilon}}$, on
$\Omega_{\varepsilon}$ is the self-adjoint operator
in $L^2(\Omega_{\varepsilon})$
associated with the quadratic form
$$
  Q[\psi]:=\|\nabla\psi\|_{L^2(\Omega_{\varepsilon})}^2
  \,, \qquad
  \dom{Q}:=\mathcal{H}_{0}^{1}(\Omega_{\varepsilon})
  \,.
$$
It was demonstrated in \cite{KRT} that $-\Delta_{D}^{\Omega_{\varepsilon}}$ is
unitarily equivalent to

\begin{equation*}
  H :=U(-\Delta_{D}^{\Omega_{\varepsilon}})U^{-1}
  =
-|g|^{-1/2}\partial_{\mu}|g|^{1/2}G^{\mu\nu}\partial_{\nu}-\varepsilon^{-2}
\partial_{u}^{2}+V,
\end{equation*}

acting in
$\mathscr{H}:=L^{2}(\Sigma\times I,|g|^{1/2}\dd x^1\ldots\dd x^d)$
with $\dom{H^{1/2}}=U[\dom{Q}]$, where
$U:\psi\mapsto\mathrm{e}^J (\psi\circ\mathscr{L})$
with
\begin{align*}
 & J := \frac{1}{4} \ln\frac{|G|}{|g|}
  = \frac{1}{2}\ln\left[1+\sum_{\mu=1}^{d-1}(-\varepsilon u)^{\mu}
  \binom{d-1}{\mu}K_{\mu}\right] , \\
 & V := |g|^{-1/2} \partial_{i}|g|^{1/2} G^{ij} \partial_{j} J
 + J_{,i} G^{ij} J_{,j}\, .
\end{align*}

\begin{remark}\label{rem:metric_cont}
Under our regularity assumption on $\Sigma$, the entries of $(g^{\mu\nu})$ are
$C^{l-1}$-functions, the entries of $(L^{\mu}_{\ \nu})$ and $(G^{\mu\nu})$ are
$C^{l-2}$-functions,
and $V$ is only a $C^{l-4}$ function in any local
coordinates. In some more detail, the entries of $(G^{\mu\nu})$ are polynomials
in the parameter $\varepsilon$ with $C^{l-2}$-smooth coefficients and
$\lim_{\varepsilon\to 0}G^{\mu\nu}=g^{\mu\nu}$. This may be easily observed,
e.g. from the Cayley-Hamilton theorem that states
 $$\prod_{i=0}^{d-1}(\kappa_i-L)=0,$$
 and so
 \begin{equation}\label{eq:Cal_Ham}
 0=\prod_{i=0}^{d-1}(\mathit{Id}-\varepsilon u L+(\varepsilon u\kappa_i
-1)\mathit{Id})=\sum_{i=0}^{d-1}A_i(\mathit{Id}-\varepsilon u L)^i,
 \end{equation}
 where $A_{i}$ is a product of a polynomial in $\{\varepsilon u\kappa_{j},\
j=1,\ldots,d-1\}$ and the identity map, $\mathit{Id}$. In particular,
 $A_{d-1}=\mathit{Id}$ and $A_0=\prod_{i=1}^{d-1}(\varepsilon u\kappa_i-1)
\mathit{Id}$. Under the assumption \ref{eq:eps_bound}, $A_0\neq 0$. Consequently
we may compute
 $(\mathit{Id}-\varepsilon u L)^{-1}$
from \eqref{eq:Cal_Ham}
as a finite sum and then substitute the result to the formula for
$(G^{\mu\nu})$, which is easy to derive from \eqref{eq:layer_metric}.
\end{remark}

\subsection{Effective Hamiltonian}
Let $(E_m,\chi_m),\ m\in\N \equiv \{1,2,\dots\}$,
be the $m$th eigenpair
of $-\partial_{u}^{2}$ in $L^2(I)$,
subject to the Dirichlet boundary condition on $\partial I$.
We have $E_m=(m\pi/2)^2$ and, in particular, we choose

\begin{equation}\label{chi1}
  \chi_{1}(u) := \cos(\pi u/2).
\end{equation}

For small values of $\varepsilon$, $H$ behaves like
\begin{equation*}
  H_{0}:= h_{\mathrm{eff}}-\varepsilon^{-2}\partial_{u}^{2}
  \ \simeq \
  h_{\mathrm{eff}} \otimes 1  + 1 \otimes (-\varepsilon^{-2}\partial_{u}^{2})
\end{equation*}
acting on $\mathscr{H}=L^2(\Sigma,|g|^{1/2}\dd x^1\ldots\dd x^{d-1})\otimes
L^2(I,\dd u)$ with the form domain $\dom{H_0^{1/2}}:= \dom{H^{1/2}}$.
Here the so-called effective Hamiltonian $h_{\mathrm{eff}}$ reads
\begin{equation}\label{h.eff}
 h_{\mathrm{eff}}=
-\Delta_{g}+V_{\mathrm{eff}},\quad\dom{h_{\mathrm{eff}}}=\dom{-\Delta_g},
\end{equation}
where
$-\Delta_{g} := -|g|^{-1/2}\partial_{\mu}|g|^{1/2}g^{\mu\nu}\partial_{\nu}$
is the Laplace-Beltrami operator on $\Sigma$ subject to
Dirichlet boundary conditions and
\begin{equation*}
 V_{\mathrm{eff}}:=-\frac{1}{2}\sum_{\mu=1}^{d-1}\kappa_{\mu}^{2}+\frac{1}{4}
\left(\sum_{\mu=1}^{d-1}\kappa_{\mu}\right)^2=\binom{d-1}{2}K_2-\left(\frac{d-1}
{2}\,K_1\right)^2\, .
\end{equation*}
To be more precise, we reproduce here Theorems 6.3 and 7.1 of \cite{KRT} in a
slightly modified form.

\begin{theorem}[\cite{KRT}]\label{Thm.KRT}
Let $H_{\mathrm{ren}}:=H-\varepsilon^{-2} E_1$,
$H_{0,\mathrm{ren}}:=H_0-\varepsilon^{-2} E_1$.
Assume \ref{eq:eps_bound} and
\begin{equation}\tag*{$\langle A2\rangle$}\label{eq:res_conv_ass}
|\nabla_{\!g}\kappa_{\mu}|_{g}
  \,, \
  \Delta_{g}\kappa_{\mu}  \in L^{\infty}(\Sigma),
\end{equation}
where $|\nabla_{\!g}f|_{g} := \sqrt{ f_{,\mu} g^{\mu\nu} f_{,\nu} }$.
Then  for any $k\in\mathrm{Res}\,H_{0,\mathrm{ren}}$,
positive constants~$\varepsilon_0$ and~$C$ exist such that,
for all $\varepsilon<\varepsilon_0$,
$k\in\mathrm{Res}\,H_{\mathrm{ren}}$ and
\begin{equation}\label{eq:res_conv}
   \|(H_{\mathrm{ren}}+k)^{-1}-(H_{0,\mathrm{ren}}+k)^{-1}\|\leq C\varepsilon .
\end{equation}
Similarly,  for any $k\in\mathrm{Res}\,h_{\mathrm{eff}}$, positive constants
$\tilde\varepsilon_0$ and $\tilde C$ exist such that, for all
$\varepsilon<\tilde\varepsilon_0$,
$k\in\mathrm{Res}\,H_{\mathrm{ren}}$ and
\begin{equation*}
   \|(H_{\mathrm{ren}}+k)^{-1}-(h_{\mathrm{eff}}+k)^{-1}\oplus 0\|\leq \tilde
C\varepsilon .
\end{equation*}
\end{theorem}

\noindent Here  $(h_{\mathrm{eff}}+k)^{-1}\oplus 0$ acts on
$\mathscr{H}_{1}\oplus\mathscr{H}_{1}^{\bot}$,
where $\mathscr{H}_{1}$ is spanned by the functions of the form
$f\otimes\chi_1$
with $f\in L^2(\Sigma,|g|^{1/2}\dd x^1\ldots\dd x^{d-1})$.

\section{Convergence of eigenfunctions}\label{Sec.conv}

\subsection{$L^2$-convergence}
Denote the eigenvalues of $H_{0,\mathrm{ren}}$
in a non-decreasing order (counting
multiplicity) by $\{\lambda_{n}^{0}\}_{n=1}^\infty$.
Let $\mu=\lambda_{k}^{0}=\ldots =\lambda_{k+p-1}^{0}$ be a $p$-degenerated
eigenvalue of $H_{0,\mathrm{ren}}$,
$k,p \geq 1$.
Due to \eqref{eq:res_conv}, one can prove (see again \cite{KRT} for details)
that for all sufficiently small $\varepsilon$, the number of eigenvalues of
$H_{\mathrm{ren}}$ in the contour $\Gamma$ of radius smaller than the isolation
distance of $\mu$  and centred at $\mu$ equals $p$. Moreover, there exists a
family of linearly independent eigenfunctions $\psi_{n}$,
with $n=k,\dots,k+p-1$, of~$H_{\mathrm{ren}}$ associated with
these eigenvalues of~$H_{\mathrm{ren}}$,
which we denote by
$\lambda_k, \dots, \lambda_{k+p-1}$
(they are not necessarily sorted in a non-decreasing order,
but they are counted according to multiplicities), and a base
$(\psi_k^0,\ldots,\psi_{k+p-1}^0)$
of the eigenspace associated with the eigenvalue $\mu$ of~$H_{0,\mathrm{ren}}$
such that the following result holds (see also \cite[Corol.~8.1]{KRT}).

\begin{theorem}
With the notation introduced above
and under the hypotheses of Theorem~\ref{Thm.KRT},
\begin{equation*}
 |\lambda_n-\lambda_{n}^{0}|=\mathcal{O}(\varepsilon),\quad
\|\psi_n-\psi_{n}^{0}\|=\mathcal{O}(\varepsilon),
\end{equation*}
as $\varepsilon\to 0$.
\end{theorem}

\begin{remark}[Eigenpairs of $H_{0,\mathrm{ren}}$] \label{rem:H_0_spec} It is
obvious that the eigenpairs $(\lambda_{n}^{0},\psi_{n}^{0})$ are of the form
$(\sigma_i+\varepsilon^{-2}(E_j-E_1),\phi_i\otimes\chi_j)$, where
$(\sigma_i,\phi_i)$ is an eigenpair of $h_{\mathrm{eff}}$. Thus for any $N\in\N$
there is $\varepsilon(N)$ such that for all $\varepsilon<\varepsilon(N)$ and
$n=1,\ldots,N$, we have $\lambda_{n}^{0}=\sigma_n$ and
$\psi_{n}^{0}=\phi_{n}\otimes\chi_1$.
\end{remark}

\subsection{$\mathcal{H}^{1}$-convergence}
Subtracting eigenvalue equations for the operators
$H_{\mathrm{ren}}$ and $H_{0,\mathrm{ren}}$,
we arrive at the following equation for $\psi:=\psi_{n}-\psi_{n}^{0}$
\begin{equation}\label{eq:psi_diff}
\begin{split}
 -|g|^{-1/2}\partial_{\mu}|g|^{1/2}G^{\mu\nu}\partial_{\nu}\psi&+\varepsilon^{-2
}(-\partial_{u}^{2}-E_{1})\psi=|g|^{-1/2}\partial_{\mu}|g|^{1/2}a^{\mu\nu}
\partial_{\nu}\psi_{n}^{0}\\
 &+(\lambda_{n}-V)\psi+(V_{\mathrm{eff}}-V+\lambda_{n}-\lambda_{n}^{0})\psi_{n}^
{0},
\end{split}
\end{equation}
where $a^{\mu\nu}:=G^{\mu\nu}-g^{\mu\nu}$. Next we multiply (using the scalar
product of $\mathscr{H}$) the both sides of this result by $\psi$ to obtain
\begin{equation*}
\begin{split}
\langle\partial_{\mu}\psi,G^{\mu\nu}\partial_{\nu}\psi\rangle+&\varepsilon^{-2}
(\|\partial_{u}\psi\|^2-E_{1}\|\psi\|^2)=-\langle\partial_{\mu}\psi_{n}^{0},a^{
\mu\nu}\partial_{\nu}\psi\rangle\\&+\langle(\lambda_{n}-V)\psi,
\psi\rangle+\langle(V_{\mathrm{eff}}-V+\lambda_{n}-\lambda_{n}^{0})\psi_{n}^{0},
\psi\rangle.
\end{split}
\end{equation*}
By the Cauchy-Schwarz and Young inequalities we have
\begin{equation}\label{eq:product_est}
\begin{split}
\langle\partial_{\mu}\psi,&G^{\mu\nu}\partial_{\nu}\psi\rangle+\varepsilon^{-2}
(\|\partial_{u}\psi\|^2-E_{1}\|\psi\|^2)
\leq \frac{\delta}{2}\|\nabla_{\! a}\psi_{n}^{0}\|^2\\
&+\frac{1}{2\delta}\|\nabla_{\! a}\psi\|^2+\|\lambda_{n}-V\|_{\infty}\|\psi\|^2
+\|V_{\mathrm{eff}}-V+\lambda_{n}-\lambda_{n}^{0}\|_{\infty}\,\|\psi_{n}^{0}\|\,
\|\psi\|.
\end{split}
\end{equation}
with any $\delta>0$.

Estimates~\eqref{eq:metric_est} imply
$$-(1-C_{+}^{-1})g^{\mu\nu}\leq  a^{\mu\nu}\leq(C_{-}^{-1}-1)g^{\mu\nu}.$$
For $\delta\leq 1/2$, we have
$$G^{\mu\nu}-\frac{1}{2\delta}a^{\mu\nu}\geq
g^{\mu\nu}+\left(1-\frac{1}{2\delta}\right)(C_{-}^{-1}-1)g^{\mu\nu}.$$
If $\varepsilon\varrho_{m}^{-1}\leq 1-\sqrt{2/3}$, then $(C_{-}^{-1}-1)\leq
1/2$.  Consequently we may set $\delta=(C_{-}^{-1}-1)$ to conclude that
$$G^{\mu\nu}-\frac{1}{2\delta}a^{\mu\nu}\geq
\frac{1}{2}g^{\mu\nu}+(C_{-}^{-1}-1)g^{\mu\nu}\geq\frac{1}{2}g^{\mu\nu}.$$
Now, from \eqref{eq:product_est} it follows that

\begin{equation*}
\begin{split}
 \frac{1}{2}\|\nabla_{\!
g}\psi\|^2+\varepsilon^{-2}(\|\partial_{u}\psi\|^2-E_{1}\|\psi\|^2)&\leq\frac{C_
{-}^{-1}-1}{2}\|\nabla_{\! a}\psi_{n}^{0}\|^2+\mathcal{O}(\varepsilon^2)\\
 &\leq\frac{(C_{-}^{-1}-1)^2}{2}\|\nabla_{\!
g}\psi_{n}^{0}\|^2+\mathcal{O}(\varepsilon^2)
 \\
 &=\mathcal{O}(\varepsilon^2).
\end{split}
\end{equation*}

Here we used that
$\|\psi_{n}^{0}\|<\infty,\, \|\nabla_{\! g}\psi_{n}^{0}\|<\infty$,
and

\begin{equation}\label{eq:pot_bound}
\begin{aligned}
\|\lambda_{n}-V\|_{\infty}&=\mathcal{O}(1),
&&& (C_{-}^{-1}-1)&=\mathcal{O}(\varepsilon),
\\
\|V_{\mathrm{eff}}-V+\lambda_{n}-\lambda_{n}^{0}\|_{\infty}
&=\mathcal{O}(\varepsilon),
&&& \|\psi\| &=\mathcal{O}(\varepsilon),
\end{aligned}
\end{equation}

as $\varepsilon\to 0$. Since for all $\psi\in\mathcal{H}_{0}^{1}(\Sigma\times
I)$, $\|\partial_{u}\psi\|^2\geq E_{1}\|\psi\|^2$
by Fubini's theorem and the Poincar\'{e} inequality

\begin{equation}\label{Poincare}
  \forall \varphi \in \mathcal{H}_{0}^{1}(I) \,, \qquad
  \int_I |\partial_{u}\varphi|^2\, \dd u
  \geq E_1 \int_I |\varphi|^2 \, \dd u \,,
\end{equation}

we finally obtain
\begin{equation}\label{eq:H_1_convergence}
 \|\nabla_{\! g}\psi\|=\mathcal{O}(\varepsilon),\quad
\|\partial_{u}\psi\|=\mathcal{O}(\varepsilon).
\end{equation}

\begin{remark}
Let  $\mathcal{K}$ be a compact set in the range of a chosen local coordinate
map on $\Sigma$. Recall that the entries of $(g_{\mu\nu})$ are continuous.
Then due to the
local boundedness and positive definiteness
of $g$ we have
$$\inf_{\mathcal{K}}\|g\|^{-1}>0,\quad \inf_{\mathcal{K}}|g|>0,$$
where $\|g\|:=\|g\|_{\C^{d-1}\to\C^{d-1}}$.
Moreover, the matrix estimate $g_{\mu\nu}\leq\|g\|\,\delta_{\mu\nu}$ implies
$$\|g\|^{-1}\,\delta^{\mu\nu}\leq g^{\mu\nu}.$$
Since
$$
\begin{aligned}
  \langle\partial_{\mu}\psi,g^{\mu\nu}\partial_{\nu}\psi\rangle
  &\geq\langle\|g\|^ {-1}\,\delta^{\mu\nu}\partial_{\mu}\psi,
  \partial_{\nu}\psi\rangle
  \\
  &\geq \inf_{\mathcal{K}}\|g\|^{-1}\,
  \inf_{\mathcal{K}}|g|^{1/2}\int_{\mathcal{K}\times I}
  \sum_{i=1}^{d-1}|\partial_{i}\psi|^2\dd x,
\end{aligned}
$$
we conclude that
$$\int_{\mathcal{K}\times
I}\sum_{i=1}^{d-1}|\partial_{i}\psi|^2=\mathcal{O}(\varepsilon^2),$$
due to \eqref{eq:H_1_convergence}.
\end{remark}

\subsection{Interior $\mathcal{H}^2$-convergence}\label{sec:H2_conv}
In this section we assume that $\psi_{n}$ and $\psi_{n}^{0}$ are sufficiently
regular so that all their derivatives up to the $3$rd order exist in the
classical sense and are interchangeable. This holds true if
\begin{equation}\tag*{$\langle A3\rangle$}\label{eq:as3}
 \Sigma\in C^l,\ \partial\Sigma\in C^l,\quad\text{with }l>d/2+6,
\end{equation}
as one may see from Remark \ref{rem:metric_cont}, the elliptic regularity, and
the Sobolev embedding theorem. Our proof of the $\mathcal{H}^2$-convergence of
the eigenfunctions is motivated by the proof of the interior regularity for
elliptic operators (see \cite[Theo.~1 in $\S$6.3.1]{evans}). The major
difference is that we have to treat $\varepsilon^{-2}$-terms in  a special
manner.
In fact, due to the product character of the domain, we will prove the
regularity up to the transverse part of  the boundary, $\Sigma\times\partial I$.

Let $W$ be an open bounded subset of $\R^{d-1}$ such that the closure of $W$
(denoted by $\overline{W}$)
lies in the range of a local coordinate map on $\Sigma$. Furthermore, let $U$
stand for an open precompact subset containing $\overline{W}$ and contained in
the range of the same coordinate map. Then there exists a smooth function
$\zeta$ (known as the bump function for $\overline{W}$ supported in $U$) with
the following properties: $0\leq\zeta\leq1$,
$\zeta = 1$ on $\overline{W}$,
and $\mathrm{supp}~\zeta\subset U$. For $k=1,2,\ldots,d$, set
\begin{equation}\label{eq:test_funct}
v:=-\partial_{k}\zeta^2\partial_{k}\psi\quad(\text{no summation over }k).
\end{equation}
Using integration by parts carefully, we obtain

\begin{align}
  \langle -\partial_{\mu}G^{\mu\nu}\partial_{\nu}\psi,v\rangle_{U\times I}
&=\langle\partial_{k}G^{\mu\nu}\partial_{\nu}\psi,\partial_{\mu}\zeta^2\partial_
{k}\psi\rangle_{U\times I} \label{eq:per_partes}
  \\
 \langle \psi,v\rangle_{U\times I}
  &=\langle\partial_{k}\psi,\zeta^2\partial_{k}\psi\rangle_{U\times I}
\label{eq:per_partes2}
  \\
  \langle -\partial_{u}^{2}\psi,v\rangle_{U\times I}
&=\langle\partial_{u}\partial_{k}\psi,\zeta^2\partial_{u}\partial_{k}
\psi\rangle_{U\times I},\label{eq:per_partes3}
\end{align}

where $\langle\cdot,\cdot\rangle_{U\times I}$ is the scalar product of
$L^{2}(U\times I,\dd x_1\ldots\dd x_d)$. The boundary terms vanish because
$\psi=\partial_{\mu}\psi=\partial_{\mu}\partial_{\nu}\psi=0$ on $U\times\partial
I$ and $\zeta=\partial_{j}\zeta^2\partial_{k}\psi=0$ on $\partial U\times I$.

If we  multiply (on $L^{2}(U\times I,\dd x_1\ldots\dd x_d)$) the both sides of
($\ref{eq:psi_diff}$) by $v$ we have
$$A_{1}+\varepsilon^{-2}A_{2}=B,$$
where
\begin{align*}
 A_{1}&:=\langle\partial_{k}G^{\mu\nu}\partial_{\nu}\psi,
 \partial_{\mu}\zeta^2\partial_{k}\psi\rangle_{U\times I},\nonumber\\
 A_{2}&:=\langle\partial_{u}\partial_{k}\psi,\zeta^2\partial_{u}\partial_{k}
\psi\rangle_{U\times I}-E_{1}\langle\partial_{k}\psi,
\zeta^2\partial_{k}\psi\rangle_{U\times I},\label{eq:poincare}\\
 B&:=\langle-(|g|^{-1/2})_{,\mu}G^{\mu\nu}|g|^{1/2}\partial_{\nu}\psi+f_{
\varepsilon}+(\lambda_{n}-V)\psi,v\rangle_{U\times I},\nonumber
\end{align*}
with
\begin{equation*}
 f_{\varepsilon}:=|g|^{-1/2}\partial_{\mu}|g|^{1/2}a^{\mu\nu}\partial_{\nu}\psi_
{n}^{0}+(V_{\mathrm{eff}}-V+\lambda_{n}-\lambda_{n}^{0})\psi_{n}^{0}.
\end{equation*}
Note that, as $\partial_{u}\zeta=0$ everywhere and $\partial_{\mu}\psi=0$ on
$U\times\partial I$, $A_{2}\geq 0$ if $k=1,\ldots,d-1$, due to Fubini's theorem
and the Poincar\'{e} inequality~\eqref{Poincare}.

\begin{remark}
In virtue of Remark~\ref{rem:metric_cont},
$$\sup_{U\times I}|a^{\mu\nu}|=\mathcal{O}(\varepsilon), \qquad \sup_{U\times
I}|\partial_\mu a^{\mu\nu}|=\mathcal{O}(\varepsilon).$$
This together with \eqref{eq:pot_bound} implies that
$$\int_{U\times I}|f_{\varepsilon}|^2=\mathcal{O}(\varepsilon^2).$$
\end{remark}

One immediately has
\begin{equation*}
 A_{1}=A_{1}'+A_{1}'',
\end{equation*}
where
\begin{equation*}
\begin{split}
 &A_{1}':=\langle
G^{\mu\nu}\partial_{\nu}\partial_{k}\psi,\zeta^2\partial_{\mu}\partial_{k}
\psi\rangle_{U\times I}\\
 &\begin{split}A_{1}'':=&\langle
G^{\mu\nu}_{,k}\partial_{\nu}\psi,2\zeta\zeta_{,\mu}\partial_{k}\psi\rangle_{
U\times I}+\langle
G^{\mu\nu}_{,k}\partial_{\nu}\psi,\zeta^2\partial_{\mu}\partial_{k}\psi\rangle_{
U\times I}\\
  &+\langle G^{\mu\nu}\partial_{k}\partial_{\nu}\psi,2\zeta\zeta_{,\mu}
\partial_{k}\psi\rangle_{U\times I}.
 \end{split}
\end{split}
\end{equation*}

Each part of this decomposition may be estimated separately. We have
\begin{equation}\label{eq:A_1'_est}
 A_{1}'\geq\gamma\langle\zeta^2\delta^{\mu\nu}\partial_{\nu}\partial_{k}\psi,
\partial_{\mu}\partial_{k}\psi\rangle_{U\times I}=\gamma\int_{U\times I}\zeta^2
|\tilde{D}\partial_{k}\psi|^2,
\end{equation}
where $\tilde{D}\psi:=(\partial_1 \psi,\ldots,\partial_{d-1}\psi)$ and the
positive constant $\gamma$ may be chosen as a uniform bound for all
$\varepsilon\in(0,\varepsilon_{U})$ with some fixed
$\varepsilon_{U}<\varrho_{m}$. In view of Remark~\ref{rem:metric_cont}, it is
also clear that the quantities
$$\sup_{U\times I}|G^{\mu\nu}| \qquad \text{and} \qquad \sup_{U\times
I}|G^{\mu\nu}_{,k}|$$
are uniformly bounded on $(0,\varepsilon_{U})$. Therefore, for
$\varepsilon<\varepsilon_{U}$, we can estimate
\begin{equation}\label{eq:A_1''_est}
 \begin{split}
 |A_{1}''|&\leq C\int_{U\times
I}\left(\zeta|\tilde{D}\psi||\partial_{k}\psi|+\zeta|\tilde{D}\psi||\tilde{D}
\partial_{k}\psi|+\zeta|\partial_{k}\tilde{D}\psi||\partial_{k}\psi|\right)\\
 &\leq 2C\int_{U\times
I}\Big(\delta\zeta^2|\tilde{D}\partial_{k}\psi|^2+\frac{1}{4\delta}\big(|\tilde{
D}\psi|^2+|\partial_{k}\psi|^2\big)\Big),
 \end{split}
\end{equation}
where $0<\delta\leq1/2$ and $C=C(W,U,\zeta,\Sigma)$.

Next, for all $\varepsilon\in(0,\varepsilon_{U}')$ with
$\varepsilon_{U}'<\varepsilon_{U}$ sufficiently small, we obtain
\begin{equation}\label{eq:B_est}
 \begin{split}
 |B|&\leq C'\int_{U\times I}(|\tilde{D}\psi|+|f_{\varepsilon}|+|\psi|)|v|\\
 &\leq C'\int_{U\times
I}\Big(\alpha|v|^2+\frac{3}{4\alpha}\big(|\tilde{D}\psi|^2+|f_{\varepsilon}
|^2+|\psi|^2\big)\Big),
 \end{split}
\end{equation}
where $\alpha>0$, and $C'=C'(W,U,\zeta,\Sigma)$. We  also have
\begin{equation}\label{eq:v_est}
 \int_{U\times I}|v|^2\leq C''\left(\int_{U\times
I}|\partial_{k}\psi|^{2}+\int_{U\times I}\zeta^2|\partial_{k}^{2}\psi|^2\right)
\end{equation}
with $C''=C''(W,U,\zeta)$.

Now we will distinguish two cases:

$\bullet \boldsymbol{k=1,2,\ldots,d-1:}$
Using \eqref{eq:A_1'_est}, \eqref{eq:A_1''_est}, \eqref{eq:B_est}, and
\eqref{eq:v_est}, the  estimate
\begin{equation}\label{eq:reg_est}
A_{1}'-|A_{1}''|+\varepsilon^{-2}A_{2}\leq |B|
\end{equation}
yields
\begin{multline}\label{eq:H^2_main_est}
  (\gamma-2C\delta)\int_{U\times
I}\zeta^2|\tilde{D}\partial_{k}\psi|^2-C'C''\alpha\int_{U\times
I}\zeta^2|\partial_{k}^{2}\psi|^2\\
 +\varepsilon^{-2}\left(\int_{U\times
I}\zeta^2|\partial_{u}\partial_{k}\psi|^2-E_{1}\int_{U\times
I}\zeta^2|\partial_{k}\psi|^2\right)\\
 \leq \Big(\frac{C}{2\delta}+C'C''\alpha\Big)\int_{U\times
I}|\partial_{k}\psi|^2+\Big(\frac{C}{2\delta}+\frac{3C'}{4\alpha}\Big)\int_{
U\times I}|\tilde{D}\psi|^2\\
 +\frac{3C'}{4\alpha}\int_{U\times I}(|f_{\varepsilon}|^2+|\psi|^2).
\end{multline}
All the terms on the right hand side are of the order
$\mathcal{O}(\varepsilon^2)$. Therefore, with~$\delta$ and~$\alpha$ small enough
we have
\begin{align}
  &\int_{U\times I}\zeta^2|\tilde{D}\partial_{k}\psi|^2
  =\mathcal{O}(\varepsilon^2),\label{eq:mixed_der_conv_D}\\
  &\int_{U\times I}\zeta^2|\partial_{u}\partial_{k}\psi|^2
  =\mathcal{O}(\varepsilon^{2}).\label{eq:mixed_der_conv}
\end{align}

$\bullet \boldsymbol{k=d:}$
From \eqref{eq:mixed_der_conv} and the inequality
$$A_{1}'+\varepsilon^{-2}\int_{U\times I}\zeta^2 |\partial_{u}^{2}\psi|^{2}\leq
|B|+|A_{1}''|+\varepsilon^{-2}E_{1}\int_{U\times I}\zeta^2|\partial_{u}\psi|^2$$
we show in a similar manner as above that
\begin{equation}\label{eq:uu_der_conv}
\int_{U\times I}\zeta^2|\partial_{u}^{2}\psi|^2=\mathcal{O}(\varepsilon^2).
\end{equation}

\subsection{$\mathcal{H}^2$-convergence up to the boundary}
In this subsection we will additionally assume that $\Sigma$ admits a uniformly
positive definite metric, and so estimate \eqref{eq:A_1'_est} remains valid up to the boundary.

Under \ref{eq:as3}, $\partial\Sigma$ is definitely $C^2$-smooth, which means
that for every boundary point $x_{\shortparallel,0}$ from the range of a local
boundary coordinate map (denote this  range by $R$) there exists a ball,
$B_{x_{\shortparallel,0}}(r)$, and $\alpha\in\{1,2,\ldots, d-1\}$ such that
$$R\cap B_{x_{\shortparallel,0}}(r)=\{(x_1,x_2,\ldots,x_{d-1})\in
B_{x_{\shortparallel,0}}(r)\,|\,
x_{\alpha}\geq\gamma(x_1,\ldots,\hat{x}_{\alpha},\ldots,x_{d-1})\},$$
where $\gamma\in C^2(\R^{d-2})$. This makes it possible to pass to  new local
coordinates with a straightened boundary;
in these coordinates (denoted by the
same symbols) defined on some open half-ball of radius $\delta$, $U(\delta)$,
we have $x_\alpha=0$ and $\psi=0$ on the straight part of $\partial
U(\delta)\times I$. Consequently,
$\partial_{j}\psi=\partial_{j}\partial_{l}\psi\equiv 0$ on
$\{x_{\shortparallel}\in\partial U(\delta)\,|\, x_\alpha=0\}\times I$, for
$j,l\neq\alpha$.

Now we may continue in the similar manner as when proving the convergence on the
``inner'' domains. More concretely, we set $U=U(\delta)$ and introduce a new
bump function that equals $1$ on a half-ball that has the common centre as
$U(\delta)$ but a smaller radius and it is completely contained in $U(\delta)$.
Beware that~\eqref{eq:per_partes} does not hold for $k=\alpha$ anymore.
Nevertheless it still holds true for $k\neq\alpha$,
and so for  $k\in\{1,\ldots,\hat{\alpha},\ldots,d-1\}$,
\eqref{eq:mixed_der_conv_D} and \eqref{eq:mixed_der_conv} remain valid too.  If
we prove \eqref{eq:mixed_der_conv} for $k=\alpha$, we will also have
\eqref{eq:uu_der_conv}.
Moreover if we prove that
$$\int_{U(\delta)\times
I}\zeta^2|\partial^2_{\alpha}\psi|^2=\mathcal{O}(\varepsilon^2),$$
we shall obtain the local $\mathcal{H}^{2}$-convergence up to the boundary.

But this may be done as follows. In the decomposition
\begin{equation*}
 \langle-\partial_{\mu}G^{\mu\nu}\partial_{\nu}\psi,
-\partial_\alpha\zeta^2\partial_\alpha\psi\rangle_{U(\delta)\times
I}=\tilde{A}_{1}'+\tilde{A}_{1}'',
\end{equation*}
with
\begin{align*}
 \tilde{A}_{1}':=&\langle
G^{\alpha\alpha}\partial^{2}_{\alpha}\psi,\zeta^2\partial^{2}_{\alpha}
\psi\rangle_{U(\delta)\times I},\\
 \begin{split}\tilde{A}_{1}'':=&\langle
G^{\alpha\alpha}_{,\alpha}\partial_{\alpha}\psi,\zeta^2\partial^{2}_{\alpha}
\psi\rangle_{U(\delta)\times I}+\langle
G^{\alpha\alpha}\partial^{2}_{\alpha}\psi,2\zeta\zeta_{,\alpha}\partial_{\alpha}
\psi\rangle_{U(\delta)\times I}\\
 &+\langle
G^{\alpha\alpha}_{,\alpha}\partial_{\alpha}\psi,2\zeta\zeta_{,\alpha}\partial_{
\alpha}\psi\rangle_{U(\delta)\times
I}+\sum_{\mu\neq\alpha\vee\nu\neq\alpha}\langle\partial_{\mu}
G^{\mu\nu}\partial_{\nu}\psi,\zeta^2
\partial^{2}_{\alpha}\psi\rangle_{U(\delta)\times I}\\
 &+\sum_{\mu\neq\alpha\vee\nu\neq\alpha}\langle\partial_{\mu}
G^{\mu\nu}\partial_{\nu}\psi,2\zeta\zeta_{,\alpha}\partial_{\alpha}\psi\rangle_{
U(\delta)\times I},
 \end{split}
\end{align*}
we may estimate $\tilde{A}'_{1}$ from below and $|\tilde{A}''_{1}|$ from above
in the same way as we have estimated $A'_{1}$ and $|A''_{1}|$, respectively.
Finally, the inequality
$$\tilde{A}_{1}'-|\tilde{A}_{1}''|+\varepsilon^{-2}A_{2}\leq |B|$$
leads to an equation that is similar to \eqref{eq:H^2_main_est} which in turn
yields the desired result.

\subsection{Interior $\mathcal{H}^k$-convergence and $C^0$-convergence}
In this section we show that under some stronger regularity assumptions on
$\Sigma$, for any $k\geq 3$,
\begin{equation*}
 \|\psi\|_{\mathcal{H}^k(W\times I)}=\mathcal{O}(\varepsilon),
\end{equation*}
whenever we have this result for $k-1$. Within this section $W$ and $U$ are the
same as in Section \ref{sec:H2_conv}. Let us assume that
\begin{equation}\tag*{$\langle A4\rangle$}\label{eq:H^k_assumptions}
\Sigma\in C^l\quad\text{with }l\in\N,\, l>2(k+1)+\frac{d}{2},
\end{equation}
which implies that all derivatives of $|g|^{\pm 1/2}$ up to the order $k-1$ are
bounded on any compact set, all derivatives of $G^{\mu\nu}$ up to the same order
are uniformly bounded in $\varepsilon$ (on some right neighbourhood of zero) on
any compact set, and
$$
 \sup_{U\times I}|D^\alpha V-D^\alpha V_{\mathrm{eff}}|
  =\mathcal{O}(\varepsilon)
  \,, \qquad
 \sup_{U\times I}|D^\alpha a^{\mu\nu}|
 =\mathcal{O}(\varepsilon) \,,
$$
where $\alpha$ is a $d$-dimensional multiindex with $|\alpha|=k-1$. In fact, the
implications above hold true also under a weaker regularity assumption
than~\ref{eq:H^k_assumptions}, but we need \ref{eq:H^k_assumptions} for a
sufficient regularity of~$\psi$ anyway
(cf.\ the definition of~$v$ below).

One may suggest proceeding
as in the proof of the $\mathcal{H}^2$-convergence but start with
$$v=(-1)^{|\alpha|}D^{\alpha}\zeta^2 D^{\alpha}\psi$$
instead of \eqref{eq:test_funct}. Here
$D^\alpha=\partial^{\alpha_{1}}_{1}\ldots\partial^{\alpha_{d}}_{d}$. However in
this way we obtain
\begin{equation}\label{eq:k_Sobolev_est}
 \|\partial_k D^\alpha \psi\|_{W\times I}=\mathcal{O}(\varepsilon)
\end{equation}
only if $\alpha_d=0$. This restriction is necessary for diminishing of boundary
terms during the integration by parts (cf.\ \eqref{eq:per_partes},
\eqref{eq:per_partes2}, and \eqref{eq:per_partes3}).
Also remark that in comparison to
Section~\ref{sec:H2_conv}
we have to treat the term denoted by~$B$ more carefully.
At first we integrate by parts $(|\alpha|-1)$-times to get

\begin{align*}
  B=\ &\langle-(|g|^{-1/2})_{,\mu}G^{\mu\nu}|g|^{1/2}
  \partial_{\nu}\psi+f_{\varepsilon}+(\lambda_{n}-V)\psi,v\rangle_{U\times I}
  \\
  =\ &-\big\langle -D^{\tilde{\alpha}}(|g|^{-1/2}),_{\mu}G^{\mu\nu}|g|^{1/2}
  \partial_\nu\psi+D^{\tilde{\alpha}}f_\varepsilon
  \\
  & \qquad +D^{\tilde{\alpha}}
  [(\lambda_n-V)\psi], \,
  2\zeta\zeta,_\varrho D^\alpha \psi+\zeta^2
  \partial_\varrho D^\alpha \psi\big\rangle_{U\times I}
  \,.
\end{align*}

Here $\varrho\in\{1,\ldots, d-1\}$ is such that $\alpha_\varrho\neq 0$ and  the
multiindex $\tilde{\alpha}$ is given by the decomposition
$\alpha=\tilde{\alpha}+(0,\ldots,0,1,0,\ldots 0)$, where, in the latter
multiindex, the only non-zero entry stands on the $\varrho$th position.
Next we find a bound for $|B|$ in a similar manner as in \eqref{eq:B_est} and
finally, in  \eqref{eq:reg_est}, we put on the left-hand-side only that term
from this bound that contains
$$\int_{U\times I}\zeta^2 |\partial_\rho D^\alpha\psi|^2.$$

Now, with \eqref{eq:k_Sobolev_est} in hand, let us focus on higher derivatives
that contain at least the second derivative with respect to~$u$.
We differentiate~\eqref{eq:psi_diff} with respect to $x_\gamma$ and then rewrite
the result as
\begin{equation*}
-\partial_\gamma\partial_{u}^{2}\psi=\varepsilon^2\partial_\gamma
|g|^{-1/2}\partial_\mu |g|^{1/2}G^{\mu\nu}\partial_\nu \psi+E_1 \partial_\gamma
\psi+\varepsilon^2 \partial_\gamma f_\varepsilon +\varepsilon^2 \partial_\gamma
(\lambda_n-V) \psi.
\end{equation*}
Next we take $L^2(W\times I)$-norm of both sides to get
\begin{multline*}
 \|\partial_\gamma\partial_{u}^{2}\psi\|_{W\times I}\leq  \varepsilon^2
\|G^{\mu\nu}\|_\infty \|\partial_\gamma\partial_\mu\partial_\nu \psi\|_{W\times
I}+\varepsilon^2 C''' \|\psi\|_{\mathcal{H}^2(W\times I)}\\
 +E_1 \|\partial_\gamma \psi\|_{W\times I}+\varepsilon^2 (\mathrm{vol}(W\times
I))^{1/2} \sup_{W\times I}|\partial_{\gamma}f_\varepsilon|,
\end{multline*}
where $C'''=C'''(W,\Sigma,n)$.
Using \eqref{eq:k_Sobolev_est} and the above mentioned consequences of
\ref{eq:H^k_assumptions} we conclude that
\begin{equation*}
  \|\partial_\gamma\partial_{u}^{2}\psi\|_{W\times I}=\mathcal{O}(\varepsilon)
\end{equation*}
as $\varepsilon\to 0$.
Differentiating \eqref{eq:psi_diff} with respect to $u$ we obtain, in a similar
manner,
\begin{equation*}
  \|\partial_{u}^{3}\psi\|_{W\times I}=\mathcal{O}(\varepsilon).
\end{equation*}
It is clear that one may use the same strategy
to estimate also all higher derivatives by a successive differentiation.

\begin{theorem}\label{theo:Hk_conv}
Assume \ref{eq:eps_bound}, \ref{eq:res_conv_ass}, and
\begin{equation}\tag*{$\langle A5\rangle$}\label{eq:reg_ass_final}
\Sigma\in C^l\quad\text{with }l\in\N,\,l>\lfloor 3d/2\rfloor+4
\end{equation}
(there may be even an equality for $d$ odd).
 Then for any open and bounded subset $W$ such that its closure lies in the
range of a local coordinate map on~$\Sigma$, we have
\begin{equation*}
 \|\psi\|_{\mathcal{H}^k(W\times I)}=\mathcal{O}(\varepsilon)
\end{equation*}
with $k=\lfloor d/2\rfloor+1$.
Moreover,
\begin{equation*}
 \|\psi\|_{C^{0,\gamma}(\overline{W\times
I})}\equiv\|\psi\|_{C(\overline{W\times I})}+\sup_{x,y\in W\times I,x\neq
y}\left\{\frac{|\psi(x)-\psi(y)|}{|x-y|^{\gamma}}\right\}=\mathcal{O}
(\varepsilon),
\end{equation*}
where $0<\gamma<1$ for $d$ even and $\gamma=1/2$ for $d$ odd.
\end{theorem}

The second statement follows directly from
the general Sobolev embedding
$\mathcal{H}^k(W \times I) \hookrightarrow C^{0,\gamma}(\overline{W \times I})$,
cf.~\cite[Thm.~5.4]{adams}.

\subsection{Interior $C^{0,1}$-convergence in the transverse direction}
This subsection is strong\-ly inspired by a similar one in~\cite{FK4}.
However, we had to somehow refine the supersolution used within the proof.

Let $B_{1},B_{2}$ be two
$(d-1)$-dimensional
balls with the common centre
$x_{\shortparallel,0}$
and radii $r_{1},r_{2}$, respectively, such that $r_{1}<r_{2}$ and $B_{2}$ lies
inside the range of a local coordinate map on $\Sigma$. Then
$$h(x_{\shortparallel}):=\chi_{\overline{B_{2}}\setminus
B_{1}}(x_{\shortparallel})(|x_{\shortparallel}-x_{\shortparallel,0}|-r_{1})^3
\ \in \ C^{2}(\overline{B_{2}}).$$

Now, rewrite \eqref{eq:psi_diff} as
$$M\psi=F_{\varepsilon}$$
with
\begin{align*}
 &M :=
  -\varepsilon^2 |g|^{-1/2}\partial_{\mu}|g|^{1/2}G^{\mu\nu}\partial_{\nu}
  -\partial_{u}^{2} \,, \\
 &F_{\varepsilon} :=
  \varepsilon^2 (\lambda_{n}-V)\psi+\varepsilon^2 f_{\varepsilon}+E_{1}\psi \,.
\end{align*}
Let us introduce, on $B_{2}\times I$,
$$w(x_{\shortparallel},u) :=
\beta\left(\mathrm{e}^{-(1+h(x_{\shortparallel}))^{-1}}-\mathrm{e}^{-|u\pm
2|}\right),$$
where $\beta>0$ will be determined below. We have
\begin{equation*}
 Mw=\beta\big(-\varepsilon^2
|g|^{-1/2}\partial_{\mu}|g|^{1/2}G^{\mu\nu}\partial_{\nu}\mathrm{e}^{-(1+h(x_{
\shortparallel}))^{-1}}+\mathrm{e}^{-|u\pm 2|}\big)\geq
\frac{\beta}{2}\mathrm{e}^{-3}
\end{equation*}
for all $\varepsilon$ smaller than some sufficiently small $\varepsilon_{1}$.
Furthermore,
\begin{equation}\label{eq:boundary_est}
w|_{\partial B_{2}\times
I}(x_{\shortparallel},u)\geq\beta\big(\mathrm{e}^{-(1+(r_{2}-r_{1})^3)^{-1}}
-\mathrm{e}^{-1}\big)=:\beta L
\end{equation}
and $w\geq 0$ on $\overline{B_{2}\times I}$.

If we set (under the assumption that the both norms are finite)
$$
  \beta:=
\max\left\{2\mathrm{e}^3\|F_{\varepsilon}\|_{C(\overline{B_{2}\times
I})},L^{-1}\|\psi\|_{C(\overline{\partial B_{2}\times I})} \right\},$$
then
\begin{align*}
 Mw&\geq M(\pm\psi) &&\text{in}\quad B_{2}\times I \,,\\
 w&\geq\pm\psi &&\text{on}\quad \partial(B_{2}\times I) \,,
\end{align*}
which implies
$$|\psi|\leq w\quad \text{on } \overline{B_{2}\times I}$$
in virtue of the maximum principle \cite[Chap.2, Theo. 6]{PW}. Namely for all
$(x_{\shortparallel},u)\in B_{1}\times I$,
\begin{equation*}
 |\psi(x_{\shortparallel},u)|\leq
w(x_{\shortparallel},u)=\beta\big(\mathrm{e}^{-1}-\mathrm{e}^{-|u\pm
2|}\big)=\beta\mathrm{e}^{-1}\big(1-\mathrm{e}^{\mp
u-1}\big)\leq\beta\mathrm{e}^{-1}|u\pm 1|.
\end{equation*}

Under the assumptions  of Theorem \ref{theo:Hk_conv},
$\beta=\mathcal{O}(\varepsilon)$, and we have
\begin{theorem}
Assume \ref{eq:eps_bound}, \ref{eq:res_conv_ass}, and \ref{eq:reg_ass_final}.
Let $B$ be an open ball that lies inside the range of a local coordinate map on
$\Sigma$. Then constants $\varepsilon_{2}>0$ and $K$ that depend on $\Sigma$ and
$B$ (and in particular on $\dist(\partial B,\partial\Sigma)$) exist such that
for all $\varepsilon<\varepsilon_{2}$,
\begin{equation}\label{eq:trans_der}
 \sup_{(x_{\shortparallel},u)\in B\times
I}\frac{|\psi(x_{\shortparallel},u)|}{|u\pm 1|}\leq K\varepsilon.
\end{equation}
\end{theorem}

\begin{remark}[$C^{0,1}$-convergence in the transverse direction up to the
boundary for $d=2,3$]\label{rem:C_conv} Once we have $\mathcal{H}^2$-convergence
up to the boundary, we can easily modify argument above to prove also
$C^{0,1}$-convergence in the transverse direction up to the boundary in the case
$d=2,3$. It is sufficient to change $B_{j}$ for some half-balls
in the range of a local boundary coordinate map.
\eqref{eq:boundary_est} then holds only on the non-flat component of $\partial B_{2}$.
However on the remaining component of the
boundary, $\psi=0$, thus it is also bounded above by $w(\geq 0)$.
\end{remark}

\section{Local convergence of nodal sets}\label{Sec.nodal}
In this section we fix $N\in\N$ and consider only $\varepsilon<\varepsilon(N)$,
where $\varepsilon(N)$ is given in Remark \ref{rem:H_0_spec}.

\begin{lemma}\label{lem:cut}
Let $\mathcal{N}\equiv\mathcal{N}(\phi_n)$ be the nodal set of $\phi_{n}$ and
$B$ be an open ball inside the range of a local coordinate map on $\Sigma$ which
is split by $\mathcal{N}$
to exactly two (open) non-empty components
$D_{1}$ and $D_{2}$ such that $\phi_{n}(x_{\shortparallel})\neq 0$
for all $x_{\shortparallel}\in D_{i}$. Then $\phi_{n}$ is positive on $D_{1}$
and negative on $D_{2}$ (or vice versa, but we will set the notation in the
manner that the former is always true).
 Moreover, if there exists a ball in $D_{2}$ that touches $\mathcal{N}$ at
$x_{\shortparallel}\in\mathcal{N}\cap B$  we have
 $$\frac{\partial\phi_{n}}{\partial
n_{x_{\shortparallel}}}(x_{\shortparallel})>0,$$
 where $n_{x_{\shortparallel}}$ is the normal vector to $\mathcal{N}$ at
$x_{\shortparallel}$ pointing out from $D_{2}$.
\end{lemma}
\begin{proof}
Assume that $\phi_{n}$ does not change its sign on $B$, e.g. $\phi_{n}\leq 0$.
Thus $\phi_{n}$ attains its maximum $(=0)$ at all points of $\mathcal{N}\cap B$.
We have
$$\Delta_{g}\phi_{n}-(V_{\mathrm{eff}}-\sigma_n)_{+}\phi_{n}=-(V_{\mathrm{eff}}
-\sigma_n)_{-}\phi_{n}\geq 0$$
on $B$.
The maximum principle \cite[Chap.~2, Theo.~6]{PW} now says that if $\phi_{n}$
attains a non-negative maximum $M$ at an interior point of~$B$
then $\phi_{n}\equiv M=0$ on~$B$.
But this implies that $\phi_n\equiv 0$ on $\Sigma$ which is not possible,
because $\phi_n$ is supposed to be an eigenfunction.

Theorem~8 of \cite[Chap.~2]{PW} applied on~$D_{2}$
then implies the second assertion of the lemma.
\end{proof}

Recall that in general the nodal set for a solution $u$ of an elliptic equation
in $\Omega\subset\R^m$ decomposes into a disjoint union of a $(m-1)$-dimensional
manifold (that contains all the points where $|\nabla u|> 0$) and a closed
countably $(m-2)$-rectifiable subset (that contains all the points where
$|\nabla u|=0$), cf.~\cite{Caffarelli-Friedman_1985} and
\cite{Hardt-Simon_1989}. The latter component is called \emph{the singular set}.
Since the eigenfunction $\phi_n$ is at least $C^3$-smooth under our regularity
assumptions, the former component of $\mathcal{N}(\phi_n)$ is surely a
$C^3$-smooth manifold, and so the assumptions of  Lemma \ref{lem:cut} are
fulfilled away from the singular set.

In particular, for any $z_{\shortparallel}\in \mathcal{N}(\phi_n)$ such that
$|\nabla{\phi_n}(z_{\shortparallel})|>0$, there exists a ball $B$ such that
$z_{\shortparallel}\in B$, $|\nabla{\phi_n}(x_{\shortparallel})|>0$ for all
$x_{\shortparallel}\in B$ , and Lemma \ref{lem:cut} holds true with this $B$.
We will denote by $\tau_\delta$ a closed tubular neighbourhood
about a piece of~$\mathcal{N}(\phi_n)$ of radius~$\delta$ contained in $B$. Let
$D_{1}$ and $n_{x_{\shortparallel}}$ be as in Lemma \ref{lem:cut}.
Take $x_{\shortparallel}\in\tau_\delta\cap D_{1}$, then there exists
$x_{\shortparallel}'\in\tau_\delta\cap\mathcal{N}(\phi_n)$ with the property
$x_{\shortparallel}-x_{\shortparallel}'=|x_{\shortparallel}-x_{\shortparallel}
'|n_{x_{\shortparallel}'}$
($x_{\shortparallel}'$ is not given uniquely but we always have
$|x_{\shortparallel}-x_{\shortparallel}'|\geq
\dist(x_{\shortparallel},\mathcal{N}(\phi_n))$).
Using the Taylor series expansion, we have
\begin{equation*}
 \phi_{n}(x_{\shortparallel})=\phi_{n}(x_{\shortparallel})-\phi_{n}(x_{
\shortparallel}')=\frac{\partial\phi_{n}}{\partial
n_{x_{\shortparallel}'}}|x_{\shortparallel}-x_{\shortparallel}'|+\mathcal{O}(|x_
{\shortparallel}-x_{\shortparallel}'|^2).
\end{equation*}
Using the explicit form of the remainder in the expansion and the regularity of
$\phi_{n}$, we infer that for $\delta$ small enough there exists a positive
constant $A$ such that
\begin{equation}\label{eq:phi_N_bound}
|\phi_{n}(x_{\shortparallel})|\geq
A\,\dist(x_{\shortparallel},\mathcal{N}(\phi_n))\quad\text{for all
}x_{\shortparallel}\in\tau_\delta.
\end{equation}

Since $\psi_{n}^{0}=\phi_{n}\otimes\chi_{1}$,
where~$\chi_{1}$ is given by~\eqref{chi1},
we have
\begin{equation*}
|\psi_{n}^{0}(x_{\shortparallel},u)|\geq
A\,\dist(x_{\shortparallel},\mathcal{N}(\phi_{n}))\dist(u,\partial I),\quad
\forall (x_{\shortparallel},u)\in\tau_\delta\times I.
\end{equation*}
Putting this result together with \eqref{eq:trans_der} we have
\begin{equation*}
 \frac{\psi_{n}\,\sgn\phi_{n}}{\dist(u,\partial I)}\geq
A\,\dist(x_{\shortparallel},\mathcal{N}(\phi_{n}))-K\varepsilon,
\end{equation*}
which implies that, in $\tau_\delta\times I$,
\begin{equation*}
\sgn\psi_{n}(x_{\shortparallel},u)=\sgn\phi_{n}(x_{\shortparallel})
\end{equation*}
whenever $\dist(x_{\shortparallel},\mathcal{N}(\phi_{n}))>KA^{-1}\varepsilon$.
Consequently, if $\varepsilon<AK^{-1}\delta$ then, for any $u\in I$, $x \mapsto
\psi_n(x,u)$ changes its sign on $\tau_\delta$.
We may extend this result along a compact piece of~$\mathcal{N}(\phi_n)$ to
obtain
\begin{theorem}\label{theo:sign_change}
 Assume \ref{eq:eps_bound}, \ref{eq:res_conv_ass}, and \ref{eq:reg_ass_final}.
Let $\tau_\delta$ be a tubular neighbourhood of radius~$\delta$ (with $\delta$
so small that \eqref{eq:phi_N_bound} holds true in $\tau_\delta$ with some
positive constant $A$) about a compact piece of~$\mathcal{N}(\phi_n)$ that lies
inside the range of a local coordinate map on $\Sigma$ and that does not
intersect the singular set of~$\mathcal{N}(\phi_n)$. Then there exists a
positive constant $\tilde{A}$ such that, for any $u\in I$,
$x \mapsto \psi_n(x,u)$ changes its sign on $\tau_\delta$, whenever
$\varepsilon<\tilde{A}\delta$.
\end{theorem}

This immediately implies Theorem \ref{Thm.Conj}, because $\mathcal{N}(\phi_n)$
can not equal to its singular set.

\section{Some more results in special cases}\label{Sec.special}
By the theorem of Courant's, $\phi_n$ has at most $n$ nodal domains. Let us
denote them $N_i,\, i=1,\ldots,m$, where $m\leq n$.
Consider the following special cases.

\subsection{$\boldsymbol{\Sigma}$ compact}

For $1>\delta>0$, let us define
$$
  N_i(\delta):=\{x\in N_i \,|\, \dist{(x,\partial
  N_i})\geq\delta\}
$$
and
$m_i(\delta):=\inf_{N_i(\delta)}|\phi_n|\cos{((1-\delta)\pi/2)}$.
Clearly,
$m_i(\delta)>0$. Using Theorem~\ref{theo:Hk_conv} we infer that there exist
positive constants $\varepsilon_\mathrm{lim}$ and $\tilde{K}$ such that, for all
$\varepsilon\leq\varepsilon_\mathrm{lim}$, we have
\begin{equation}\label{eq:unif_conv}
\|\psi\|_{C(\overline{\cup_{i=1}^{m}N_i(\delta)\times
I})}\leq\tilde{K}\varepsilon.
\end{equation}
If $d=2,3$ we have even stronger result due to Remark \ref{rem:C_conv},
\begin{equation*}
\|\psi\|_{C(\overline{\Sigma\times I})}\leq\tilde{K}\varepsilon,\quad
\forall\varepsilon\leq\varepsilon_\mathrm{lim},
\end{equation*}
with $\tilde{K}$ and $\varepsilon_\mathrm{lim}$ being $\delta$-independent (on
the contrary to the generic case).
Using \eqref{eq:unif_conv} we obtain
$$\psi_n \sgn{\phi_n}\geq|\phi_n\chi_1|-\tilde{K}\varepsilon\geq
m_i(\delta)-\tilde{K}\varepsilon$$
in $N_i(\delta)\times (-1+\delta,1-\delta)$. We conclude that for
\begin{equation}\label{eq:dom_conv_bound}
\varepsilon<\tilde{K}^{-1}\min{\{m_i(\delta),\ i=1,\ldots, n\}},
\end{equation}
$\sgn{\psi_n}(x_{\shortparallel},u)=\sgn{\phi_n}(x_{\shortparallel})$ in
$\cup_{i=1}^{m}N_i(\delta)\times (-1+\delta,1-\delta)$. This means that, for all sufficiently small
$\varepsilon$, the number of nodal domains of $\psi_n$ cannot be smaller than $m$ and that $N_i(\delta)\times
(-1+\delta,1-\delta)$ is contained in a nodal domain of $\psi_n$, which  will be henceforth called the
$i$th nodal domain of $\psi_n$.
Indeed, if the number of nodal domains of $\psi_n$ was smaller that $m$, it
would contradict Theorem \ref{theo:sign_change}. We conclude that the Lebesgue measure of the symmetric
difference of the $i$th nodal domain of $\psi_n$ and $N_i\times I$ tends to zero
as $\varepsilon\to 0$, and that $\mathcal{N}(\psi_n)$ lies in
$L(\delta):=\Sigma\times I\setminus\cup_{i=1}^{m}N_i(\delta)\times(-1+\delta,1-\delta)$
for all $\varepsilon\leq\varepsilon_\mathrm{lim}$.

\subsection{$\boldsymbol{n=2, \Sigma$ compact and closed,
$\boldsymbol{\mathcal{N}(\phi_n)}}$ has empty singular set}

For any second eigenfunction there are exactly two nodal domains. Since $\Sigma$
is compact and the singular set of $\mathcal{N}(\phi_2)$ is empty, we can use
Theorem~\ref{theo:sign_change} along the whole nodal set $\mathcal{N}(\phi_2)$.
Then for all $\varepsilon<\min{\{\tilde{A}_{i}\delta_i\}}$,
$\mathcal{N}(\psi_2)$ is localised in the Cartesian product of the tubular neighbourhood of radius
$\max{\{\delta_i\}}$ about $\mathcal{N}(\phi_2)$ times $I$. So the nodal set of $\psi_2$
converges in the Hausdorff sense to the set $\mathcal{N}(\phi_2)\times I$, as
$\varepsilon\to 0$. Similarly, both the nodal domains of $\psi_2$ converge in
the Hausdorff sense to the Cartesian product of the nodal domains of $\phi_2$
times $I$.

In fact, one can locate
the nodal set and the nodal domains of~$\psi_n$ in a
much more precise way
if $d=2$ (see~\cite{FK4}).
The reason is that in that case we have much more explicit knowledge about
$\phi_n$, since it is just an eigenfunction of some Sturm-Liouville operator.

\subsection*{Acknowledgement}
One of the authors (M.T.) would like thank to the Institute Mittag-Leffler
(Djursholm, Sweden), where this work was started,
for its kind hospitality. The work has been partially supported
by the project RVO61389005
and the grants No.\ 13-11058S and 14-06818S
of the Czech Science Foundation (GA\v{C}R).

\input{nodal11.bbl}

\end{document}

config: arxiv


TITLE: How to solve a system of linear equations without storing the matrix?
QUESTION [7 upvotes]: I have a procedurally defined Hermitian matrix $M$, i.e. I can get any matrix element by calling a black box function (e.g. a library function), and a vector $Y$. And I have to solve a system of linear equations:
$M\cdot X=Y$.
But $Y$ is such that having $n$ elements, it takes about half of available RAM, another half would be for $X$, so if I try to store $M$, it'll take $n^2$ space, i.e. even if I double the RAM space, this will still place $n-2$ matrix columns/rows into swap.
At the same time, all the algorithms which I've found need saving large amounts ($\geq n^2$) of data while solving the system.
So, the question: are there any algorithms to solve systems of linear equations which don't require me to store the matrix, and still are fast enough (maybe not $O(n^3)$, but at least not much slower)?

REPLY [6 votes]: This looks like a situation where the Kaczmarz method could work.
What you do to maintain an approximate solution and then project cyclically onto the hyperplanes which are given by the $k$-th equation. More precisely: If you have the $m$-the iterate $X^m$ and use the $k$-th equation, then you have the next iterate
$$X^{m+1} = X^M + \frac{Y_k - a_k^T\cdot X^m}{\|a_k\|^2}a_i$$
where $a_k$ is the $k$-th row of $A$ and $Y_k$ is the $k$-the entry of $Y$.
Hence, you only need one row of $A$, one entry of $Y$ and the current iterate $X^m$ to perform one iteration, i.e. $2n+1$ space. Also the iteration complexity is very low (it's $\mathcal{O}(n)$) but you usually need a lot of iterations.
This methods is widely used in discrete tomography and is also an instance of the "Projection onto convex sets" method.
Recently it has been shown by Strohmer and Vershynin that a randomized version of this method has favorable convergence properties (when you pick each column with a probability proportional to its norm). Also "block iterative" versions work, i.e. you take a hyperplane of higher codimension to project on. So, if you have some memory left, you could also take some more rows of $A$ at once... See, e.g. here.


TITLE: Arranging word FACETIOUS but vowels must be in order
QUESTION [3 upvotes]: The word FACETIOUS is the most common word in the English language which has each of the five vowels in order from left to right. How many ways can the letters in the word FACETIOUS be arranged such that all five vowels are in order from left to right?
I am wondering if there is an easier way than bashing and counting all the possiblities.

REPLY [2 votes]: There are nine spaces, $5$ vowels that must be in order, and $4$ are consonants.  You can pick any letter first and choose where to place it so lets choose to place the $4$ consonants first.
We can put the $F$ in any of the $9$ places.  We can put the $C$ in any of the remaining $8$.  We can place the $T$ and the $S$ in any of the remaining $7$ and $6$ places.  That's $9\times 8\times 7 \times 6$ ways to place the consonants.
The remaining $5$ spots must be filled with the vowels in order.  There is only one way left to do that.
So there are $9\times 8\times 7 \times 6$ ways.
....
Alternatively.  There are ${9\choose 4}$ ways to pick spots for the consonants, or ${9\choose 5}$ ways to pick spots of the vowels.  For any one of the vowel/consonants placements we can arrange-- in the four spots set for the consonants-- the consonants in $4!$ ways.  So there are ${9\choose 4}\times 4!$ or ${9\choose 5} \times 4!$ ways.
......
Or there are $9!$ ways total to do arrange the nine letters.  However the vowels can be ordered in $5!$ ways and we can only accept $1$ of the the $5!$ orders.  So of all the $9!$ ways to arrange the nine letters we can only accept $1$ out of $5!$ of them.  SO there are $\frac{9!}{5!}$ ways.
===
Method 1: gives us $9\cdot 8 \cdot 7\cdot 6=3024$ ways.
Method 2a: gives us ${9\choose 4}\cdot 4! = \frac {9!}{5!4!}4!= \frac {9!}{5!} = \frac {1\cdot 2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9}{1\cdot2\cdot3\cdot4\cdot5}=6\cdot7\cdot8\cdot9 =3024$ ways.  Method 2b: and ${9\choose 5} = \frac {9!}{4!5!} = {9\choose 4}$ gives us the same answer.
Method 3: gives us $\frac {9!}{5!} = \frac {1\cdot 2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9}{1\cdot2\cdot3\cdot4\cdot5}=6\cdot7\cdot8\cdot9 =3024$ ways.


TITLE: Sums of squares have zero upper density
QUESTION [6 upvotes]: Define the upper density of a set $A \subseteq \mathbf{N}$ to be
$$\bar{d}(A) = \limsup_{n \to \infty} \frac{|A \cap [1,n]|}{n}.$$
Let $A$ be the set of sums of two squares, i.e. $A = \{x^2 + y^2 : x,y \in \mathbf{Z}\}$.
I know that any prime congruent to 1 modulo 4 is the sum of two squares, as is the number $2$ itself. Also, for primes congruent to 3 modulo 4, I know that $p^{2n} = x^2+y^2$ for some $x,y$. Hence, since sums of two squares are closed under multiplication, any number, n, of the form:
$$n = 2^{\alpha} (\prod_{i \leq m_1} p_i^{\beta_i})^{2} \prod_{j \leq m_2} q_j^{\gamma_j},$$
Where $\alpha, \beta_i, \gamma_i, m_1, m_2$ are non-negative integers, $p_i$ is a prime congruent to 3 modulo 4, and $q_j$ is a prime congruent to 1 modulo 4.
I'm having difficulty passing to an argument about upper density about the set of all such $n$, though.

REPLY [6 votes]: Only primes $\equiv 3\pmod 4$ are "obstacles" against $n$ being the sum of two squares. For each such prime $p$ at least those numbers $n\equiv kp\pmod {p^2}$, $1\le k<p$, are not the sum of two squares. This alone would leave us with a density of
$$\prod_{p\equiv 3\pmod 4}\left(1-\frac{p-1}{p^2}\right).$$
However, we neclected higher powers of $p$, i.e. we should additionally filter out $kp^3\pmod{p^4}$, $1\le k<p$, and so on, which leads to
$$\prod_{p\equiv 3\pmod 4}\left(1-\frac{p-1}{p^2}-\frac{p-1}{p^4}-\frac{p-1}{p^6}-\ldots\right)=\prod_{p\equiv 3\pmod 4}\left(1-\frac{1}{p+1}\right).$$
The product does not converge (that is: the sequnce of partial products tends to zero) and therefore the density is zero.
How can we see that thr product diverges? The reciprocal is $\prod_{p\equiv 3}\left(1+\frac 1p\right)>\sum_{p\equiv 3}\frac 1p$. As far as I know, at least $\sum_p\frac 1p$ diverges and by the "equidistribution" of primes modulo $4$, so should $\sum_{p\equiv 3}\frac 1p$.


text:
{\bf Problem.} What is the sum of the coordinates of the midpoint of the segment with endpoints $(6, 12)$ and $(0, -6)$?
{\bf Level.} Level 2
{\bf Type.} Algebra
{\bf Solution.} The midpoint of a line segment with endpoints $(x_1, y_1), (x_2, y_2)$ is $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

So, the midpoint of the segment is $\left(\frac{6+0}{2}, \frac{12+(-6)}{2}\right)$, which simplifies to $(3,3)$. The sum of these coordinates is $\boxed{6}$.

set_name: MATH


TITLE: Taylor series of $\frac{x}{e^x-e^{-x}}$
QUESTION [1 upvotes]: My textbook says that (when $x$ approaches $0$):
$$\frac{x}{e^x-e^{-x}}=\frac{1}{2}-\frac{x^2}{12}+\cdots$$
It is also said that the result can be deduced by using $e^x=1+x+x^2/2+\cdots$.
But when I sub it in directly:
$$\begin{aligned}\frac{x}{e^x-e^{-x}}&=\frac{x}{(1+x+x^2/2+x^3/6\cdots)-(1-x+x^2/2-x^3/6+\cdots)}\\&=\frac{x}{2x+x^3/3+\cdots}\end{aligned}$$
I cannot get the right expansion. So what is the right approach?

REPLY [4 votes]: Another way:
$$\begin{split}\frac{x}{e^x-e^{-x}} &= e^{-x}\frac{x}{1-e^{-2x}} \\ &= e^{-x}\frac{x}{2x-2x^2+4x^3/3+O(x^4)}\\&=e^{-x}\frac{1}{2}\frac{1}{1-(x-2x^2/3+O(x^3))} \\&=\frac12\left(1-x+\frac 12x^2+O(x^3)\right)\left(1+\left(x-\frac23x^2\right)+\left(x-\frac23 x^2\right)^2+O(x^3)\right)\\&=\frac12\left(1-x+\frac 12x^2+O(x^3)\right)\left(1+x+\frac13x^2+O(x^3)\right)\\&=\frac12-\frac1{12}x^2+O(x^3).\end{split}$$ Here, we have used the expansion
$$\frac{1}{1-x}=1+x+x^2+\cdots.$$