text:
\begin{document}

\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{question}[thm]{Question}

\theoremstyle{definition}
\newtheorem{defn}[thm]{\bf{Definition}}

\theoremstyle{remark}
\newtheorem{rmk}[thm]{Remark}

\def\square{\hfill${\vcenter{\vbox{\hrule height.4pt \hbox{\vrule width.4pt height7pt \kern7pt \vrule width.4pt} \hrule height.4pt}}}$}

\newenvironment{pf}{{\it Proof:}\quad}{\square \vskip 12pt}

\title[Minimal Surfaces in Hyperbolic $3$-manifolds]{Minimal Surfaces in Hyperbolic $3$-manifolds}
\author{Baris Coskunuzer}
\address{UT Dallas, Dept. Math. Sciences, Richardson, TX 75080}
\email{coskunuz@utdallas.edu}

\thanks{The author is partially supported by Simons Collaboration Grant, and Royal Society Newton Mobility Grant.}

\newcommand{\Si}{S^2_{\infty}(\mathbf{H}^3)}
\newcommand{\PI}{\partial_{\infty}}
\newcommand{\BH}{\mathbf{H}}
\newcommand{\BR}{\mathbb R}
\newcommand{\BC}{\mathbb C}
\newcommand{\BZ}{\mathbb Z}
\newcommand{\BN}{\mathbb N}

\newcommand{\N}{\mathbf{N}}

\newcommand{\e}{\epsilon}
\newcommand{\wh}{\widehat}
\newcommand{\wt}{\widetilde}

\newcommand{\D}{\mathcal{D}}
\newcommand{\I}{\mathcal{I}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\U}{\mathcal{U}}
\newcommand{\B}{\mathbf{B}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\X}{\mathcal{X}}
\newcommand{\E}{\mathcal{E}}
\newcommand{\s}{\mathcal{S}}
\newcommand{\p}{\mathcal{P}}

\newcommand{\cc}{\mathfrak{C}}
\newcommand{\T}{\mathbf{T}}

\begin{abstract}
We show the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic $3$-manifolds except some special cases.
\end{abstract}

\maketitle

\section{Introduction}

The existence of minimal surfaces in general $3$-manifolds is one of the classical problems of geometric analysis. There has been numerous foundational results on the existence of closed embedded minimal surfaces in compact $3$-manifolds, or positive curvature case. By \cite{Pi}, every closed $3$-manifold contains a smooth, embedded, closed minimal surface. By geometric measure theory \cite{Fe}, we know that for any compact manifold, there exists an area minimizing surface in every homology class. By \cite{MSY}, it is known that every isotopy class contains a minimal surface in such manifolds. Furthermore, by \cite{MSY} and many others, it is also known the existence of minimal surfaces in positive curvature case.

For the negative curvature case, the situation is highly different. Even in the constant negative curvature (hyperbolic) case, the answer is unknown.

\begin{question} \label{mainques} Which complete hyperbolic $3$-manifolds contain a smoothly embedded closed minimal surface?
\end{question}

In the tree diagram below, green boxes represent known cases, and the red boxes represent the unknown cases for the existence problem.  Note that for some trivial cases like $M=\BH^3$ or $M$ is a Fuchsian Schottky manifold, $M$ contains no closed minimal surface by maximum principle. In this paper, we answer the question by dealing with all red boxes except some special cases listed below.

The summary of the known cases (green boxes) is as follows: For closed hyperbolic $3$-manifolds, Almgren-Pitts min-max theory \cite{Pi} gives a positive answer. For noncompact hyperbolic $3$-manifolds, there are 2 cases: Finite volume, and infinite volume. Recently, Rosenberg et al have showed the existence of minimal surfaces in finite volume noncompact hyperbolic $3$-manifolds \cite{CHMR, HW2, CL}, and finished this case. Hence, the remaining case is the infinite volume hyperbolic $3$-manifolds.

\

\begin{tikzpicture} [level distance=.7in,sibling distance=12em,
man/.style={rectangle,draw,fill=green!20},
woman/.style={rectangle,draw,fill=red!20},
edge from parent fork down,
every node/.style = {shape=rectangle, rounded corners=.8ex,	draw, align=center},
grandchild/.style={grow=down,xshift=1em,anchor=west,
	edge from parent path={(\tikzparentnode.south) |- (\tikzchildnode.west)}},
first/.style={level distance=6ex},
second/.style={level distance=12ex}]

\node {Hyperbolic 3-manifolds}
child { node { Infinite Volume  }
	child { node[woman] {Geom. Infinite}
		child[grandchild,first] {node[woman] {Bounded Geom.}}
		child[grandchild,second] {node[woman] {Unbounded Geom.}}	}
	child { node[man] {Geom. Finite}}
}
child { node[man] { Finite Volume  }
	child[grandchild,first] { node[man] {Closed Manifolds}}
	child[grandchild, second] { node[man] {Cusped Manifolds}}
}

;

\end{tikzpicture}

\

Infinite volume complete hyperbolic $3$-manifolds can be divided into two classes: {\em Geometrically finite} and {\em geometrically infinite}. The class of the manifold is determined by its ends. In particular, if $M$ is an infinite volume hyperbolic $3$-manifold, then there is a compact codimension-$0$ submanifold $\C_M$, {\em compact core}, where $M\simeq int(\C_M)$ (See Figure \ref{endsfig}). Then, the ends of $M$ can be thought as the components of $M-\C_M$ (assuming no cusps). Hence, if $\partial \C_M=S_1\cup...\cup S_k$, then each end $\E_i\simeq S_i\times (0,\infty)$ by Marden's Tameness Conjecture \cite{Ag,CG}.

We call an end $\E_i$ geometrically finite, if $S_i$ can be chosen convex surface, and $\E_i$ can be foliated by equidistant surfaces $\{S_i^t\}$. The geometry of these ends are very simple, as the equidistant surfaces from $\partial \C_M$ in $M-\C_M$ are convex surfaces flaring out to infinity. This is why this case is called geometrically finite.  If all ends of $M$ is geometrically finite, then we call the manifold $M$ is {\em geometrically finite}. Equivalently, $M$ is geometrically finite if the convex core $\cc(M)$ has finite volume. Otherwise, we call $M$ is geometrically infinite, i.e if there exists an end which is not geometrically finite. The existence of minimal surfaces in most geometrically finite cases is straightforward as the area minimizing representative of any homology class must stay in the compact convex core.

Therefore, the interesting case is when $M$ contains a geometrically infinite end. In this case, the convex core is no longer compact, and contains all the geometrically infinite ends. The reason why this case is subtle is that the area minimizing sequence of surfaces can escape to infinity. Hence, the standard techniques do not apply here.

There are two classes of geometrically infinite ends: {\em Bounded geometry} and {\em unbounded geometry}. We call an end $\E_i$ has bounded geometry if there is a positive lower bound for injectivity radius. We call an end $\E_i$ has unbounded geometry otherwise. The crucial distinction between bounded and unbounded geometry is that in the ends with unbounded geometry, there exists a sequence of short geodesics $\{\alpha_n\}$ where $|\alpha_n|\searrow 0$ and $\alpha_n$ escapes to infinity.

Before stating our main result, we need to define  {\em exceptional manifolds}:

\begin{itemize}
	\item \textit{Type I:} $M$ is a product manifold with bounded geometry, i.e. $M\simeq S\times \BR$ for some closed surface of genus $\geq 2$.
	\item \textit{Type II:} $M$ has exactly one end, and $H_2(M)$ is trivial.
\end{itemize}

\begin{thm}
	Let $M$ be a complete hyperbolic $3$-manifold with finitely generated fundamental group. Then, if $M$ is not an exceptional manifold as listed above, $M$ contains a smoothly embedded closed minimal surface.
\end{thm}

\vspace{.2cm}

\noindent {\em Outline of the Proof:} The main problem in these manifolds is that the area minimizing sequence of surfaces might escape to infinity, and give an empty limit. In this paper, we will use various trapping techniques to construct a minimal surface. In particular, we will trap the surface between two short geodesics, or between two cusps, or between one cusp and convex surface. When trapping does not work, we will use min-max methods.

\vspace{.2cm}

\noindent {\em Unbounded Geometry Case:} Let $\E$ be an end with unbounded geometry. We will trap a minimal surface between two short geodesics in $\E$ (Theorem \ref{ubdthm}). In particular, we will consider the area minimizing surface among the surfaces separating two short geodesics $\alpha$ and $\beta$ far out in the end. When the geodesics are sufficiently short, then the Margulis tubes near the geodesics becomes a barrier for area minimizing surfaces \cite{Ha,HW}. This means the area minimizing surface $\Sigma$ in that class should be away from the geodesics. Hence, $\Sigma$ is a smoothly embedded stable minimal surface in $M$.

\vspace{.2cm}

\noindent {\em Bounded Geometry Case:} In this case, we will use a completely different strategy. In particular, if one of the ends have unbounded geometry, then we can construct a minimal surface in the unbounded end as above. So, we will assume every end of $M$ has bounded geometry (Theorem \ref{mainthm}). In this case,  we will consider the area minimizing sequence of surfaces in each end. If they are escaping to infinity in the end, we will construct a mean concave surface in that end by using shrinkwrapping \cite{CG}. If the area minimizing surface does not escape to infinity in one end, and assuming $M\not\simeq S\times \BR$, then by using the topology of $M$, we can trap an area minimizing surface in the homology class of that end in the compact part. If the area minimizing sequence escapes to infinity in all ends, then we construct mean concave surfaces $\Sigma_i$ in each $\E_i$ so that we have a bounded domain $\Omega$ in $M$ with smooth mean concave boundary. Then, by using \cite{Mo}, we get a smoothly embedded min-max surface $T$ with $T\cap\Omega\neq \emptyset$. Hence, in this case we get an unstable minimal surface. In all other cases, we get a stable minimal surface. Note that in each case, we will deal with the cusps by using different techniques.

\vspace{.2cm}

\noindent {\em Exceptional Manifolds:} As listed above, we have 2 exceptional families for the existence question. In Type I, $M$ is a product manifold with bounded geometry, i.e. $M\simeq S\times \BR$ where $S$ is a closed surface of genus $\geq 2$.  This case is quite delicate, as one needs to rule out the existence of mean convex foliation in $M\simeq S\times \BR$. Finally in Type II, we have examples like $M=\BH^3$, and Schottky manifolds. In these cases, since the topology of the manifold is quite trivial, the area minimization methods do not work. Also, the structure of the ends prevents the use of min-max methods. In Section 5, we will discuss these exceptional cases in detail.

Organization of the paper is as follows. In Section 2, we will give basic definitions, and relevant results which will be used in the remainder of the paper. First, in Section 3, we deal with the unbounded geometry case. Then, in Section 4, we deal with the bounded geometry case, and prove the main result. In Section 5, we will discuss the exceptional cases, and give some refinements of the main result in these cases. In Section 6, we will discuss further questions, and give concluding remarks.

\subsection{Acknowledgements} I would like to thank Yair Minsky, David Gabai, Antoine Song, and Brian White for very valuable conversations.

\section{Preliminaries} \label{Prelim}

In this section, we will cover the basic definitions and the known results on the problem.

\subsection{Classification of Hyperbolic $3$-Manifolds and Existence Results.} \

\vspace{.2cm}

Throughout the paper, we will only deal with complete, orientable hyperbolic $3$-manifolds with finitely generared fundamental group. To go over the previous results on the existence of minimal surfaces, we will follow the tree diagram in the introduction. For more details on the classification of hyperbolic $3$-manifolds, see the survey paper by Minsky \cite{Mi}.\\

\noindent {\bf Finite Volume Hyperbolic $3$-manifolds:} These are the hyperbolic $3$-manifolds with finite volume. These can be classified by compactness:

\begin{itemize}
	\item {\bf Closed Manifolds:} By foundational result of Almgren-Pitts min-max theory \cite{Pi}, every closed Riemannian manifold contains a smoothly embedded minimal surface. The minimal surfaces obtained by this method are \textit{unstable} by construction. See also \cite{MR}. On the other hand, if $M$ is Haken, one can obtain area minimizing surfaces in the homology class of the incompressible surface \cite{MSY,Ha2}. The minimal surfaces obtained this way are \textit{stable}.

	\vspace{.2cm}

	\item {\bf Cusped Manifolds:} In the noncompact case, the finite volume hyperbolic $3$-manifolds is homeomorphic to the interior of a compact $3$-manifold where every boundary component is a torus. The ends of these manifolds are called cusps. Cusps topologically a solid torus where the core circle is removed.

	Recently, Rosenberg et al showed that all these manifolds contain a a minimal surface \cite{CHMR} by using min-max methods. They prove that minimal surfaces cannot go very deep in the cusps, hence they obtain smoothly embedded closed minimal surfaces. Again,  these minimal surfaces are also \textit{unstable}. Recently, Huang-Wang obtained a similar result for such manifolds with different techniques \cite{HW2}. Their surfaces are least area in their homotopy class, and hence {\em stable}.

	Note also that recently, Chambers-Liokumovich obtained a very general existence result for any finite volume noncompact $3$-manifold \cite{CL} by generalizing Almgren-Pitts min-max techniques. Hence, the minimal surfaces they obtain is \textit{unstable}, too.
\end{itemize}

\noindent {\bf Infinite Volume Hyperbolic $3$-manifolds:} The classification of infinite volume complete hyperbolic $3$-manifolds are done by the geometry of the ends \cite{Th}.

Let $M$ be a complete hyperbolic $3$-manifold with infinite volume. Let $\C_M$ be a {\em compact core} of $M$. In particular, $\C_M$ is a compact codimension-$0$ subset of $M$ where $M$ is homeomorphic to interior of $X$. Then assuming no cusps, $M-\C_M=\bigcup_{i=1}^N \E_i$ where $\E_i\simeq S_i\times(0,\infty)$ for some component $S_i$ of $\partial \C_M$ which is a closed surface of genus $\geq 2$ \cite{Ag,CG}. Here, we call $\E_i$ {\em an end of $M$} (See Figure \ref{endsfig}). The geometry of the ends $\{\E_1, \E_2,...,\E_n\}$ describes how complicated the manifold $M$ is. Note that throughout the paper, if $M$ has cusps, we will not treat the cusp regions as an end of $M$, but we will call them cusps.

\begin{figure}[h]

	\relabelbox  {\epsfxsize=3.5in

		\centerline{\epsfbox{ends2.eps}}}

	\relabel{1}{\footnotesize $\C_M$} \relabel{2}{\footnotesize $M$}  \relabel{3}{\footnotesize $\E_1$} \relabel{4}{\footnotesize $\E_2$} \relabel{5}{\footnotesize $\E_3$}
	\relabel{6}{\scriptsize $S_1$} \relabel{7}{\scriptsize $S_2$} \relabel{8}{\scriptsize $S_3$}

	\endrelabelbox

	\caption{\label{endsfig} \small $M$ is an infite volume hyperbolic $3$-manifold with $3$ ends. The shaded region is the compact core $\C_M$.}

\end{figure}

An end $\E_i$ is {\em geometrically finite end}, if we can take the boundary component $S_i\subset \partial \C_M$ convex, i.e. $\C_M$ is in the convex side of $S_i$. Geometrically finite ends has very simple geometry, which can be foliated by equidistant surfaces to $S_i$. In particular, if $S^t_i$ is the $t$-equidistant surface to $S_i$ in $\E_i$, then $\{S^t_i\}$ foliates $\E_i$ where $S^t_i$ is a convex surface whose area is increasing exponentially in $t$.

If an end is not geometrically finite, we call it {\em geometrically infinite end}. In particular, let $\cc(M)$ be the convex core of $M$ which is the smallest convex set in $M$ where the inclusion $\cc(M)\hookrightarrow M$ is a homotopy equivalence. An equivalent definition is that $\E_i$ is geometrically infinite end in $M$ if $\E_i\subset \cc(M)$. Hence, every component of $M-\cc(M)$ is a geometrically finite end.

Geometrically infinite ends are classified into two categories as follows: Ignoring the cusps, if the geometrically infinite end $\E_i$ has positive injectivity radius, then we call $\E_i$ has {\em bounded geometry}. Otherwise, we call $\E_i$ has {\em unbounded geometry}. One crucial property of the ends with unbounded geometry is that there exists a sequence of geodesics exiting the end. In other words, if $\E_i$ is geometrically infinite end, then there exists a sequence of geodesics $\{\alpha^i_n\}$ in $\E_i$ such that $\alpha^i_n\to \infty$, i.e. $\alpha_n^i$ escapes every compact subset of $M$ \cite{Bo}. Furthermore, we can choose this sequence of geodesics $\{\alpha^i_n\}$ such that $\|\alpha^i_n\|\searrow 0$ where $\|.\|$ is the length. We call such a sequence $\{\alpha^i_n\}$ {\em arbitrarily short geodesics}.

Now, we are ready to continue to list the results in complete hyperbolic $3$-manifolds with infinite volume.

\begin{itemize}
	\item {\bf Geometrically Finite Manifolds:} A complete hyperbolic $3$-manifold $M$ is called {\em geometrically finite} if the convex core $\cc(M)$ has finite volume. In other words, every end of $M$ is a geometrically finite end. In this case, assuming nontrivial $H_2(M)$, the existence of a minimal surface in $M$ is straightforward as follows. Consider the area minimizing surface $T$ in the nontrivial homology class of $\xi\in H_2(M)$ \cite{Ha2,Fe}. As any area minimizing surface in $M$ must stay in the convex core $\cc(M)$, $T \subset \cc(M)$. Since $\cc(M)$ is compact, $T$ cannot escape to infinity. Hence, there exists a smoothly embedded stable minimal surface $T$ in $M$. Quasi-Fuchsian Manifolds are examples of geometrically finite manifolds. Note that one can also use the solutions of asymptotic Plateau problem in the universal cover $\BH^3$ in order to get least area surface in such $M$ \cite{An,Co2}. Notice that in this case, we are excluding some Type II Exceptional Manifolds, e.g. Schottky Manifolds (See Section \ref{type3sec}).

	\vspace{.2cm}

	\item {\bf Geometrically Infinite Manifolds with Bounded Geometry:} We call a complete hyperbolic $3$-manifold $M$ is geometrically infinite with bounded geometry, if $M$ has a positive injectivity radius bound, and a geometrically infinite end. In other words, ignoring the cusps, there exists $\rho_0>0$ such that the shortest geodesic $\beta$ in $M$ has length $\|\beta\|>\rho_0$. In particular, $M$ has a geometrically infinite end, and every such end has bounded geometry. The infinite cover of a mapping torus with pseudo-Anosov monodromy (Cannon-Thurston manifolds) are examples of such manifolds \cite{CT}. There is no general existence result for this case.

	\vspace{.2cm}

	\item {\bf Geometrically Infinite Manifolds with Unbounded Geometry:} We call a complete hyperbolic $3$-manifold $M$ is geometrically infinite with unbounded geometry, if $M$ has a geometrically infinite end with unbounded geometry. In particular, $M$ a sequence of geodesics $\{\alpha_n\}$ which escapes every compact subset of $M$ and $\|\alpha_n\|\searrow 0$. There is no general existence result for this case, either.

\end{itemize}

Hence, when the complete hyperbolic $3$-manifolds $M$ are finite volume, or geometrically finite, we have the existence results for minimal surfaces. Hence, the only remaining cases for Question \ref{mainques} are geometrically infinite hyperbolic $3$-manifolds.

\subsection{Cusps and Margulis Tubes.} \

\vspace{.2cm}

In this part, we will go over the basic definitions, and relevant results on cusps and short geodesics in hyperbolic $3$-manifolds. For further details, see \cite{BP, Ma}.

Any finite volume, complete, non-compact hyperbolic $3$-manifold $M$ is homeomorphic to the interior of a compact $3$-manifold $N$ where $\partial N$ is a collection of tori. In particular, the ends of a finite volume, complete hyperbolic $3$-manifolds are homeomorphic to $T^2\times[0,\infty)$. One can think of these ends as a solid torus where the core circle is removed. Such ends are called {\em cusps}. Informally, one can use the analogy between a punctured surface with hyperbolic structure, where the end has $S^1\times[0,\infty)$ structure near the puncture (See Figure \ref{cuspfig}).

\begin{figure}[b]

	\relabelbox  {\epsfxsize=3.5in

		\centerline{\epsfbox{cusps.eps}}}

	\relabel{1}{\footnotesize $\gamma_1$} \relabel{2}{\footnotesize $\gamma_2$}  \relabel{3}{\footnotesize $X_1$} \relabel{4}{\footnotesize $X_2$} \relabel{5}{\footnotesize $X_3$}

	\endrelabelbox

	\caption{\label{cuspfig} \small $2$-dimensional analogous picture for Margulis Tubes and Cusps. $\gamma_1$ and $\gamma_2$ short geodesics, and shaded regions represent Margulis Tubes and Cusps.}

\end{figure}

The geometry of the cusps are well understood, as they are generated by parabolic isometries in the representation of the fundamental group of the manifold. In particular, every cusp is foliated by constant mean curvature 1 tori, induced by the corresponding horospheres in the universal cover $\BH^3$.

On the other hand, a similar structure is also true for neighborhoods of sufficiently short geodesics in $M$. Let $\epsilon_3$ be the Margulis constant for hyperbolic $3$-manifolds, which is known to be between $0.1$ and $0.6$.
Let $M_{[0,\epsilon_3)}$ is the part of $M$ where the injectivity radius is smaller than $\e_3$. We call $M_{[0,\epsilon_3)}$ is the {\em thin part} of $M$. By Margulis Lemma, the structure of the thin parts of the hyperbolic $3$-manifold is well understood. It is either homeomorphic to $T^2\times[0,\infty)$ (cusp), or homeomorphic to a solid torus where the core circle is a short geodesic (See Figure \ref{cuspfig} for an analogous picture). These solid torus neighborhoods $\N_r(\gamma)$ of short geodesics $\gamma$ are called {\em Margulis Tubes}. The diameter of the maximal tubes depends on the length of the geodesic. In particular, if $l=\mbox{length}(\gamma)\searrow 0$, then $r\nearrow \infty$ (roughly $r\sim -\log l$). Their geometric structure is very similar to the cusps. In a way, one can think of cusps as Margulis tubes for "length $0$ geodesics".

Note that cusps and short geodesics can appear in not only finite volume hyperbolic $3$-manifolds, but also infinite volume hyperbolic $3$-manifolds. Even though cusps can topologically be considered as an end of the $3$-manifold, because of their very special structure, we will not call them an end of the manifold, instead we will call them {\em cusps}. Hence, when we say $\E$ is an end  of a hyperbolic $3$-manifold $M$, we always mean $\E\simeq S\times [0,\infty)$ where $S$ is a closed surface of genus $\geq 2$.

Note also that even for a complete hyperbolic 3-manifold $M$ with finitely generated fundamental group, there can be infinitely many components in the thin part of $M$ for every positive $\e$ smaller than the Margulis constant \cite{BO}. However, Sullivan proved that a complete hyperbolic 3-manifold $M$ with finitely generated fundamental group has only finitely many cusps \cite{Su}. So, by choosing the compact core $\C_M$ accordingly, we can assume that there is no cusp region in the ends, i.e. $\E_i\simeq S_i\times [0,\infty)$

In \cite{CHMR}, the authors showed the following strong result about minimal surfaces in cusp neighborhoods when proving the existence of minimal surfaces in finite volume hyperbolic $3$-manifolds. The following lemma states that minimal surfaces in a hyperbolic $3$-manifold cannot go very deep in a cusp.

\begin{lem} \label{cusplem} \cite[Proposition 8]{CHMR} \cite{CHMR2}
Let $M$ be a complete hyperbolic $3$-manifold, and let $X=T^2\times[0,\infty)$ be a cusp in $M$. Let $\Sigma$ be a closed minimal surface in $M$ whose Morse index is bounded below by $k$. Then, there exists $C_X(k)>0$ such	that $\Sigma\cap X\subset T^2\times[0, C_X(k)]$.
\end{lem}

Note that all the surfaces we are going to construct in this paper will be stable except where we use Corollary \ref{minmaxcor} where we have a index bound. Therefore, the lemma above will suffice for our purposes. For different versions of the lemma above, see also \cite[Theorem 5.9]{HW2} and \cite[Lemma 2.5]{Ha}.

On the other hand, we have a similar result for Margulis Tubes near short geodesics. The following lemma states that area minimizing surfaces cannot get very close to the short curves because of the structure of Margulis tubes. Note that in this lemma, "$\Sigma$ does not link $\gamma$" means that $\Sigma$ and $\gamma$ have representatives $\Sigma',\gamma'$ in their homotopy class so that $\Sigma'\cap \gamma'=\emptyset$.

\begin{lem} \label{tubelem} Let $M$ be a complete hyperbolic $3$-manifold. Then, there exists a sufficiently small $l_M>0$  and $\delta(l_M)>0$ with the following property: Let $\gamma$ be a short geodesic in $M$ with $\mbox{length}(\gamma)<l_M$. Let  $\N_\delta(\gamma)$ be its $\delta$ neighborhood of $\gamma$ in the Margulis Tube. Let $\Sigma$ be a closed area minimizing surface in $M$ which does not link $\gamma$. Then, $\Sigma\cap \N_{\delta}(\gamma)=\emptyset$.
\end{lem}

\begin{pf} Let $\gamma$ be a short geodesic in $M$. Let $Y=N_r(\gamma)$ be the maximal tube around $\gamma$ in $M$. Let $\wt{Y}$ be the universal cover of $Y$. Consider $\wt{Y}$ in $\BH^3$. Then, if the infinite geodesic $\wt{\gamma}$ is the cover of $\gamma$, then $\wt{Y}$ be the $r$ neighborhood of $\wt{\gamma}$ in $\BH^3$. In other words, $\wt{Y}$ is an infinite solid cylinder in $\BH^3$ (See Figure \ref{tubefig}-left). Let $\Omega$ be a fundamental domain corresponding $\partial Y$ in  $\partial \wt{Y}$. In particular, $\Omega$ is a finite height (comparable to length of $\gamma$) annulus in the infinite cylinder $\partial \wt{Y}$.

Now, let $\Sigma$ be an area minimizing surface in $M$. By assumption, $\Sigma$ does not link $\gamma$. Let $\Delta=\Sigma\cap Y$. Then, by \cite[Lemma 3.9]{HW}, $\Delta$ is a collection of boundary parallel disks or annuli. Hence, $\mu=\partial \Delta=\Sigma\cap\partial Y$ is a collection of nullhomotopic curves, or pair of curves in the homotopy class of the core curve (corresponding to annular components in $\Delta$). Then, $\wt{\mu}$ in $\Omega\subset \partial \wt{Y}$ will be collection of null-homotopic curves, or annuli homotopic to $\Omega$. Similarly, let $\wt{\Delta}$ be the lift of $\Delta$ in $\wt{Y}$ with $\partial \wt{\Delta}=\wt{\mu}$.

\begin{figure}[h]
	\begin{center}
		$\begin{array}{c@{\hspace{.3in}}c}

		\relabelbox  {\epsfysize=2in \epsfbox{tube1.eps}}
		\relabel{1}{\scriptsize $\BH^3$}
		\relabel{2}{\scriptsize $\wt{Y}$}
		\relabel{3}{\scriptsize $\wt{\gamma}$}
		\endrelabelbox &

		\relabelbox  {\epsfysize=2in \epsfbox{tube2.eps}}
		\relabel{1}{\scriptsize $\wt{\Delta}$}
		\relabel{2}{\scriptsize $\wt{Y}$}
		\relabel{3}{\scriptsize $\C$}
		\relabel{4}{\scriptsize $\Omega$}
		\relabel{5}{\scriptsize $\wt{\gamma}$}
		\relabel{6}{\scriptsize $\BH^3$}
		 \endrelabelbox \\
	\end{array}$

\end{center}
\caption{\label{tubefig} \footnotesize In the left, we have the lift of the short curve $\gamma$ and the maximal tube $Y=N_r(\gamma)$ in $\BH^3$. The figure in the right shows the least area spherical catenoid $\C$, and the annulus $\Omega$ in $\partial \wt{Y}$. The least area surface $\wt{\Delta}$ stays in the other side of $\C$.}
\end{figure}

Now, in order to prove the lemma, it is enough to show that there exists $\delta>0$ such that $N_\delta(\wt{\gamma})\cap \wt{\Delta}=\emptyset$. Now, fix a least area spherical catenoid $\C$ given in \cite[Theorem 1.2]{Wa}. Then, for $|\gamma|$ sufficiently short, the radius $r$ of maximal tube will be sufficiently large so that the annulus $\Omega$ is in the other side of the catenoid $\C$ (See Figure \ref{tubefig}-right). In particular, $\C$ separates $\BH^3$ into two regions such that $\BH^3-\C=U^+\cup U^-$ where $\PI U^+$ is an annulus in $\Si$. Then, by choosing the right lift $\Omega$ in $\partial \wt{Y}$, and choosing sufficiently small geodesic $\gamma$, we can make sure that $\Omega\subset U^+$. Then, by Meeks-Yau exchange roundoff trick $\wt{\Delta}\subset U^+$. In particular, if $\Sigma\cap \C\neq \emptyset$, we can swap the disks and annuli in $\Sigma$ with the ones in $\C$, and get the same area surfaces as $\C$ is also least area. However, new surface have folding curves, which we can reduce the area by pushing to convex side \cite{MY}. This gives a contradiction. Then, for $0<\delta<d(\wt{\gamma},\C)$, the proof follows.
\end{pf}

\begin{rmk} Hass proved a similar result for cusp regions in \cite{Ha}. Also, Huang and Wang obtained a similar result for Margulis tubes by using the complex length of the short geodesic \cite{HW}.
\end{rmk}

\subsection{Shrinkwrapping.} \label{shrinksec} \

\vspace{.2cm}

Throughout the paper, we will frequently use shrinkwrapping technique to obtain "defective" minimal surfaces. The shrinkwrapping technique was introduced by Calegari and Gabai in \cite{CG} where they prove the celebrated {\em Marden's Tameness Conjecture}. A simpler version of this technique can also be seen at \cite{Co1}.

In particular, shrinkwrapping is "tightening" a given "incompressible surface" $S$ outside of a finite collection of geodesic curves $\Gamma$ in a hyperbolic $3$-manifold. At the end of this "tightening" process, one obtains an isotopy of $S$ into a "defective minimal surface" $T$, where $T$ is minimal except along some curve segments along $T\cap \Gamma$. We call such a surface $T$ $\Gamma$-minimal, i.e. $T$ is smooth with mean curvature $0$ everywhere except $T\cap\Gamma$. The geometry of the surface near these singular segments is also well-understood along the lines of {\em thin obstacle problem}. Here is the main technical result of \cite{CG}.

\begin{lem} [Shrinkwrapping] \label{shrinklem} \cite[Theorem 1.10]{CG}
Let $M$ be a complete, orientable hyperbolic $3$-manifold, and let $\Gamma$ be a finite collection of pairwise disjoint simple closed geodesics in $M$. Furthermore, let $S \subset M-\Gamma$ be a closed embedded $2$-incompressible surface rel. $\Gamma$ which separates some component of $\Gamma$ from another. Then S is homotopic to a $\Gamma$-minimal surface T via a homotopy $F : S \times[0, 1] \to M$ such that

\begin{enumerate}
\item $F(S \times 0) = S$,
\item $F(S \times t) = S_t$ is an embedding disjoint from $\Gamma$ for $0\leq t< 1$,
\item $F(S \times 1) = T$,
\item if $T'$ is any other surface with these properties, then $\mbox{area}(T) \leq \mbox{area}(T')$
\end{enumerate}

\end{lem}

\begin{rmk}
	In the original statement of above lemma, there is a cusp-free assumption for $M$. However, after the recent proof of Lemma \ref{cusplem}, there is no need for such assumption, as it was recently shown that area minimizing surfaces cannot go very deep in the cusps by area comparison arguments.
\end{rmk}

\begin{rmk}
	Note that \cite[Remark 1.33]{CG} gives a nice alternative proof of the lemma above by using techniques of \cite{HS}, and the thin obstacle problem.
\end{rmk}

Here, $S$ is $2$-incompressible relative to $\Gamma$ if any essential compressing disk for $S$ intersects $\Gamma$ at least twice. For further details on the structure near $T\cap\Gamma$, see \cite[Section 1.6 and 1.7]{CG}.

On the other hand, getting such a set of geodesics $\Gamma$ for a given surface $S$ in an end $\E$ is  straightforward by \cite[Theorem 4.1]{CG} and \cite[Remark 4.2]{CG}. In particular, as a result of \cite{CG}, we know that every end of a complete hyperbolic $3$-manifold is homeomorphic to $S\times[0,\infty)$ for some closed surface $S$ of genus $\geq 2$ (Topological Tameness). By Bonahon's result \cite{Bo}, every end $\E$ contains a sequence of geodesics $\{\alpha_i\}$ exiting the end $\E$, i.e. $\alpha_i\to \infty$. By \cite[Remark 4.2]{CG}, for sufficiently large $i$, we can find a suitable finite subset $\Gamma_i$ of $\{\alpha_i\}$ such that $S_i=S\times i$ is $2$-incompressible relative to $\Gamma_i$. In particular, we can "shrinkwrap" any such level set $S_i$ in $\E$ into a defective minimal surface $T_i$ by Lemma \ref{shrinklem} (See Figure \ref{shrinkfig}).

\begin{figure}[h]

	\relabelbox  {\epsfxsize=3.5in

		\centerline{\epsfbox{shrinkwrap.eps}}}

	\relabel{1}{\footnotesize $M$} \relabel{2}{\footnotesize $S\times 0$}  \relabel{3}{\footnotesize $\alpha_1$} \relabel{4}{\footnotesize $S\times 1$} \relabel{5}{\footnotesize $S\times m$}
	\relabel{6}{\footnotesize $T_1$} \relabel{7}{\footnotesize $T_2$}  \relabel{8}{\footnotesize $T_m$} \relabel{9}{\footnotesize $\E$}
	\relabel{10}{\footnotesize $M$} \relabel{11}{\footnotesize $\E$}
	\relabel{12}{\footnotesize $S\times 2$}
	\endrelabelbox

	\caption{\label{shrinkfig} \small By shrinkwrapping, we deform $S\times k$ into a $\Gamma$-minimal surface $T_k$ in the end $\E$.}

\end{figure}

Before finishing our discussion on shrinkwrapping surfaces, we need to give the regularity results near their "defective" parts. As the lemma above describes, the surface $T$ is minimal except the {\em coincidence set} $L:=T\cap\Gamma$. The behavior of $T$ near $L$ is well-understood, and it is called {\em Thin Obstacle Problem} in the literature. The following lemma summarizes these regularity results. For further details, see \cite[Section 1.7]{CG}.

\begin{lem} \label{thinlem} [Thin Obstacle Problem] \cite[Lemma 1.31]{CG}
	Let $T$ be a $\Gamma$-minimal surface in $M$ as described above. Then, there exists a parametrization of $T$ with $u:S\to M$   such that the derivative $du$ along local sheets of $T$ is continuous from each side along the coincidence set $L$, and continuous at noninterior points.
\end{lem}

\begin{rmk} Note that Calegari-Gabai needed a finer understanding of $T$ near coincidence set to prove $T$ is indeed a $CAT(-1)$ surface. In this paper, we only need that $T$ has tangent planes in both sides of coincidence sets, which follows from the lemma above.
\end{rmk}

\subsection{Bounded Diameter Lemmas.} \label{bddsec} \

\vspace{.2cm}

In this part, we will give two important lemmas about diameters of closed minimal surfaces in hyperbolic $3$-manifolds. The first one is about shrinkwrapping surfaces and incompressible surfaces. The second lemma is about area minimizing surfaces in a homology class.

First, we will show that diameter of a closed minimal surface $S$ in a hyperbolic $3$-manifold $M$ can be bounded by a constant depending on the genus $g$ of $S$. Furthermore, by \cite[Lemma 1.15]{CG}, the result applies to shrinkwrapped surfaces, too.

\begin{lem} \label{boundiamlem} [Bounded Diameter Lemma]
Let $M$ be a hyperbolic $3$-manifold. Let $\Gamma$ be a disjoint collection of $\delta$-separated embedded geodesics. Let $\rho_0>0$ be the injectivity radius of $M$. If $S\subset M-\Gamma$ is a 2-incompressible $\Gamma$-minimal
surface, then there is a constant $C = C(g,\rho_0, \delta)\in \BR^+$ such that  $\mbox{diam}(S) \leq C$.
\end{lem}

\begin{pf} As $S$ is $2$-incompressible, for any point $p\in S\cap (M-M_{[0,\rho_0)})$, $S$ contains an embedded $\rho$-disk $\B_\rho(p)$ where $\rho=\min(\rho_0/2, \delta/2)$.

Notice that by Gauss' Theorem, as $S$ is a minimal surface, $K_S<K_M=$ where $K_M$ is the sectional curvature. Then, we have $-\int_S K_S\geq \mbox{area}(S)$. Then, by Gauss-Bonnet, $\int_S K_S=2\pi \chi(S) - \int_{\partial S}\kappa \ dl$ where $\kappa$ is the geodesic curvature. By applying this to $B_\rho(p)$ in $S$, we get
$\mbox{area}(B_\rho(p))\geq 2\pi(\cosh(\rho)-1) \geq \pi\rho^2$.

Recall that for any closed minimal surface $S$ in $M$, $\mbox{area}(S)\leq 2\pi(2g-2)$ by Gauss-Bonnet where $g$ is the genus. Then, let $\mathcal{P}$ be the maximal collection of points in $S$ which are $2\rho$ apart, and let $N_\rho$ be the number of points in $\mathcal{P}$. Then, we get $N_\rho\cdot \pi \rho^2\leq 2\pi(2g-2)$, which gives an upper bound for $N_\rho$, i.e. $N_\rho\leq \dfrac{4(g-1)}{\rho^2}$. Hence, we have $\mbox{diam}(S)\leq N_\rho\cdot 2\rho=\dfrac{8(g-1)}{\rho}$. The proof follows.
\end{pf}

\begin{rmk} Note that the same proof works for any incompressible, closed, embedded minimal surface $S$ in a hyperbolic $3$-manifold $M$.
\end{rmk}

Now, we modify the proof of the above lemma for area minimizing surfaces in a homology class of $M$. Notice that the main difference is that the area minimizing representative may not be incompressible. The main ingredients are Lackenby's general bounded diameter result on closed minimal surfaces in general $3$-manifolds \cite{La}, and genus bound for stable minimal surfaces in hyperbolic $3$-manifolds \cite{BD}.

\begin{lem} \label{boundiamlem2} Let $M$ be a hyperbolic $3$-manifold with $inj(M)=\rho_0>0$. Let $S$ be an embedded closed surface with nontrivial homology, and let $\xi$ be a homology class of $S$ in $M$. If $\Sigma$ is an area minimizing surface in $\xi$, then there exist a constant $C(\xi,\rho_0)>0$ such that   $\mbox{diam}(\Sigma_i) \leq C(\xi,\rho_0)$.
\end{lem}

\begin{pf} By assumption, $\mbox{area}(\Sigma)\leq\mbox{area}(S)$. As $\Sigma$ is area minimizing in $\xi$, it is also stable. Then, by \cite[Lemma 3.3]{BD}, we have $\pi|\chi(\Sigma)|\leq \mbox{area}(\Sigma) \leq 2\pi|\chi(\Sigma)|$. Hence,  $\pi|\chi(\Sigma)|\leq \mbox{area}(\Sigma)\leq\mbox{area}(S)$ gives an upper bound for $|\chi(\Sigma)|$ and the genus of $\Sigma$.

Furthermore, by \cite[Proposition 6.1]{La}, $\mbox{diam}(\Sigma)<|\chi(\Sigma)|f(\kappa,\rho)$ where $\rho$ is the lower bound for injectivity radius, and $\kappa<0$ is the upper bound for the sectional curvature of $M$. By above, we have an upper bound for $\chi(\Sigma)$. Hence, we have $\mbox{diam}(\Sigma) \leq C=|\chi(\Sigma)|f(\kappa,\rho)$. As $\kappa=-1$, and $\rho=\rho_0$ in our case, the constant $C$ only depends on the homology class $\xi$, and $\rho_0$, i.e. $C=C(\xi,\rho_0)$. The proof follows.
\end{pf}

\subsection{Min-Max Surfaces in Non-Compact Manifolds.} \

\vspace{.2cm}

In this section, we will give a recent min-max result which is the key ingredient to finish off a very important case in our main result. By using Almgren-Pitts Min-Max Theory, Montezuma and Song independently constructed embedded minimal surfaces in non-compact manifolds. Their result is more general, but we give the adapted version here.

\begin{lem} \label{minmaxlem} \cite{Mo, So}
	Let $M$ be a complete non-compact $3$-manifold. If $M$ contains a bounded open set $\Omega$ such that $\overline{\Omega}$ is a smooth strictly mean-concave boundary, then there exists a finite area embedded minimal surface $\Sigma$ in $M$ intersecting $\Omega$.
\end{lem}

\begin{rmk} \label{minmaxrmk} Note that Song's recent result \cite[Corollary 5]{So} is more general, and indeed we are using his version here.  Montezuma proved a very similar result in \cite{Mo}, but his version does not apply to the case when $M$ has a cusp.
\end{rmk}

Notice that if there is no cusp in $M$, then the finite area minimal surface is automatically closed. However, if $M$ has a cusp, the finite area minimal surface may not be closed, as it can have a non-compact part in the cusp. However, hyperbolic $3$-manifolds and cusps are special. By combining Lemma \ref{cusplem} and above lemma, we have the following stronger result for hyperbolic $3$-manifolds.

\begin{cor} \label{minmaxcor}
Let $M$ be a complete hyperbolic $3$-manifold. If $M$ contains a bounded open set $\Omega$ such that $\overline{\Omega}$ is a smooth strictly mean-concave boundary, then there exists a closed embedded minimal surface $\Sigma$ in $M$ intersecting $\Omega$.
\end{cor}

\begin{pf} To prove this corollary, we will look into Song's result above in the hyperbolic $3$-manifold case. In the proof of Theorem 4 and Corollary 5 in \cite{So}, Song obtains the minimal surface as a limit of closed minimal surfaces in larger and larger domains exhausting the manifold. In particular, let $\{U_i\}$ be sequence of domains exhausting $M$ such that $U_i\subset U_{i+1}$ and $M=\bigcup U_i$. Then, Song obtains a closed minimal surface $S_i$ in $U_i$ in two different cases in \cite[Theorem 4]{So}. Then, the minimal surface mentioned in the Theorem 4 and Corollary 5 in \cite{So} is the limit minimal surface $S=\lim S_i$. Furthermore, the closed minimal surfaces $\{S_i\}$ are stable in one case, and Morse index at most 1 in the other case.

Furthermore, when $M$ is hyperbolic, minimal surfaces with bounded index cannot go deep in the cusp regions by Lemma \ref{cusplem}. That means the closed minimal surfaces in the sequence $\{S_i\}$ stay away from the cusps. Then, the limit finite area minimal surface $S=\lim S_i$ must be a closed surface, too.
\end{pf}

\subsection{Area Minimizing Surfaces in $3$-Manifolds.} \

In this section, we will quote famous existence results for area minimizing representatives in a homotopy or homology class. These results will play crucial role in our construction. The first one is the celebrated result of geometric measure theory.

\begin{lem} \label{AMSlem} \cite{Fe,Ha2}  Let $M$ be a closed or mean convex compact orientable $3$-manifold. Let $\xi$ be a nontrivial homology class in $H_2(M)$. Then there exists a smoothly embedded surface $\Sigma$ which has the smallest area among the surfaces in $\xi$.
\end{lem}

The second result is the existence of least area representative in a homotopy class by Meeks-Simon-Yau.

\begin{lem} \label{LAlem} \cite{MSY} Let $S$ be an embedded surface in a Riemannian $3$-manifold $M$. After series of compressions, isotopies, and collapsings of boundaries of $I$-bundles to their cores, $S$ can be realizes as a union (possibly empty) of disjoint embedded minimal surfaces.
\end{lem}

The third one is more specific version of the above result, i.e. least area representatives of incompressible surfaces by Hass-Scott.

\begin{lem} \label{incomplem} \cite{HS}  Let $M$ be a closed or mean convex compact orientable, irreducible $3$-manifold. Let $S$ be an incompressible surface in $M$. Then, there exists a smoothly embedded least area surface $\Sigma$ in the isotopy class of $S$.
\end{lem}

Note that we quoted these results in a way which we are going to use in the remainder of the paper. Some results that were stated for closed manifolds can be applied to compact mean convex manifolds by using maximum principle \cite[Section 6]{HS}.

\section{Geometrically Infinite Manifolds with Unbounded Geometry}

In this section, we will prove the existence of minimal surfaces in geometrically infinite complete hyperbolic manifolds with unbounded geometry. As mentioned in the introduction, we will construct an area minimizing surface in a restricted class in the end with unbounded geometry. Therefore, the minimal surfaces we obtain will be stable.

\begin{thm} \label{ubdthm}
Let $M$ be a complete hyperbolic $3$-manifold with finitely generated fundamental group. If $M$ contains a geometrically infinite end $\E$ with unbounded geometry, then there exists a smoothly embedded closed minimal surface in $\E$, and hence in $M$.
\end{thm}

\begin{pf} Recall that as $M$ has finitely-generated fundamental group, it has finitely many cusps. So, we can choose every end $\E_i$ cusp-free, i.e. $\E_i\simeq S_i\times[0,\infty)$.

Let $\E\simeq S\times [0,\infty)$ be an end of $M$ with unbounded geometry. Then, there exists a sequence of geodesics $\{\alpha_n\}$ in $\E$ such that $\alpha_n$ escapes to infinity, and $\|\alpha_n\|\searrow 0$, i.e. arbitrarily short geodesics case.

Let $l_M>0$ be the constant in Lemma \ref{tubelem}. Fix a constant $0<\rho_0<\frac{l_M}{2}$. Let $\alpha$ be a geodesic in $\E$ with $\mbox{length}(\alpha)<l_M$, and let $\delta_\alpha>0$ be the corresponding constant in Lemma \ref{tubelem}. Let $\A=\{\alpha_i\}$ be the set of short geodesics with $inj(\T_{\frac{\delta_i}{3}}(\alpha_i))<\rho_0$ where $inj(.)$ is the injectivity radius and $\T_\delta(\alpha)$ is the solid torus, $\delta$-neighborhood of $\alpha$. Let $\E'=\E-\bigcup_\A \T_{\frac{\delta_i}{3}}(\alpha_i)$. Then, $inj(\E')\geq \rho_0$.

Let $C_0$ be the bounded diameter constant for $\E'$ in Lemma \ref{boundiamlem}. Then, we can find $K>3 C_0$ and a pair of short geodesics $\alpha^+$ and $\alpha^-$ in $\A$ such that  $\alpha^-\subset S\times [2K,3K]$  and  $\alpha^+\subset S\times [4K,5K]$.

Let $M'= S\times[K,6K]\cap \E'$. Since $\E$ is cusp-free, so is $M'$. Therefore, $M'$ is a compact manifold with boundary. First, we modify the metric near $\partial M'$ so that $M'$ becomes mean convex. Then, we construct an area minimizing surface $\Sigma$ in $M'$, and show that $\Sigma$ is away from the regions where we changed the metric near $\partial M'$.

Let $\A'$ be the geodesics in $\A$, also belonging to $S\times[K,6K]$. Then, $M'$ is compact manifold with boundary, say $\partial M'=S\times \{K,6K\}\bigcup_{\A'}Z_i$ where $Z_i=\partial \T_{\frac{\delta_i}{3}}(\alpha_i)$ is a $2$-torus. We call $S\times\{K\}= \partial^- M'$ and  $S\times\{6K\}= \partial^+ M'$.

Now, let $Y_i=\N_{\frac{\delta_i}{2}}(\alpha_i)\cap M'$ be the collar neighborhood of the boundary tori $Z_i$ in $M'$, i.e. $Y_i=\N_{\frac{\delta}{6}}(Z_i)$. Now, we modify the metric in these collar neighborhoods $Y_i$ as in \cite[Lemma 1.18]{CG} so that $M'$ becomes mean convex near $Z_i$ for any $\alpha_i\in \A'$. Now, fix small $\e>0$, and again modify the metric in $S\times[K, K+\e)$ and  $S\times(6K-\e, 6K]$ so that $M'$ becomes mean convex near $\partial^\pm M'$. Hence, $M'$ is a compact mean convex manifold, and let $\hat{M}$ be the manifold $M'$ with this new modified metric.

Now, let $Z^\pm=\partial\T_{\frac{\delta^\pm}{3}}(\alpha^\pm)$, and let $\tau$ be a proper arc in $\hat{M}\cap S\times[2K, 5K]$ connecting the boundary components $Z^+$ and $Z^-$. Fix $t_0\in (3K,4K)$, and let $S_0=S\times\{t_0\}$ be the level surface as above. Then, $S_0$ is a closed surface in the compact, mean convex manifold $\hat{M}$. Notice that by construction $\langle S_0,\tau\rangle=1$, where $\langle \ ,\ \rangle$ is the algebraic intersection number.

By construction, $S_0$ is incompressible in $\hat{M}$. Then, by Lemma \ref{incomplem}, there exists a least area representative $\Sigma$ in the homotopy class of $S_0$. Furthermore, since $\langle S_0,\tau\rangle=1$ and $\Sigma$ is in the homotopy class of $S_0$, then $\langle \Sigma,\tau\rangle=1$, too. This implies $\Sigma\cap \tau \neq\emptyset$. Therefore, by Bounded Diameter Lemma (Lemma \ref{boundiamlem}), $\Sigma\subset S\times [2K-C_0,5K+C_0]$. Hence, $\Sigma$ is away from $\partial^\pm M$. Here, we abuse the notation by assuming the parametrization $S\times[0,\infty)$ reflects the distance, but by choosing $K$ sufficiently large, this will not constitute a problem.

Furthermore, $\Sigma$ is away from the collar neighborhoods $Y_i$ for any $i$ by Lemma \ref{tubelem}, as $\Sigma$ is area minimizing, i.e. $\Sigma\cap T_{\delta_i}(\alpha_i)=\emptyset$. Even though we modified the metric in $Y_i$, the area comparison argument still applies here, as the metric is still same in Margulis Tube $\T_i(\alpha_i)$ away from $Y_i$.

This means $\Sigma$ is away from all boundary components of $\hat{M}$.  Hence, $\Sigma$ is completely in the part of $\hat{M}$ where the hyperbolic metric has not been modified. This implies $\Sigma$ is a minimal surface in the original manifold $M$. Note that as $\Sigma$ is area minimizing in $\hat{M}$, it is a stable minimal surface in $M$. The proof follows.
\end{pf}

\begin{rmk}
	Note that in the proof above, we basically give a simplified version of a shrinkwrapping technique described in Section \ref{shrinksec}. The same result can be proved with similar arguments by using the Shrinkwrapping Technology in \cite{CG}. In particular, by applying \cite[Theorem 4.1]{CG} and \cite[Remark 4.2]{CG} to $\E, S_0$ and $\{\alpha_n\}$, one can get a shrinkwrapping surface $\Sigma$ of $S_0$. Then, by using Bounded Diameter Lemma, and Lemma \ref{tubelem} as above, one can argue that $\Sigma$ is a smooth minimal surface in $M$.
\end{rmk}

\section{Geometrically Infinite Manifolds with Bounded Geometry}

In this section, we will deal with the remaining cases, and prove our main result. In particular, if a complete, infinite volume hyperbolic $3$-manifold $M$ has no end with unbounded geometry, then all ends of $M$ are either geometrically finite, or a geometrically infinite end with bounded geometry. As geometrically finite ends also have bounded geometry, we combined the remaining cases under one name, "bounded geometry". Of course, if the manifold $M$ has cusps, $M$ no longer has the bounded geometry property.

First, we will deal with a very important case. If $M$ has more than one cusp, then by using similar trapping argument in the proof of Theorem \ref{ubdthm}, we obtain the following result.

\begin{lem} \label{cusplem2} Let $M$ be a complete hyperbolic $3$-manifold where every end has bounded geometry. If $M$ has more than one cusps, then $M$ contains a smoothly embedded closed minimal surface.
\end{lem}

\begin{pf}	Let $\{X_1, X_2, ..., X_N\}$ be the cusp regions of $M$ where $X_i\simeq T^2_i\times [0,\infty)$. Let $C_i$ be the constant in Lemma \ref{cusplem} for $X_i$. Let $Y_i=T^2_i\times (2C_i,\infty)$. Let $M'=M-\bigcup_{i=1}^N Y_i$. Furthermore, for some small $\e>0$, modify the metric in $W_i=T^2_i\times(2C_i-\e,2C_i]$ so that $M'$ becomes mean convex. Since every end of $M$ has bounded geometry, $inj(M')=\rho_0>0$. Let $K_1$ be the constant in the first bounded diameter lemma (Lemma \ref{boundiamlem}). Let $K_2$ be the constant in the second bounded diameter lemma (Lemma \ref{boundiamlem2}). Let $K_0=\max\{K_1,K_2\}$.

Let $X_1$ and $X_2$ be two cusps. Let $\tau$ be a proper arc in $M'$ connecting $T^2_1\times\{2C_1\}$ and $T^2_2\times\{2C_2\}$. Let $\s_0$ be an embedded surface in $M'$ separating $X_1$ and $X_2$. By using Morse Theory, it is straightforward to get such a surface. Now, for each end $\E_i \simeq S_i\times[0,\infty)$, choose a surface $\wh{S}_i=S_i\times\{c_i\}$ with $d(\wh{S}_i,\tau)>2K_0$. Let $M''=M'-\bigcup_i S_i\times(c_i,\infty)$. Again, modify the metric in $S_i\times (c_i-\e,c_i]$ in $M''$ so that $M''$ is a compact mean convex manifold. Then, by Lemma \ref{LAlem}, there exists a least area surface  $\Sigma$ in the homotopy class of $\s_0$. Furthermore, as $\langle \s_0,\tau \rangle =1$, we have $\langle \Sigma,\tau \rangle =1$. Hence, $\Sigma$ is nonempty. Moreover, as $\Sigma\cap \tau\neq \emptyset$, by bounded diameter lemma, $d(\wh{S}_i,\Sigma)>K_0$. Similarly, by Lemma \ref{cusplem}, in the cusp regions, $\Sigma\cap T^2_i\times(C_i,2C_i]=\emptyset$. This shows $\Sigma$ is completely in the part of $M''$ where the metric is not modified. This implies $\Sigma$ is a minimal surface in $M$. Since $\Sigma$ is area minimizing in $M''$, it is a stable minimal surface in $M$. The proof of Step 1 follows.
\end{pf}

Before giving our main result, we need to define {\em exceptional manifolds:} The following hyperbolic $3$-manifolds are exceptional manifolds:

\begin{itemize}
	\item \textbf{Type I:} $M$ is a product manifold with bounded geometry, i.e. $M\simeq S\times \BR$ for some closed surface of genus $\geq 2$.
	\item \textbf{Type II:} $M$ has exactly one end, and $H_2(M)$ is trivial.
\end{itemize}

Now, we can state our main result.

\begin{thm} \label{mainthm}
	Let $M$ be a complete hyperbolic $3$-manifold with finitely generated fundamental group. Further assume that $M$ is not an exceptional manifold as listed above. Then, there exists a smoothly embedded, closed minimal surface $\Sigma$ in $M$.
\end{thm}

\begin{pf} Let $M-\C_M=\bigcup_{i=1}^N \E_i$ where $\C_M$ is the compact core of $M$, and $\{\E_i\}$ are the ends of $M$. As there can only be finitely many cusps, we assume $\E_i\simeq S_i\times [0,\infty)$ where $S_i$ is a closed surface of genus $\geq 2$ in $\partial\C_M$.

If one of the ends have unbounded geometry, then Theorem \ref{ubdthm} gives a minimal surface. If all ends are geometrically finite, and $M$ is not Type II exceptional manifold, then we have an area minimizing surface of the nontrivial homology class in the convex core $\cc(M)$ as explained in Section \ref{Prelim}. Hence, in the remaining of the proof, we will assume that $M$ has at least one geometrically infinite end with bounded geometry, and there is no geometrically infinite end with unbounded geometry.

First, by Lemma \ref{cusplem2}, if $M$ has more than one cusp, then we are done. So, from now on, we will assume either $M$ is cusp-free, or $M$ has exactly one cusp.

If $M$ has a cusp $X$, we will modify the manifold $M$ as follows. Let $X\simeq T^2\times [0,\infty)$. Let $C$ be the constant in Lemma \ref{cusplem} for $X$. Let $Y=T^2\times (2C,\infty)$. Let $M'=M-Y$. Furthermore, for some small $\e>0$, modify the metric in $W=T^2\times(2C-\e,2C]$ so that $M'$ becomes mean convex. Since every end of $M$ has bounded geometry, $inj(M')=\rho_0>0$. Let $C_1$ be the constant in the first bounded diameter lemma for $M'$ (Lemma \ref{boundiamlem}). Let $C_2$ be the constant in the second bounded diameter lemma for $M'$ (Lemma \ref{boundiamlem2}). Let $C_0=\max\{C_1,C_2\}$.

Now, for every end $\E_i$, we will construct a special surface $\Sigma_i$ in $\E_i$.
If an end $\E_i$ is geometrically finite, we modify the compact core $\C_M$ so that $\Sigma_i\subset \partial \C_M$ is a convex surface towards $\C_M$. If an end $\E_i$ is geometrically infinite, then we will use the exiting sequence of geodesics $\{\alpha^i_n\}$ in $\E_i$ to define $\Sigma_i$.

Let $\E$ be a geometrically infinite end in $M$, and $\{\alpha_n\}$ be an exiting sequence of geodesics in $\E$. 	Parametrize $\E\simeq S\times [0,\infty)$ so that $\alpha_n\subset S\times(n-1, n)$ for any $n\in \BN$. Further assume that $d(\alpha_n,\alpha_{n+1})>2C_0$.

Let $\Delta_m=\{\alpha_1, \alpha_2,...,\alpha_m\}$. Let $S_n=S\times\{n\}$. By \cite[Remark 4.2]{CG}, for sufficiently large $m_0$, $S_{m_0}$ is $2$-incompressible relative to $\Delta_{m_0}$. Take the shrinkwrapping	surface $\Sigma$ of $S_{m_0}$ given by Lemma \ref{shrinklem} (See Figure \ref{shrinkfig}).

Notice that by bounded diameter lemma (Lemma \ref{boundiamlem}) $\Sigma$ can  intersect at most one geodesic in $\Delta_{m_0}$ as $d(\alpha_n,\alpha_{n+1})>2C_0$ for any $n$. Hence, we have three mutually exclusive cases:

\begin{enumerate}
	\item 	$\Sigma\cap \alpha_{m_0}\neq \emptyset$ \quad ($\Sigma$ is Concave),
	\item  $\Sigma\cap \alpha_{m_0-1}\neq \emptyset$ \quad ($\Sigma$ is Convex),
	\item  $\Sigma\cap \Delta_m=\emptyset$.
\end{enumerate}

If we are in case (3), then $\Sigma$ is a smooth embedded minimal surface in $M$, and we are done. So, we will omit this case.

We will call a shrinkwrapping surface $\Sigma$ in the end $\E$ "{\em concave}" if the dihedral angle at the coincidence set is less than $\pi$. In particular, the component of $\E-\Sigma$ near infinity is mean convex. From the point of view the compact core $\C_M$, $\Sigma$ would be a "concave" surface. Similarly, we will call a shrinkwrapping surface $\Sigma$ in the end $\E$ "{\em convex}" if the dihedral angle at the coincidence set is greater than $\pi$. In particular, the component of $\E-\Sigma$ containing the compact core $\C_M$ is mean convex (See Figure \ref{shrinkfig2}).

If we are in case (1), we claim that $\Sigma$ is a {\em concave shrinkwrapping surface}. If we are in case (2), we claim that $\Sigma$ is a {\em convex shrinkwrapping surface}. In particular, we claim that the intersecting "obstacle" geodesic ($\alpha_{m_0}$ or $\alpha_{m_0-1}$) always stays in the convex side of $\Sigma$.

We will finish the proof of the theorem in 3 steps:

\vspace{.2cm}

\noindent {\bf Step 1:} Case (1) induces a Concave Shrinkwrapping Surface, and Case (2) induces a Convex Shrinkwrapping Surface.

\vspace{.2cm}

\noindent {\em Proof of Step 1:} Assume that $\Sigma\cap \alpha_{m_0}\neq \emptyset$ (Case 1). Recall that $\Sigma$ is obtained by shrinkwrapping the surface $S\times{m_0}$ in the end $\E$ which separates the geodesics $\alpha_{m_0}$ and $\alpha_{m_0-1}$. In particular, $\Sigma$ is a limit of smooth embedded surfaces $T_n$ which are minimal except a small ($r_n\searrow 0$) neighborhood of geodesic collection $\Delta_{m_0}$. Let $\tau$ be a proper curve segment connecting $\alpha_{m_0}$ and $\alpha_{m_0-1}$. Then, the algebraic intersection number $\langle S_{m_0}, \tau\rangle=1$. By construction, $T_n$ and $\Sigma$ is homotopic to $S_{m_0}$, then we have $\langle \Sigma, \tau\rangle=1$ and $\langle T_n, \tau\rangle=1$ for any $n$ by Lemma \ref{shrinklem}. This implies $\Sigma$ and $T_n$ separates $\alpha_{m_0-1}$ and $\alpha_{m_0}$ in $\E$.

\begin{figure}[t]

	\relabelbox  {\epsfxsize=3.5in

		\centerline{\epsfbox{endshrink.eps}}}

	\relabel{1}{\footnotesize $\C_M$} \relabel{2}{\footnotesize $M$}  \relabel{3}{\footnotesize $\E_1$} \relabel{4}{\footnotesize $\E_2$} \relabel{5}{\footnotesize $\E_3$}
	\relabel{6}{\scriptsize $S_1$} \relabel{7}{\scriptsize $S_2$} \relabel{8}{\scriptsize $S_3$}
	\relabel{9}{\scriptsize $\Sigma_1$} \relabel{10}{\scriptsize $\Sigma_3$}

	\endrelabelbox

	\caption{\label{shrinkfig2} \small In the figure above, $\E_1$ has a convex shrinkwrapping surface $\Sigma_1$, and $\E_3$ has a concave shrinkwrapping surface $\Sigma_3$. $\E_2$ is geometrically finite, and $\Sigma_2=S_2$.}

\end{figure}

Now, let $\N_n(x)$ be the normal vector on $T_n$ pointing towards infinity in $\E$. If $\Sigma=\lim T_n$ intersects $\alpha_{m_0}$, we claim that the geodesic $\alpha_{m_0}$ stays in the convex side of $\Sigma$. By Lemma \ref{thinlem}, $\Sigma$ has well-defined normal vector everywhere except $\Sigma-\alpha_{m_0}$.

Let $L=\Sigma\cap\alpha_{m_0}$ be the coincidence set. Consider $r$-tubular neighborhood $N_r(L)$ of $L$ in $M$ for some small $r>0$. Let $T=\Sigma\cap N_r(L)$.
Let $T^+$ and $T^-$ be the components of $T-L$. Notice that $L\subset \overline{T^\pm}$.
Let $p$ be a point in $L$. By using $T^+$ and $T^-$, we can define the normal vectors $N^\pm(p)$ to $T^\pm$ at $p$, pointing towards to the component of $\E-T$ containing $\alpha_{m_0}$ (the component containing infinity). By \cite[Lemma 1.27]{CG}, $T$ has a well-defined tangent cone, and let $\alpha(p)$ is the angle between $N^+(p)$ and $N^-(p)$. In particular, $\alpha(p)$ is the angle of the tangent cone of a great bigon in \cite[Lemma 1.27]{CG}.

We claim that $\alpha(p)\leq \pi$ for any $p\in L$. Assume that there exists a point $p_0 \in L$ with $\alpha(p_0)>\pi$. Then there exists a small neighborhood $I'$ of $p_0$ in $L$ such that for any point $p'\in I'$, $\alpha(p')>\pi$. Notice that for $r>0$ small, $T$ is a topological disk. Furthermore, $T$ is minimal except along $L$. Notice that $\Sigma$ is least area surface satisfying shrinkwrapping properties by Lemma \ref{shrinklem} - Condition (4). This implies $T$ is a least area disk with thin obstacle $L$. In particular, there cannot be a smaller area disk with the same boundary staying in the same side of $\alpha_{m_0}$. However, by construction of $T$, $L$ is a folding curve for this "least area" disk. We cannot push $T$ to convex side to decrease area along $L$ in general because of the shrinkwrapping condition. However, near $I'\subset L$, $\alpha_{m_0}$ stays in the concave side of $T$, so we can push $T$ along the folding curve $I'$ to the convex side without breaking the shrinkwrapping condition. Then, by \cite{MY}, we can decrease the area of $T$ by pushing $T$ along $I'$ to the convex side, and  by keeping the boundary of $T$ fixed. However, this contradicts to $T$ being least area satisfying the shrinkwrapping conditions. Hence, we conclude that $\alpha(p)\leq \pi$ along $L$, and hence $\Sigma$ is a concave shrinkwrapping surface.

The similar idea works in Case 2 by changing the orientation of the normal vectors $N^\pm(p)$ towards to the compact core $\C_M$. Then, if there is an interval $I'\subset L$ where $\alpha(p)>\pi$ along $I'$, we can push $T$ to the other side of $\alpha_{m_0-1}$ and decrease the area. This gives a contradiction.  Again, we conclude that $\alpha(p)\leq \pi$ along $L$, and hence $\Sigma$ is a convex shrinkwrapping surface. Step 1 follows. \hfill $\Box$

\vspace{.2cm}

We repeat this process in every geometrically infinite end $\E_i$, and obtain a convex or concave shrinkwrapping surface $\Sigma_i$ in $\E_i$. Now, assuming the trivial case never happens, we have a convex or concave shrinkwrapping surface $\Sigma_i$ in every end $\E_i$ of $M$. We separate the remaining part of the proof into two steps: Every $\Sigma_i$ is concave, or at least one $\Sigma_i$ is convex.

\vspace{.2cm}

\noindent {\bf Step 2:} If $M$ contains at least one geometrically finite end, or one of the shrinkwrapping surfaces is convex, then $M$ contains a smoothly embedded minimal surface.

\vspace{.2cm}

\noindent {\em Proof of Step 2:} Let $\E_{i_0}$ be the geometrically finite end, or the geometrically infinite end with convex shrinkwrapping surface. Our aim is to cut the manifold $M$ from each end with surfaces $\wh{S}_i$ deep in $\E_i$ for $i\neq i_0$, and from $\Sigma_{i_0}$ in $\E_{i_0}$ so that we get a compact mean convex manifold $M'$ by modifying the metric near boundary for $i\neq i_0$. Then, using a nontrivial homology class $\xi$ in $M'$, we want to obtain a smoothly embedded surface $\Sigma$ in $\xi$ in $M'$. Then, we will show that $\Sigma$ is completely in the hyperbolic part of $M'$.

Let $C_M$ be the compact core of $M$. If $\E_{i_0}$ is a geometrically finite end, recall that we modified the compact core $\C_M$ so that the component $T_{i_0}$ of $\partial \C_M$ corresponding to $\E_{i_0}$ is convex. In this case, let $\wh{S}_{i_0}=T_{i_0}$.

Similarly, if $\E_{i_0}$ is a geometrically infinite end which contains a convex shrinkwrapping surface $\Sigma_{i_0}$, then we take $\wh{S}_{i_0}=\Sigma_{i_0}$. Also, we modify the compact core $\C_M$ so that $\Sigma_{i_0}\subset \partial \C_M$.

Now, for $i\neq i_0$, in every end $\E_i\simeq S_i\times[0,\infty)$, fix a level surface $\wh{S}_i=S_i\times \{c_i\}$ such that $d(\C_M, \wh{S}_i)>2C_0$. For $i=i_0$, let $\wh{S}_{i_0}=T_{i_0}$.

Now, let $U_{i_0}$ be the component of $\E_{i_0}-\wh{S}_{i_0}$ containing infinity. Let $M'=M- (\bigcup_{i\neq i_0} S_i\times(c_i,\infty)\cup U_{i_0})$. For $i\neq i_0$, change the metric in the tubular neighborhood $N_\e(\wh{S}_i)$ in $M'$ so that $M'$ becomes mean convex. If there is a cusp $X$,  we cut the cusp deep, and modify the metric near boundary torus as described above so that $M'$ becomes a {\em compact mean convex manifold}. Now, we finish the proof in the following 4 cases.

\vspace{.2cm}

\noindent {\em Case 1: $M$ has more than 2 ends.} Let $\xi$ be the homology class of $S_{i_0}$. If $M$ has more than 2 ends, then $\xi$ is nontrivial in $H_2(M')$. Then, by Lemma \ref{LAlem}, we have a least area surface $\Sigma$ in the homotopy class of $S_{i_0}$. $\Sigma$ is nonempty as the homology class $\xi$ is nontrivial. Furthermore, as $M$ has more than 2 ends, for any surface $S'$ in the homotopy class of $S_{i_0}$, we have $S'\cap \C_M\neq \emptyset$. Hence, $\Sigma\cap \C_M \neq \emptyset$.

As $M'$ is mean-convex, $\Sigma$ is smoothly embedded in $M'$. Furthermore, since $\Sigma\cap \C_M \neq \emptyset$, then $d(\Sigma, \wh{S_i})>C$ for any $i\neq i_0$. Similarly, as $\Sigma$ is area minimizing, $d(\Sigma, T^2_j\times\{c_j\})>2\e$ by Lemma \ref{cusplem}. This implies $\Sigma$ is completely in the part of $M'$ where the metric is unmodified. This implies $\Sigma$ is a smoothly embedded minimal surface in the original manifold $M$. Hence, If $M$ has more than 2 ends, we are done.

In the following cases, we will use a similar idea. All we need to make sure that the homology class $\xi$ of $S_{i_0}$ is not trivial, and the least area representative $\Sigma$ intersect the compact core $\C_M$.

\vspace{.2cm}

\noindent {\em Case 2: $M$ has a cusp.} If $M$ has any cusps, then we can choose $\wh{S}_i$ deep in $\E_i$ so that $M'$ has the cusp. Again, in this case, the homology class $\xi$ of $S_{i_0}$ is nontrivial, and any least area representative $\Sigma$ of $\xi$ must intersect the compact core $\C_M$. Then, as in the previous paragraph, we get a smoothly embedded minimal surface $\Sigma$ in $M$. Hence, if $M$ has a cusp, we are done.

\vspace{.2cm}

\noindent {\em Case 3: $M$ is cusp-free with 2 ends.} Now, assume that $M$ is cusp-free, and has exactly 2 ends. Then, the homology class  $\xi$ of $S_{i_0}$ is non-trivial in $H_2(M')$. Again, let $\Sigma$ be the least area representative of the homotopy class of $S_{i_0}$. Now, by assumption $M$ is not Type I Exceptional Manifold, i.e. $M$ is not a product. Hence, $\Sigma$ must intersect the compact core $\C_M$. Then, as above, we obtain a smoothly embedded minimal surface $\Sigma$ in $M$. Notice that the assumption that $M$ is not a product manifold is crucial here, as otherwise $\Sigma$ may not intersect $\C_M$, and we cannot make sure $\Sigma$ is away from $\partial M'$ where we modify the metric.

\vspace{.2cm}

\noindent {\em Case 4: $M$ is cusp-free with 1 end.} Now, assume that $M$ is cusp-free, and has exactly 1 end. Then, by assumption of Step 2, $\E_{i_0}$ is the only end of $M$. Since $M$ is cusp-free, and $\wh{S}_{i_0}$ is already mean convex, $M'$ is codimension-$0$ submanifold of $M$, i.e. $M'$ is a compact hyperbolic manifold with boundary. In other words, we do not need to modify the metric anywhere as $M'$ is already compact and mean convex. Now, since $M$ is not Type II Exceptional Manifold, then $H_2(M)$ is not trivial. Then, so is $H_2(M')$. Let $\xi$ be the nontrivial homology class of $M'$. Let $\Sigma$ be the least area representative of $\xi$ in $M'$. As $M'$ is mean convex, $\Sigma$ is away from $\partial M'$. Hence,  $\Sigma$ is a smoothly embedded stable minimal surface in the original manifold $M$. The proof of Step 2 follows. \hfill $\Box$

\vspace{.2cm}

\noindent {\bf Step 3:} If every end $\E_i$ of $M$ contains a concave shrinkwrapping surface $\Sigma_i$ for $1\leq i \leq N$, then $M$ contains a smoothly embedded minimal surface.

\vspace{.2cm}

\noindent {\em Proof of Step 3:} In this step, our aim is to use Lemma \ref{minmaxlem} to obtain a smoothly embedded minimal surface in $M$. In particular, we want to get an open bounded set $\Omega$ in $M$ where $\overline \Omega$ is a strictly mean concave manifold.

Now, by assumption, every end $\E_i$ of $M$ contains a concave shrinkwrapping surface $\Sigma_i$. In particular, they are minimal everywhere except some curve segments. Along these curve segments, the dihedral angle is less than $\pi$, i.e. convex towards the component containing infinity. Hence, when we evolve $\Sigma_i$ under the level set flow, by the main theorem (and Section 11) of \cite{Wh}, $\Sigma_i$ immediately becomes a smooth, strictly mean concave (normal vector points to $\infty$ side of $\E_i$) surface $\Sigma_i'$ in the positive side ($\infty$ side of $\E_i$) of $\Sigma_i$. We will take $\Sigma_i'$ as our smooth, strictly mean concave surfaces in the end $\E_i$ for each $i$. Let $Z=\bigcup_i\Sigma_i'$.

Now, $Z$ separates a bounded open domain $\Omega$ in $M$. Furthermore, $\overline{\Omega}$ has smooth strictly concave boundary $Z$. Hence, by  Lemma \ref{minmaxlem} and Corollary \ref{minmaxcor}, $M$ contains a smoothly embedded unstable minimal surface $\Sigma$ with $\Sigma\cap\Omega\neq \emptyset$. Step 3 follows. \hfill $\Box$.

\vspace{.2cm}

This finishes the proof of the theorem. To recap, if $M$ has geometrically finite end, and $M$ is not a product, then Step 2 gives a minimal surface. By Step 1, we have a convex or concave shrinkwrapping surface in every geometrically infinite end of $M$. If all ends of $M$ is geometrically infinite, and one of them contains a convex shrinkwrapping surface, then again Step 2 finishes the proof. If all ends are geometrically infinite, and all of them contains a concave shrinkwrapping surface, then Step 3 gives a smoothly embedded minimal surface in $M$. The proof of the theorem follows.
\end{pf}

\begin{rmk} To recap our main result, the following is the organization of the  cases we dealt with:

	\begin{itemize}
		\item When $M$ has a geometrically infinite end with unbounded geometry (Theorem \ref{ubdthm}).
		\item When $M$ has more than one cusps (Lemma \ref{cusplem2}).
		\item When $M$ has bounded geometry with more than one end (Thm \ref{mainthm}).
	\end{itemize}

	Recall that finite volume, and closed hyperbolic manifold cases were already proved in \cite{CHMR,HW2,Pi}.
\end{rmk}

\section{Exceptional Manifolds} \label{Xsec}

In this section, we will focus on the exceptional manifolds, and see what can be derived more about these cases by using the proof of the main result.

First, we recall the exceptional manifolds:

\begin{itemize}
	\item Type I: $M$ is a product manifold with bounded geometry.
	\item Type II: $M$ has exactly one end, and $H_2(M)$ is trivial.
\end{itemize}

\subsection{Type I: Product Manifolds with Bounded Geometry.} \

\vspace{.2cm}

Let $M$ be hyperbolic $3$-manifold which is a product, i.e. $M\simeq S\times \BR$ for some closed surface of genus $\geq 2$. Then, in the following cases, we can deduce the existence of a smoothly embedded closed minimal surface in $M$.

\begin{itemize}
	\item \textit{$M$ is geometrically finite:} In this case, $M$ has a compact convex core $\cc(M)$, and the area minimizing representative $\Sigma$ of the homotopy class of $S$ stays in $\cc(M)$. As $\cc(M)$ is convex, $\Sigma$ is smoothly embedded area minimizing surface in $M$.

	\item\textit{ One end of $M$ is geometrically finite, and other end contains a convex shrinkwrapping surface (Theorem \ref{mainthm} - Step 1):} In this case, cut the manifold $M$ from convex surface $T^+$ in the geometrically finite end $\E^+$, and from the convex shrinkwrapping surface $\Sigma^-$ in $\E^-$. Then, we obtain a compact mean convex manifold $M'\simeq S\times [-1,1]$. Then, there is an area minimizing representative $\Sigma$ of $S$ in $M'$ by Lemma \ref{AMSlem}. Hence, $\Sigma$ is a smoothly embedded stable minimal surface in $M$.

	\item \textit{Both ends of $M$ contains a convex shrinkwrapping surface:} The proof is same with the previous case by replacing $T^+$ with $\Sigma^+$.

	\item \textit{Both ends of $M$ contains a concave shrinkwrapping surface:} This case follows from the proof of Theorem \ref{mainthm} - Step 3.

\end{itemize}

Hence, by eliminating the above cases, there is only one case remaining where the existence of smoothly embedded closed minimal surface is unknown:

\vspace{.2cm}

\noindent {\bf $\diamond$ Very Exceptional Type I Manifold:} {\em $M$ is a geometrically infinite product manifold such that one end $\E^+$ contains a convex shrinkwrapping surface (or geometrically finite), and the other end $\E^-$ contains concave shrinkwrapping surface.}

\vspace{.2cm}
In particular, this case is quite delicate because of the following situation: If $M$ contains a mean convex foliation $\{\Sigma_t\}$, then there is no minimal surface in $M$ by maximum principle. It is not known such a product hyperbolic manifold exist. So, to rule out this case, one needs to rule out the existence of such a mean convex foliation \cite{So,Gr}.

\subsection{Type II: One End with trivial $H_2(M)$.} \label{type3sec} \

\vspace{.2cm}

This case is quite interesting. First notice that trivial $H_2(M)$ implies that such manifolds are cusp-free. Furthermore, in the trivial case $M=\BH^3$, we know that there is no smoothly embedded closed minimal surface in $M$. The reason for this, the concentric $r$-spheres $S_r=\partial\B_r(0)$ with mean curvature $\coth{r}>1$ foliates $\BH^3$. If $\Sigma$ was a smoothly embedded minimal surface in $\BH^3$, then the largest $r$ with $S_r\cap \Sigma\neq \emptyset$ would give the first point of touch, and give a contradiction with maximum principle.

Other interesting case in Type II Exceptional Manifolds is the {\em Schottky Manifolds}. These are genus $g\geq 2$ handlebodies with geometrically finite ends \cite{Ma}. It is possible to use a similar idea to prove the nonexistence of a minimal surface in a special class of these manifolds, namely Fuchsian Schottky Manifolds.

\begin{lem} \label{FuchsianSchottky} Let $M$ be a Fuchsian Schottky $3$-manifold. Then, there exists no closed embedded minimal surface in $M$.
\end{lem}

\begin{pf} Since $M$ is Schottky, $\pi_1(M)\simeq \mathbf{F}_g$ where $\mathbf{F}_g$ is a free group with $g$ generators for some $g\geq 2$. In particular, $M$ is a genus $g$ handlebody, which is in the homotopy type of a disk with $g$ punctures.

As $M$ is also Fuchsian, then the representation of the fundamental group of the manifold is in $Isom(\BH^2)=PSL(2,\BR)$, i.e. $\Gamma=\rho(\pi_1(M))\subset Isom(\BH^2)\subset Isom(\BH^3)$. Hence, there exists a hyperbolic plane $\p_0$ in $\BH^3$ fixed by $\Gamma$. Then, we have a completely geodesic surface $\Sigma_0=\p_0/\Gamma$ in $M$. Furthermore, let $\Lambda=\PI \p_0$ be the asymtotic boundary of $\p_0$ which is a round circle in $S^2_{\infty}(\mathbf{H}^3)$. Now, consider the foliation of $\BH^3$ by constant mean curvature $H$-planes $\{\p_H\}$ with $\PI \p_H=\Lambda$ where $H\in(-1,1)$.

By construction, $\Lambda$ is preserved by $\Gamma$, and $\p_H$ is the unique $H$-plane with $\PI\p_H=\Lambda$ for $H\in(-1,1)$. Then, $\p_H$ is preserved by $\Gamma$, i.e. for any $\varphi\in\Gamma$, we have $\varphi(\p_H)=\p_H$. This induces a foliation $\{\Sigma_H\mid H\in (-1,1)\}$ where $\Sigma_H=\p_H/\Gamma$. Intuitively, one can imagine this foliation as follows: $M$ is a genus $g$ handlebody, and  $\Sigma_0$ is the middle slice (largest cross-section) of $M$. Hence, $\Sigma_0$ is topologically a disk with $g$ holes. Then, upper half of $M-\Sigma_0$ is the foliation $\{\Sigma_H\mid H>0\}$, and the lower half of $M-\Sigma_0$ is the foliation $\{\Sigma_H\mid H<0\}$.

Let $S$ be a closed embedded minimal surface in $M$. Consider the last point of touch $H_0=\sup\{H\mid \Sigma_H\cap S\neq \emptyset\}$. As $S$ is closed, $H_0<1$. If $H_0<0$, then we can use the last point of touch in the other side. So, we will assume $H_0\in [0,1)$. However, this implies $\Sigma_{H_0}$ and $S$ has tangential intersection while one lies in the other side of the other. This is a contradiction by maximum principle. The proof follows.

\end{pf}

Hence, this proves that in the simplest class of Schottky manifolds, there is no closed embedded minimal surface. On the other hand, for a general Schottky manifold, while constructing a least area surface is not possible, it might still be possible to use some variation of min-max methods to construct a closed embedded minimal surface. However, as the ends are geometrically finite in Schottky Manifolds, the current min-max methods are not sufficient to obtain such a minimal surface.

\section{Final Remarks}

\subsection{Further Questions.} \

\begin{question}
	If we remove "closed" condition from the minimal surface, which hyperbolic 3-manifolds admit an embedded minimal surface?
\end{question}

In this problem, we allow noncompact complete properly embedded minimal surfaces. In particular, the ends of the minimal surface lives in the ends of the ambient hyperbolic $3$-manifold. For the convex cocompact (geometrically finite and cusp-free) hyperbolic $3$-manifolds, \cite{AM} gives a positive answer to this question in various cases.

\begin{question}
	Other than the trivial example $M=\BH^3$, and Fuchsian Schottky Manifolds, which complete hyperbolic $3$-manifolds do not have any closed embedded minimal surface?
\end{question}

As mentioned above, Schottky manifolds are potential examples for the nonexistence question. On the other hand, it might be possible that in the  Very Exceptional Type I Manifold case, the existence of minimal surfaces can be proven by improving the min-max techniques in \cite{So}. Hence, we conjecture that there is no closed embedded minimal surface only in Type II Exceptional Manifolds.

\begin{question}
	Are there nonproperly embedded minimal surfaces in complete hyperbolic $3$-manifolds?
\end{question}

Related to Question 1 above, one can ask if there exists a nonproperly embedded minimal surfaces in a complete hyperbolic $3$-manifold. When $M=\BH^3$, we constructed such a nonproperly embedded minimal surface in \cite{Co3}. Here, the question is whether there exists such examples in hyperbolic $3$-manifolds with nontrivial topology. This is known as \textit{Calabi-Yau Conjecture} in general \cite{CM}. In particular, such an example would induce a group invariant nonproperly embedded minimal surface in $\BH^3$.

\subsection{Stability} \

Notice that all the minimal surfaces we constructed are obtained by using area minimization with a trapping argument except one case: Hyperbolic $3$-manifold with geometrically infinite ends with bounded geometry where all ends contains a concave shrinkwrapping surface (Theorem \ref{mainthm} - Step 3). In this case, we used a min-max method \cite{Mo,So}. Hence, this is the only case we get an {\em unstable minimal surface}. In all other cases, the minimal surfaces we get are {\em stable}.

\subsection{Number of Minimal Surfaces} \

After the resolution of Yau's Conjecture \cite{MN, So2}, number of minimal surfaces in $3$-manifolds became a very attractive topic. In particular, Song and Marques-Neves proved that any closed $3$-manifold contains infinitely many closed embedded minimal surfaces \cite{MN, So2}. Furthermore, Song has proved that any finite volume non-compact hyperbolic $3$-manifold contains infinitely many closed embedded minimal surfaces \cite{So}. However, in the infinite volume case, the problem is completely open.

\begin{question}
Which infinite volume hyperbolic $3$-manifolds contain infinitely many embedded minimal surfaces?
\end{question}

When $M$ has a geometrically infinite end with unbounded geometry, our techniques gives not infinite, but any number of minimal surfaces. Notice that in Theorem \ref{ubdthm}, we obtained a minimal surface in an end $\E$ with unbounded geometry by trapping the surface between two short geodesics in the end. By iterating this process farther at the end, one can get any number of minimal surfaces in these manifolds. On the other hand, in the geometrically finite case, Huang-Wang obtained a similar result for quasi-Fuchsian manifolds \cite{HW3}.

Note also that Song has recently proved an astonishing dichotomy on the number of minimal surfaces for non-compact complete manifolds. He showed that if $M$ is thick at infinity, then either $M$ contains no embedded unstable minimal surface, or infinitely many embedded unstable minimal surfaces \cite[Theorem 18]{So}. Here, $M$ is \textit{thick at infinity} means that any connected, complete, finite area minimal surface in $M$ is closed. By using our bounded diameter lemmas, it is not hard to show that complete hyperbolic $3$-manifolds with no cusp are thick at infinity. Hence, if one can guarantee that there is an unstable minimal surface in a complete, cusp-free hyperbolic $3$-manifold, Song's dichotomy gives infinitely many embedded, unstable minimal surfaces.

As mentioned above, while our trapping techniques induces stable minimal surfaces in general, in one special case, we indeed obtain unstable minimal surface by using min-max methods: Hyperbolic $3$-manifold with geometrically infinite ends of bounded geometry where all ends contains a concave shrinkwrapping surface (Theorem \ref{mainthm} - Step 3). Hence, in that special case, Song's dichotomy gives infinitely many embedded minimal surfaces.

\subsection{Incompressibility} \

Unlike the many earlier existence results, the closed embedded minimal surfaces constructed in this paper may not be incompressible in the ambient hyperbolic $3$-manifold $M$ in general. In particular, in several cases, we are trapping a minimal surface between two short geodesics, two cusps, or one cusp and a convex surface. These give a restricted homology class where we minimize the area. Hence, even though some of the minimal surfaces are incompressible in some region $\Omega$ in $M$, they are not incompressible in $M$ in general. Furthermore, in min-max case (Theorem \ref{mainthm} - Step 3), we have no control on the incompressibility of the minimal surface obtained.


TITLE: Proving $f(x,y)=(e^x \cos y,e^x \sin y)$ Has No Inverse
QUESTION [0 upvotes]: By the inverse theorem we can see that the Jacobin is $e^x\neq 0$ so there is a locally inverse function, how do I show that it has not inverse function?

REPLY [2 votes]: In order for a function to have an inverse, it has to be one-to-one.
Notice that  For $$  f(x,y)=(e^x \cos y,e^x \sin y)$$
We have $$  f(x,y+2k\pi )=(e^x \cos( y+ 2k\pi) ,e^x \sin( y+2k\pi)) = f(x,y)$$
Thus your function is not one-to-one.


TITLE: For $n$ random variables with PDF $f(x;\theta)=\frac{2x}{\theta^2}, 0<x\leq\theta, \theta > 0$ Is $Var(\theta^*)$ equal to $CRLB(\theta)$?
QUESTION [1 upvotes]: Let $X_1,...,X_n$ be iid random varaibles from the following PDF $$f(x;\theta)=\frac{2x}{\theta^2}, 0<x\leq\theta, \theta > 0$$
Let $\theta^*$ be an unbiased estimator for $\theta$ based on $\hat{\theta}_{ML}$.
Is $Var(\theta^*)$ equal to $CRLB(\theta)$?
My try:
I'm mainly struggling to find $\theta^*$.
First I got that the $\hat{\theta}_{ML}$ is $X_{(n)}$
Then $\theta^*$ must be a function of $X_{(n)}$.
But I get that $E[X_{(n)}]=\frac{x^2}{\theta}$ and I'm not sure on how to calculate $\theta^*$ from it. Any suggestions would be great!

REPLY [2 votes]: There's a mistake in your computation of $E(X_{(n)})$. Observe that the CDF of $X_i$ is :
$$F_X(x) = \int_0^x \frac{2t}{\theta^2}dt = \frac{x^2}{\theta^2}$$
So that the CDF of $X_{(n)}$ is given by $$F_{X_{(n)}} (x) = F_{X}(x)^n = \frac{x^{2n}}{\theta^2}$$ And its density is simply $f_{X_{(n)}}(x) = \frac{2n x^{2n-1}}{\theta^2}$. From there, the expectation is: $$E(X_{(n)}) = \int_0^\theta 2n \cdot x \cdot \frac{x^{2n-1}}{\theta^2} = \frac{2n}{2n+1} \theta$$
You should be able to compute $\theta^*$ from this and determine whether it achieves the Cramér-Rao lower bound.