Minimal surfaces near short geodesics in hyperbolic $3$-manifolds

TL;DR

This paper investigates how the minimal surface area functional in hyperbolic 3-manifolds behaves under geometric convergence, focusing on the interaction with short geodesics and establishing continuity properties.

Contribution

It proves the lower semi-continuity and conditions for continuity of the minimal surface area functional in hyperbolic 3-manifolds, highlighting the role of short geodesics.

Findings

The functional is lower semi-continuous under geometric convergence.

Continuity holds when the minimal surface satisfies certain conditions.

Interaction between minimal surfaces and short geodesics is characterized.

Abstract

If $M$ is a finite volume complete hyperbolic $3$-manifold, the quantity $\mathcal A_1(M)$ is defined as the infimum of the areas of closed minimal surfaces in $M$. In this paper we study the continuity property of the functional $\mathcal A_1$ with respect to the geometric convergence of hyperbolic manifolds. We prove that it is lower semi-continuous and even continuous if $\mathcal A_1(M)$ is realized by a minimal surface satisfying some hypotheses. Understanding the interaction between minimal surfaces and short geodesics in $M$ is the main theme of this paper

Figures

Equations

$${\mathcal{A}}_1(M)=\inf (\{|\Sigma |,\Sigma \in \mathcal{O}\}\cup \{2|\Sigma |,\Sigma \in \mathcal{U}\})$$

$${\mathcal{A}}_1(\overline{M})\le \mathrm{liminf}{\mathcal{A}}_1(M_i).$$

$${\mathcal{A}}_1(\overline{M})=\lim {\mathcal{A}}_1(M_i).$$

$$g=e^{-2t}d{\sigma }^2+d{t}^2$$

$$F(z,\theta ,r)={\exp }_{c(z)}(r(\cos \theta \vec{\nu }(z)+\sin \theta \vec{\tau }(z)))$$

$$g=({\cosh }^{2}r)d{z}^2+({\sinh }^{2}r)d{\theta }^2+d{r}^2.$$

$$g={\sinh }^2(R-t)d{x}_1^2+{\cosh }^2(R-t)d{x}_2^2+d{t}^2=d{\sigma }_t^2+d{t}^2$$

$${\sinh }^{2}R=\frac{1}{2}\left(\frac{\sqrt{1-2k}}{k}-1\right)$$

$$\frac{1}{k_i}d(p,q)\le d({\phi }_i(p),{\phi }_i(q))\le k_{i}d(p,q)$$

$${\tilde{g}}_{i,kl}$$

$$=⟨d{\phi }_{i}d\overline{\Pi }{\tilde{X}}_{y_k},d{\phi }_{i}d\overline{\Pi }{\tilde{X}}_{y_l}{⟩}_{M_i}$$

$$=⟨d{\Pi }_{i}d{f}_i{\tilde{X}}_{y_k},d{\Pi }_{i}d{f}_i{\tilde{X}}_{y_l}{⟩}_{M_i}$$

$$=⟨d{f}_i{\tilde{X}}_{y_k},d{f}_i{\tilde{X}}_{y_l}{⟩}_{{\mathbb{H}}^3}$$

$${\cosh }^2(\rho +r_n)e^{-2r_n}d{a}^2+{\sinh }^2(\rho +r_n)e^{-2r_n}d{b}^2+d{\rho }^2$$

$$\Vert u_n-r_n\Vert \le A{\epsilon }_0\Vert \nabla u_n\Vert \le \sinh r_n\Vert \mathrm{Hess}{u}_n\Vert \le \frac{1}{C}{\sinh }^2{r}_n.$$

$$\Vert w_n\Vert$$

$$u(v_0)>h(v_0)>h(0,0)=u(0,0)=u_{v_0}(v_0)$$

$$u_{v_0}(0,0))=u(-v_0)>h(-v_0)>h(0,0)=u(0,0)$$

$$u(v_0)>\bar{r}=u_{v_0}(v_0)\text{ and }{u}_{v_0}(0,0)=u(-v_0)>\bar{r}=f(0,0)$$

$$n_p(k_1,\dots ,k_p)=\#\{i\in \{1,\dots ,p\}|k_i\in \{1,2\}\}.$$

$$b_{n,kl}(y_1,y_2,y_3)=a_{n,kl}(y_1,y_2,y_3/{\lambda }_n+s_n){\lambda }_n^{n_2(k,l)}$$

$$\frac{h_n^2(y_3/{\lambda }_n+s_n)}{h_n^2(s_n)}(d{y}_1^2+d{y}_2^2)+d{y}_3^2$$

$${\partial }_i{b}_{n,kl}={\partial }_i{a}_{n,kl}(y_1,y_2,y_3/{\lambda }_n+s_n){\lambda }_n^{n_{k,l,i}-1}$$

$$|{\partial }_i{b}_{n,kl}|\le A\frac{h_n^{n_3(k,l,i)}(y_3/{\lambda }_n+s_n)}{h_n^{n_3(k,l,i)}(s_n)}{h}_n(s_n)\to 0$$

$$\frac{1}{{\sinh }^2(R_{\ell })}({\sinh }^2(R_{\ell }-y_3)d{y}_1^2+{\cosh }^2(R_{\ell }-y_3)d{y}_2^2)+d{y}_3^2$$

$$g\ge {\sinh }^2(\rho -2a)(d{z}^2+d{\theta }^2)+d{r}^2\ge \frac{1}{4}{\sinh }^2\rho (d{z}^2+d{\theta }^2)+d{r}^2$$

$$|S\cap (N_{{\rho }_2}\setminus N_{{\rho }_1})|$$

$$\le {\int }_{{\rho }_1}^{{\rho }_2}{\int }_{{\mathbb{S}}^1}\frac{\cosh r}{\sinh R_{\ell }}({\cosh }^2(R_{\ell })z^{\prime 2}(s)+{\sinh }^2(R_{\ell }){\theta }^{\prime }(s){)}^{1/2}drds$$

$$\le \frac{{\mathfrak{s}}_0}{\sinh R_{\ell }}(\sinh {\rho }_2-\sinh {\rho }_1)$$

$$\le \frac{2{\mathfrak{s}}_0}{\sinh R_{\ell }}\cosh \frac{{\rho }_1+{\rho }_2}{2}\sinh \frac{{\rho }_2-{\rho }_1}{2}$$

$$|S\cap (N_{\rho }\setminus N_{\rho -2a})|$$

$$\le \frac{2\kappa {\mathfrak{s}}_0}{\sinh R_{\ell }}\cosh (\rho )a$$

$$\le \frac{2{\kappa }^{\prime }{\mathfrak{s}}_0}{\cosh R_{\ell }}\sinh (\rho )a$$

$$\le 8{\kappa }^{\prime }{a}^2\le \frac{8{\kappa }^{\prime }}{\pi }|\Sigma \cap Y_{\rho }|\le \frac{8{\kappa }^{\prime }}{\pi }|\Sigma \cap (N_{\rho }\setminus N_{\rho -2a})|$$

$$|\Sigma \cap N_R|$$

Full text

1. Introduction

The area of a closed minimal surface $\Sigma$ in a complete hyperbolic $3$-manifold is bounded above by $-2\pi\chi(\Sigma)$; this follows from the Gauss equation. Finding an optimal lower bound for the area is a more subtle question. Notice that in dimension $2$, there is no lower bound for the length of a closed geodesic in a hyperbolic surface. However the Margulis lemma and the monotonicity formula does give a lower bound of $2\pi(\cosh(\bar{\varepsilon})-1)$, for the area of a properly immersed minimal surface in a complete hyperbolic $3$-manifold; $\bar{\varepsilon}$ is the Margulis constant. According to explicit estimates of $\bar{\varepsilon}$, this number is at least 0.104 [11].

In a previous paper the authors proved the area is at least $2\pi$ when $\Sigma$ is a closed embedded minimal surface in a complete finite volume hyperbolic $3$-manifold of Heegaard genus at least $6$. If $\Sigma$ is non-orientable the lower area bound is $\pi$. Perhaps the main goal of the present paper it to introduce techniques to resolve the remaining cases: $2\leq\mathrm{Heegaard\ genus}\leq 5$.

In our paper [10], we introduce the quantity $\mathcal{A}_{1}(M)$, where $M$ is a compact orientable $3$-manifold. If $\mathcal{O}$ denotes the collection of all smooth orientable embedded closed minimal surfaces in $M$ and $\mathcal{U}$ the collection of all smooth non-orientable ones, $\mathcal{A}_{1}(M)$ is defined by

[TABLE]

so $\mathcal{A}_{1}(M)$ gives a lower bound for the area of any minimal surface in $M$.

The main result in [10] says that $\mathcal{A}_{1}(M)$ is the area (or twice the area) of some minimal surface in $M$. Moreover it gives some characterization of this minimal surface in terms of its index and its genus.

Let $(g_{i})_{i}$ be a sequence of smooth Riemannian metrics on $M$ which smoothly converge to $\bar{g}$. Because of the characterization of the minimal surface that realizes $\mathcal{A}_{1}(M,g_{i})$ and thanks to a compactness result by Sharp [14], it can be proved that $\liminf\mathcal{A}_{1}(M,g_{i})\geq\mathcal{A}_{1}(M,\bar{g})$. Moreover, if $\mathcal{A}_{1}(M,\bar{g})$ is realized by a non degenerate minimal surface, $\lim\mathcal{A}_{1}(M,g_{i})=\mathcal{A}_{1}(M,\bar{g})$. However one can produce examples where $\mathcal{A}_{1}$ is not upper semi-continuous (F. Morgan suggested examples of a $2$-sphere looking like a pear).

Concerning hyperbolic manifolds, our study proves that, if $M$ is hyperbolic and its Heegaard genus is at least $6$, then $\mathcal{A}_{1}(M)\geq 2\pi$ which gives a universal lower bound for the area of a minimal surface in $M$. This reasoning can be adapted to the case $M$ is a finite volume hyperbolic manifold (not necessarily compact).

In order to remove the hypothesis about the Heegaard genus, we ask the question of the continuity of $\mathcal{A}_{1}$ when the space of hyperbolic manifolds is endowed with the geometric convergence topology. Here the situation is not as above where we have a sequence of Riemannian metrics on a fixed manifold, here we have a sequence of manifolds $M_{i}$ with changing topologies. Moreover, if $(M_{i})_{i}$ is a non trivial converging sequence of hyperbolic manifolds then $M_{i}$ contains a geodesic $\gamma_{i}$ whose length goes to [math]. As a consequence, an important question for our study is to understand the behaviour of a minimal surface intersecting a neighborhood of a short geodesic.

This question has been already studied by several authors. For example, Hass [7] and Huang and Wang [8] study the geometry of minimal surfaces near a short geodesic in order to construct hyperbolic manifolds that fiber over the circle but such that the fibers can not be made minimal.

Our study of minimal surfaces near short geodesics starts with a result of Meyerhoff [11]. Basically it says that a short geodesic in $M$ of length $\ell$ has a embedded tubular neighborhood $N_{R_{\ell}}$ of radius $R_{\ell}$ and $\lim_{\ell\to 0}R_{\ell}=+\infty$.

We obtain two results concerning minimal surfaces in $N_{R_{\ell}}$. The first one deals with stable minimal surfaces in tubular neighborhood of short geodesics (Corollary 7). Basically it says that such a stable minimal surface either stays far from the short geodesic or it intersects transversely the short geodesic. Moreover in the second case, the surface must have a very large area in the $R_{\ell}$ tubular neighborhood of the geodesic.

Our second result deals with general minimal surfaces (not assumed to be stable) (Proposition 9). It says that a minimal surface in the neighborhood of a short geodesic either stays very far from the core geodesic or comes very close to it (the estimate depending on the index of the minimal surface). As above in the second case, we obtain a lower bound for the area of a minimal surface coming close to the short geodesic.

Actually these two results are very similar to results we obtained with Collin and Hauswirth in [5] concerning the geometry of minimal surfaces in hyperbolic cusps. In both cases, the argument is based on the fact that the tubular neighborhoods are foliated by equidistant tori whose diameter are small. As a consequence, an embedded minimal surface with bounded curvature can not be tangent to these equidistant surfaces.

Once the behaviour of minimal surfaces close to short geodesics is understood, we study the continuity of $\mathcal{A}_{1}$. A version of our result can be stated as follows. It is similar to the result that can be obtained for a fixed manifold with a converging sequence of metrics.

Theorem**.**

Let $M_{i}\to\overline{M}$ be a converging sequence of hyperbolic cusp manifolds. Then

[TABLE]

If $\mathcal{A}_{1}(\overline{M})$ is not realized by the area of a stable-unstable separating minimal surface, then

[TABLE]

Let us recall that ”stable-unstable” means that the first eigenvalue of the stability operator is [math]. Of course one can expect that the surface that realizes $\mathcal{A}_{1}(\overline{M})$ is never stable-unstable but we do not know how to prove this. Actually it is possible to expect that no minimal surface in a hyperbolic manifold is stable-unstable. In fact the above result is a combination of two propositions: Propositions 22 and 25

The main difficulty in the proof of Proposition 25 is to be able to control where is located a minimal surface $\Sigma_{i}$ that realizes $\mathcal{A}_{1}(M_{i})$. Actually, our study of minimal surfaces near short geodesics implies that $\Sigma_{i}$ can not enter into a tubular neighborhood of a short geodesic. So it stays in a part of $M_{i}$ where the convergence $M_{i}\to\overline{M}$ is just the smooth convergence of the metric tensor. Thus a compactness result by Sharp [14] gives the lower semicontinuity of $\mathcal{A}_{1}$. Concerning Proposition 22, we first prove that $\limsup\mathcal{A}_{1}(M_{i})$ is bounded. Thus if $\mathcal{A}_{1}(\overline{M})$ is not realized by a stable-unstable separating minimal surface $\Sigma$ then $\Sigma$ can be deformed into a minimal surface in $M_{i}$. This implies the second inequality.

Of course one can also think about hyperbolic manifolds with infinite volume and ask the following question. For which class of complete hyperbolic 3-manifolds of infinite volume can one hope for an area lower bound $2\pi$? There may not exist a closed minimal surface in $M$, but if $\mathcal{A}_{1}(M)$ is realized, can one expect it to be at least $2\pi$?

The paper is organized as follows. In Section 2.1, we recall some basic facts about the description of cusp and tubular ends of complete finite volume hyperbolic $3$-manifolds. Section 3 studies the geometry of minimal surfaces with bounded curvature in tubular ends. In Section 4, we study the general behaviour of minimal surfaces in tubular ends. In Section 5 we recall some facts about the min-max theory for minimal surfaces that we will use in the next sections. Section 6 is devoted to recall the work we made in [10] and how it should be adapted to work with non compact hyperbolic manifolds. Sections 7 and 8 are devoted to the study of the lower and upper semi-continuity of the $\mathcal{A}_{1}$ functional. Finally in Appendix A, we prove some technical results and formulas.

Preliminary remarks

Let $S$ be a smooth Riemannian surface, we will denote by $|S|$ its area.

Let $(T,d\sigma^{2})$ be a flat torus. Its universal cover is a flat $\mathbb{R}^{2}$ so we have coordinates $(x_{1},x_{2})$ such that the flat metric can be written $dx_{1}^{2}+dx_{2}^{2}$. Then $T$ is the quotient of $\mathbb{R}^{2}$ by some lattice $\Gamma$. We say that $(x_{1},x_{2})$ is an orthonormal coordinate system on $T$.

Moreover, we can choose $(x_{1},x_{2})$ such that $\Gamma$ is generated by $v_{1},v_{2}$ where $v_{1}=(a_{1},0)$ and $v_{2}=(a_{2},b_{2})$. We then say that $(x_{1},x_{2})$ is a well oriented orthonormal coordinate system.

We notice that if $(T,d\sigma^{2})$ has diameter $\delta$ then the lattice can be generated by vectors of length less than $2\delta$.

2. Hyperbolic manifolds

In this first section we recall some facts concerning the geometry of hyperbolic $3$-manifolds with finite volume also called cusp manifolds. We refer to [2] for part of this description.

2.1. The cusp and tubular ends

Let $M$ be a complete hyperbolic $3$-manifold of finite volume. For any $\varepsilon$ less than the Margulis constant, the manifold $M$ can be split into two parts: the $\varepsilon$-thick part $M_{[\varepsilon,\infty)}$ which is connected, not empty (recall that $p\in M_{[\varepsilon,\infty)}$ is any non null homotopic closed loop at $p$ has length at least $\varepsilon$) and the $\varepsilon$-thin part which may have a finite number of connected components. The connected components of the thin part are of two types: cusp ends and tubular neighborhoods of closed geodesics also called tubular ends.

Cusp ends are isometric to $E_{0}=T\times\mathbb{R}_{+}$ endowed with a metric

[TABLE]

where $d\sigma^{2}$ is a flat metric on the $2$-torus $T$. We define $E_{t}=T\times[t,+\infty)$. We notice that if $E_{0}$ is a component of the $\varepsilon$-thin part then $E_{t}$ is a component of the the $\delta\varepsilon$-thin part with $e^{-2t}\leq\delta\leq e^{-t/2}$.

For tubular ends, let $\gamma$ be a short geodesic in $M$ and consider $c$ a lift of $\gamma$ to $\mathbb{H}^{3}$. If $R$ is small the $R$-tubular neighborhood $N_{R}$ of $\gamma$ in $M$ is the quotient of the $R$ tubular neighborhood $V_{R}$ of $c$ in $\mathbb{H}^{3}$ by some loxodromic transformation $\tau$ of axis $c$ (see Figure 1).

In order to introduce some coordinate system, let $z$ denote arclength along $c$ and let $\vec{\nu}(z),\vec{\tau}(z)$ be parallel orthogonal unit normal vectorfields along $\gamma$, we introduce cylindrical coordinates in $V_{R}$ by

[TABLE]

In these coordinates, the hyperbolic metric is

[TABLE]

$N_{R}$ can be viewed as the quotient of $M_{R}=\{(t,\theta,r)\in\mathbb{R}^{2}\times[0,R]\}$ by the relations $(t,\theta,0)\sim(t,\theta^{\prime},0)$, $(t,\theta,r)\sim(t,\theta+2\pi,r)$ and $(t,\theta,r)\sim(t+\ell,\theta+\alpha,r)$ for some parameters $\ell>0$ and $\alpha$. $\ell$ is the length of the geodesic loop $\gamma$ and $\alpha$ is called the twist parameter of $\gamma$ (it is the angle of the loxodromic transformation). As above, if $N_{R}$ is a component of the $\varepsilon$-thin part, then $N_{r}$ is a component of the $\delta\varepsilon$-thin part some some $\delta\in[e^{2(r-R)},e^{(r-R)/2}]$ if $R$ and $r$ are larger than some universal constant. In the following, we denote by $S_{\bar{r}}=\partial N_{\bar{r}}$ the torus $\{r=\bar{r}\}$.

The above coordinates are called tubular coordinates. In order to be coherent with the coordinates we use on cusp ends, we will also use the coordinate system $(x_{1},x_{2},t)=(\theta,z,R-r)$ such that the metric can be written

[TABLE]

on $T\times[0,R]$ where $T$ is the quotient of $\mathbb{R}^{2}$ by the translations by $(2\pi,0)$ and $(\alpha,\ell)$ (notice that $g$ is singular on $T\times\{R\}$).

The interest of these coordinates is that any part of a cusp or tubular end can be described as $T\times[a,b]$ with some metric $d\sigma_{t}^{2}+dt^{2}$ where $d\sigma_{t}^{2}$ is a flat metric on the torus $T$. We denote by $T_{t}=T\times\{t\}$. So the family $(T_{t})_{t}$ gives a foliation of the ends by tori.

If $C$ is the torus in such an end that corresponds to $T\times\{\bar{t}\}$, the graph of a function $u:\Omega\subset C\to\mathbb{R}$ is just the surface parametrized by $\{(p,t)\in T\times\mathbb{R}|t=\bar{t}+u(p)\}$ (notice that we will often identify $C\subset M$ with $T_{\bar{t}}\in T\times\mathbb{R}$).

One question is to know what is the maximal radius $R$ that can be considered in the above discussion ($N_{R}$ being embedded). This has been estimated by Meyerhoff in [11] where the following result is proved.

Theorem 1**.**

Let $\gamma$ be a geodesic loop in a complete hyperbolic $3$-manifold. If the length $\ell$ of $\gamma$ is less than $\frac{\sqrt{3}}{4\pi}\ln^{2}(\sqrt{2}+1)$, then there exists an embedded tubular neighborhood around $\gamma$ whose radius $R$ satisfies

[TABLE]

where $k=\cosh\sqrt{\frac{4\pi\ell}{\sqrt{3}}}-1$.

In the sequel we denote by $R_{\ell}$ the solution of $\sinh^{2}R=\frac{1}{2}\left(\frac{\sqrt{1-2k}}{k}-1\right)$. When $\ell$ is small this implies that $\sinh^{2}R_{\ell}\sim\cosh^{2}R_{\ell}\sim\frac{\sqrt{3}}{4\pi\ell}$. For example, the area of $S_{R_{\ell}}$ goes to $\frac{\sqrt{3}}{2}$ as $\ell\to 0$.

Let us notice that the mean curvature of the torus $S_{r_{0}}$ with respect to $-\partial_{r}$ is $(\tanh r_{0}+\coth r_{0})/2$.

2.2. The geometric convergence

The space of cusp manifolds with volume less than $V_{0}$ is compact for geometric convergence. This convergence is defined as follows (see Sections E.1 and E.2 in [2]). Let $\Pi_{i}:\mathbb{H}^{3}\to M_{i}$ and $\overline{\Pi}:\mathbb{H}^{3}\to\overline{M}$ be the universal covers and $o$ a point in $\mathbb{H}^{3}$. We say that the pointed manifolds $(M_{i},\Pi_{i}(o))$ converge for the geometric convergence topology to $(\overline{M},\overline{\Pi}(o))$ if, for any $r$, there are $f_{i}:B(o,r)\subset\mathbb{H}^{3}\to\mathbb{H}^{3}$ which are equivariant ($\overline{\Pi}(z)=\overline{\Pi}(z^{\prime})\Leftrightarrow\Pi_{i}(f_{i}(z))=\Pi_{i}(f_{i}(z^{\prime}))$) such that $(f_{i})_{i}$ converges to the identity in the $C^{\infty}$ topology (here $B(o,r)$ denotes the geodesic ball in $\mathbb{H}^{3}$). Actually defining $\varphi_{i}$ by $\varphi_{i}(\overline{\Pi}(z))=\Pi_{i}(f_{i}(z))$, we will often use the following consequence (see Lemma E.2.2 in [2]).

Lemma 2**.**

Let $(M_{i})_{i}$ be a sequence of finite volume hyperbolic manifolds converging to $\overline{M}$ in the geometric topology. Let $\varepsilon>0$ be fixed, after eliminating some initial terms, there exists:

• •

$(\sigma_{i})_{i}$* with $\sigma_{i}>0$ and $\sigma_{i}\to 0$,*

• •

$(k_{i})_{i}$* with $k_{i}>1$ and $k_{i}\to 1$,*

• •

for all $i$ a $k_{i}$-quasi-isometric embedding $\varphi_{i}$ from a neighborhood of $\overline{M}_{[\varepsilon,\infty)}$ into $M_{i}$.

with the following properties

• •

$\varphi_{i}(\overline{M}_{[\varepsilon,\infty)})$* is contained in the interior of ${M_{i}}_{[\varepsilon-\sigma_{i},\infty)}$ and*

• •

$\varphi_{i}(\partial\overline{M}_{[\varepsilon,\infty)})$* does not meet an open neighborhood of ${M_{i}}_{[\varepsilon+\sigma_{i},\infty)}$.*

Here $k_{i}$-quasi isometry must be understood as smooth maps $\varphi_{i}$ such that

[TABLE]

When we will use these properties, we will not forget that $\varphi_{i}$ come from maps $f_{i}$ that are $C^{\infty}$ close to id.

Actually, $\varepsilon$ is always chosen small enough such that the $\varepsilon$-thin part of $\overline{M}$ contains only cusp ends. Moreover if $\varepsilon$ is small enough each connected component of ${M_{i}}_{[\varepsilon-\sigma_{i},\varepsilon+\sigma_{i}]}$ contains exactly one component of $\varphi_{i}(\partial\overline{M}_{[\varepsilon,\infty)})$ (see Theorem  E.2.4 in [2]). The description of this component of $\varphi_{i}(\partial\overline{M}_{[\varepsilon,\infty)})$ is given by the following result.

Lemma 3**.**

Let $(M_{i})_{i}$, $\overline{M}$, $\varepsilon>0$ and $\varphi_{i}$ as above. Let $C$ be a connected component of $\partial\overline{M}_{[\varepsilon,\infty)}$. Then for $i$ large, $\varphi_{i}(C)$ is a graph of a function $u_{i}$ over the corresponding component $C_{i}$ of $\partial{M_{i}}_{[\varepsilon,\infty)}$. Moreover $u_{i}\to 0$ and $C$ and $C_{i}$ are $\kappa_{i}$-quasi-isometric with $\kappa_{i}\to 1$.

Proof.

$C$ is a surface with principal curvatures $1$. Thus $\varphi_{i}(C)$ has principal curvatures close to $1$ and between $1/2$ and $3/2$.

Since $\varphi_{i}(C)\subset A_{i}$ where $A_{i}$ is a component of ${M_{i}}_{[\varepsilon-\sigma_{i},\varepsilon+\sigma_{i}]}$, $\varphi_{i}(C)$ is contained either in a cusp end of $M_{i}$ or a neighborhood of a short geodesic of $\gamma_{i}$. In the second case, there is a smallest $\delta_{i}\leq\sigma_{i}$ such that $\varphi_{i}(C)\subset{M_{i}}_{[\varepsilon-\sigma_{i},\varepsilon+\delta_{i}]}$ so $\varphi_{i}(C)$ is tangent to a boundary torus of $\partial{M_{i}}_{[\varepsilon_{i}+\delta_{i},\infty)}$. The comparison of the mean curvature at this tangency point gives the mean curvature of $\partial{M_{i}}_{[\varepsilon_{i}+\delta_{i},\infty)}$ is close to $1$. Thus the distance from $\gamma_{i}$ to $\varphi_{i}(C)$ is very large and goes to $+\infty$.

In both cases, $A_{i}$ is described as $T_{i}\times[-\alpha_{i},\beta_{i}]$ with a metric $d\sigma_{i,t}^{2}+dt^{2}$ with $\alpha_{i},\beta_{i}\to 0$ and $C_{i}=T_{i}\times\{0\}$ is the boundary of ${M_{i}}_{[\varepsilon,\infty)}$ in $A_{i}$.

Let $\gamma$ be a geodesic in $\varphi_{i}(C)$. Since $\varphi_{i}(C)$ has curvature uniformly bounded, there is $k_{0}$ such that $|\partial_{s}(\gamma^{\prime}(s),\partial_{t})|\leq k_{0}$ so $|(\gamma^{\prime}(s),\partial_{t})|\geq|(\gamma^{\prime}(0),\partial_{t})|/2$ for $0<s<s_{0}=|(\gamma^{\prime}(0),\partial_{t})|/(2k_{0})$. Looking at the $t$ coordinate along $\gamma$, we then have $\beta_{i}+\alpha_{i}\geq|t(\gamma(s_{0}))-t(\gamma(0))|\geq\frac{|(\gamma^{\prime}(0),\partial_{t})|^{2}}{4k_{0}}$. Since $\alpha_{i}+\beta_{i}\to 0$, this implies that the angle between $\varphi_{i}(C)$ and $\partial_{t}$ goes to $\pi/2$ uniformly. Since $\varphi_{i}(C)$ is embedded this implies that $\varphi_{i}(C)$ is a graph over $C_{i}$: there is a function $u_{i}:C_{i}\to\mathbb{R}$ such that $\varphi_{i}(C)=\{(p,t)\in T\times\mathbb{R}|t=u_{i}(p)\}$. Since the angle between $\varphi_{i}(C)$ and $\partial_{t}$ goes to $\pi/2$, the gradient of $u_{i}$ goes to [math]. Besides $\varphi_{i}(C)\subset A_{i}$ so $|u_{i}|$ is close to [math].

This implies that $(p,u_{i}(p))\in\varphi_{i}(C)\mapsto(p,0)\in C_{i}$ is a $\kappa_{i}$-quasi-isometry ($\kappa_{i}\to 1$) which can be composed with $\varphi_{i}:C\to\varphi_{i}(C)$ to obtain a $\kappa_{i}k_{i}$-quasi-isometry. ∎

As consequence, we have the following result.

Corollary 4**.**

Let $V_{0}$ be positive then there are $\ell_{0}$, $\delta_{0}$, $\mathfrak{s}_{0}$ such the following is true. Let $M$ be a cusp manifold with volume less than $V_{0}$ and $\gamma$ be a geodesic loop of length $\ell\leq\ell_{0}$. Then $S_{R_{\ell}}=\partial N_{R_{\ell}}$ has diameter less than $\delta_{0}$ and systole larger than $\mathfrak{s}_{0}$.

The set of flat tori with diameter less than $\delta_{0}$ and systole larger than $\mathfrak{s}_{0}$ is a compact subset of the set of flat tori.

Proof.

If it not true there is a sequence of cusp manifolds $M_{i}$ that converge to $\overline{M}$ and in $M_{i}$ there is a geodesic loop $\gamma_{i}$ of length $\ell_{i}\to 0$ such that either the diameter of $S_{R_{\ell_{i}}}$ goes to $\infty$ or its systole goes to [math].

After taking a subsequence, we can assume that the tubular ends around $\gamma_{i}$ converges to one cusp end in $\overline{M}$. Let $\varepsilon>0$ be small and consider $C$ the component of $\partial\overline{M}_{[\varepsilon,\infty)}$ inside this cusp end. Let $C_{i}$ be the component of $\partial{M_{i}}_{[\varepsilon,\infty)}$ inside the tubular end around $\gamma_{i}$. By the above lemma, $C$ and $C_{i}$ are $2$ quasi-isometric. So the area $C_{i}$ is close to that of $C$. Since the area of $S_{R_{\ell_{i}}}$ in $M_{i}$ is close to $\sqrt{3}/2$ this implies that the distance between $C_{i}$ and $S_{R_{\ell_{i}}}$ is uniformly bounded. Since the diameter and the systole of $S_{R_{\ell_{i}}}$ differ from those of $C_{i}$ by at most a uniform factor. This contradicts that either the diameter goes to $\infty$ or the systole goes to [math]. ∎

Remark 1*.*

Let us consider a particular one sided neighborhood of $\varphi_{i}(C)$ in $M_{i}$. Actually, let $\overline{A}$ be the part of the $2$-tubular neighborhood of $C$ inside $\overline{M}_{[\varepsilon,\infty)}$. Thus $\varphi_{i}(\overline{A})$ is a one sided neighborhood of $\varphi_{i}(C)$.

$\overline{A}$ can be parametrized by $T\times[-2,0]$ with the metric $g=e^{-2t}d\bar{\sigma}^{2}+dt^{2}$. Let $X:T\times[-2,0]\to\overline{M}$ be this parametrization and $(x_{1},x_{2})$ be orthonormal coordinates such that $g=e^{-2x_{3}}(dx_{1}^{2}+dx_{2}^{2})+dx_{3}^{2}$. Let us now estimate the metric $\varphi_{i}^{*}g_{i}$. We notice that $X$ lifts to an equivariant map $\widetilde{X}:\mathbb{R}^{2}\times[-2,0]\to\mathbb{H}^{3}$ i.e. $X=\overline{\Pi}\widetilde{X}$. If $\tilde{g}_{i}=\tilde{g}_{i,kl}dx_{k}dy_{l}$ we have

[TABLE]

since $\Pi_{i}$ is a local isometry. Since $f_{i}$ converges to the identity map in the $C^{\infty}$ topology this implies that $\tilde{g}_{i}\to g$ in the $C^{\infty}$ topology.

Remark 2*.*

The topology of a complete finite volume hyperbolic $3$- manifold determines its hyperbolic structure. Thus if a converging sequence $M_{i}\to\overline{M}$ is not constant, there is a subsequence whose topologies are distinct from that of $\overline{M}$. then there are short geodesics $\gamma_{i}$ in $M_{i}$ whose lengths converge to zero and whose maximal embedded tubular neighborhoods are converging to cusp ends of $\overline{M}$ (see Figure 2).

3. Transversallity in tubular ends

The aim of this section is to understand the behaviour of a minimal surface in a tubular end when we know a priori an upper bound on its curvature. A similar study was made for cusp ends in [5].

In this section, we use the tubular coordinates $(z,\theta,r)$.

3.1. An intersection property

We recall that, if $c$ is a geodesic in $\mathbb{H}^{3}$, $V_{r}$ denotes its tubular neighborhood of radius $r$. Moreover, for $r>0$, we denote $B_{r}=\partial V_{r}$.

Lemma 5**.**

Let $k_{0}$ and $\varepsilon_{0}$ be positive, then there are $r_{0}$ and $\eta_{0}$ such that the following is true. Let $c$ be a geodesic in $\mathbb{H}^{3}$. Let $r\in[0,r_{0}]$ and $p_{i}=(z_{i},\theta_{i},r)$ ($i=1,2$) be two points in $V_{r_{0}}$ such that $\theta_{2}\in[\theta_{1}+\frac{\pi}{3},\theta_{1}+\frac{2\pi}{3}]$ and $z_{2}\in[z_{1}-\eta_{0},z_{1}+\eta_{0}]$. Let $\Sigma_{i}$, $i=1,2$ be surfaces in $V_{r_{0}}$ whose curvatures are bounded by $k_{0}$, $p_{i}\in\Sigma_{i}$ and $d_{\Sigma_{i}}(p_{i},\partial\Sigma_{i})>\varepsilon_{0}$. If both $\Sigma_{i}$ are tangent to $B_{r}$ at $p_{i}$ (if $r=0$ we assume moreover that a unit normal vector to $\Sigma_{i}$ at $p_{i}$ is $\partial_{r}(z_{i},\theta_{i},0)$) then $\Sigma_{1}$ and $\Sigma_{2}$ has non empty transversal intersection.

Proof.

We look for $r_{0}\leq 2$. In $V_{2}$ the hyperbolic metric is $\cosh^{2}rdz^{2}+\sinh^{2}rd\theta^{2}+dr^{2}$. Let us change the metric in $V_{2}$ to the Euclidean metric $g_{e}=dz^{2}+r^{2}d\theta^{2}+dr^{2}$. So there are constants $\tilde{k}_{0}$ and $\tilde{\varepsilon}_{0}$ depending only on $k_{0}$ and $\varepsilon_{0}$ such that, with $g_{e}$, $\Sigma_{1}$ and $\Sigma_{2}$ have curvature bounded by $\tilde{k}_{0}$ and $d_{\Sigma_{i}}(p_{i},\partial\Sigma_{i})>\tilde{\varepsilon}_{0}$.

Thus there is $\eta_{1}>0$ such that $\Sigma_{i}$ can be described as a graph over the Euclidean disk of radius $\eta_{1}$ tangent to $\Sigma_{i}$ at $p_{i}$ (see Proposition 2.3 in [13]). Moreover if $\eta_{1}$ is chosen small enough, the gradient of the function parametrizing $\Sigma_{i}$ is less than $1/10$.

Let $r_{0}=\eta_{0}=\eta_{1}/10$. With these choices, the tangent disks of radius $\eta_{1}$ tangent to $\Sigma_{i}$ at $p_{i}$ must intersect at an angle between $\pi/3$ and $2\pi/3$ (see the schematic figure 3). Moreover since each $\Sigma_{i}$ is at a distance less than $\eta_{1}/10$ from its tangent disk, $\Sigma_{1}$ and $\Sigma_{2}$ must intersect and, as the gradient is less than $1/10$ and the angle between the disks is in $[\pi/3,2\pi/3]$, the intersection is transverse. ∎

3.2. The transversality result

The main result of the section is then the following. We recall that $S_{r}=\partial N_{r}$.

Proposition 6**.**

Let $\delta_{0}$, $k_{0}$ and $\varepsilon_{0}$ be positive, then there is $\ell_{0}>0$ and $\overline{R}$ such that the following is true. Let $\ell\leq\ell_{0}$ and $N_{R_{\ell}}$ be the hyperbolic tubular neighborhood of a geodesic loop $\gamma$ of length $\ell$ and such that the diameter of $S_{R_{\ell}}$ is less than $\delta_{0}$. Let $\Sigma$ be an embedded minimal surface in $N_{R_{\ell}}$ whose curvature is bounded by $k_{0}$. Let $\bar{r}<R_{\ell}-\overline{R}$ and $p$ be a point in $\Sigma\cap S_{\bar{r}}$ such that $d_{\Sigma}(p,\partial\Sigma)>\varepsilon_{0}$. Then $\Sigma$ is not tangent to $S_{\bar{r}}$ at $p$.

We notice that for $\bar{r}=0$, $S_{\bar{r}}$ is just the central geodesic $\gamma$ so the proposition states that $\Sigma$ can not be tangent to $\gamma$.

Proof.

We start with some $\ell_{0}$ such that $R_{\ell}>10$. Let $r_{0}\leq 1$ and $\eta_{0}$ be given by Lemma 5 for $k_{0}$ and $\varepsilon_{0}$ (we assume $\varepsilon_{0}\leq 1$). We first prove that the result is true if $\bar{r}\leq r_{0}$.

Let $\Sigma$ be a minimal surface as in the statement of the proposition and assume that $\Sigma$ is tangent at $p$ to $S_{r}$ for some $r$. We consider the lift $\widetilde{\Sigma}$ of $\Sigma$ to $\mathbb{H}^{3}$. $\widetilde{\Sigma}$ is then contained in a solid cylinder $V_{R_{\ell}}$.

The surface $\widetilde{\Sigma}$ is then an embedded minimal surface (may be non connected) which is invariant by the action of the loxodromic transformation $\tau:(z,\theta,r)\mapsto(z+\ell,\theta+\alpha,r)$. Let $p_{1}$ be a lift of $p$. We can assume that $p_{1}=(0,0,\bar{r})$; if $\bar{r}=0$, we assume that $\partial_{r}(0,0,0)$ is the unit normal vector to $\widetilde{\Sigma}$.

$S_{R_{\ell}}$ has a diameter less than $\delta_{0}$. So, for any $q$ in $B_{R_{\ell}}$, the intrinsic disk of radius $\delta_{0}$ in $B_{R_{\ell}}$ and center $q$ must contain an image of $(0,0,R_{\ell})$ by some $\tau^{n}$.

Let us consider the domain $A_{r}=\{(z,\theta,r)\in B_{r}|z\in[-\frac{\delta_{0}}{\cosh R_{\ell}},\frac{\delta_{0}}{\cosh R_{\ell}}],\theta\in[\frac{\pi}{2}-\frac{\delta_{0}}{\sinh R_{\ell}},\frac{\pi}{2}+\frac{\delta_{0}}{\sinh R_{\ell}}]\}$, $A_{R_{\ell}}$ is a square in $B_{R_{\ell}}$ whose edges have length $2\delta_{0}$. So $A_{R_{\ell}}$ contains an image of $(0,R_{\ell},0)$ by some $\tau^{n}$. This implies that $\tau^{n}$ is the composition of a vertical translation by some $z^{\prime}\in[-\frac{\delta_{0}}{\cosh R_{\ell}},\frac{\delta_{0}}{\cosh R_{\ell}}]$ and a rotation by some $\theta^{\prime}\in[\frac{\pi}{2}-\frac{\delta_{0}}{\sinh R_{\ell}},\frac{\pi}{2}+\frac{\delta_{0}}{\sinh R_{\ell}}]$.

The point $p_{2}=\tau^{n}(p_{1})=(z_{2},\theta_{2},\bar{r})$ is another lift of $p$ in $A_{\bar{r}}$. $\widetilde{\Sigma}$ is then also tangent to $B_{\bar{r}}$ at $p_{2}$. We have $|\theta_{2}-\pi/2|\leq\delta_{0}/\sinh R_{\ell}$ and $|z_{2}|\leq\delta_{0}/\cosh R_{\ell}$. So we can choose $\ell_{0}$ such that, for $\ell\leq\ell_{0}$, $\delta_{0}/\sinh R_{\ell}\leq\pi/6$ and $\delta_{0}/\cosh R_{\ell}\leq\eta_{0}$. Then we can apply Lemma 5 to the geodesic disks $\Sigma_{i}$ of radius $\varepsilon_{0}$ in $\widetilde{\Sigma}$ around $p_{i}$. Lemma 5 applies since, when $\bar{r}=0$, the unit normal vector to $\Sigma_{2}$ at $p_{2}$ is $\partial_{r}(z_{2},\theta_{2},0)$ with $|\theta_{2}-\pi/2|\leq\delta_{0}/\sinh R_{\ell}$ ($\Sigma_{2}$ is the image of $\Sigma_{1}$ by $\tau^{n}$). This gives that $\widetilde{\Sigma}$ has transverse self-intersection which is impossible. So the result is proved for $\bar{r}\leq r_{0}$.

Let us now prove that we can extend this result to the region $r_{0}\leq\bar{r}\leq R_{L}-\overline{R}$ for some $\overline{R}>0$.

If the result is not true, for any $n>0$, we can find a neighborhood $N_{R_{\ell_{n}}}$ of a closed geodesic $\gamma_{n}$ of length $\ell_{n}\leq\frac{1}{n}$ and a minimal surface $\Sigma_{n}$ in $N_{R_{\ell_{n}}}$ which is tangent to $S_{r_{n}}$ at $p_{n}$ for some $r_{n}\leq R_{\ell_{n}}-\frac{1}{4}\ln n$ (notice that $R_{\ell_{n}}-\frac{1}{4}\ln n>0$). Actually because of the first part we can assume $r_{n}>r_{0}$. In the following we denote $R_{\ell_{n}}$ by $R_{n}$.

Let $\eta_{1}=\min(r_{0}/10,\eta_{0})$ and replace the sequence $\Sigma_{n}$ by the sequence of $\eta_{1}$-geodesic disks in $\Sigma_{n}$ centered at $p_{n}$. So we can be sure that $\Sigma_{n}$ never touches the central geodesic $\gamma_{n}$ and stays outside of $N_{r_{0}-\eta_{1}}$.

We lift $\Sigma_{n}$ to $M_{R_{n}}$ endowed with the metric (1). This gives us a minimal surface $\widetilde{\Sigma}_{n}$ which is doubly periodic and may be non connected. $\widetilde{\Sigma}_{n}$ is doubly periodic by translation in the $(z,\theta)$ parameters by two vectors $v_{1}^{n},v_{2}^{n}$. Since $T_{R_{n}}$ has diameter less than $\delta_{0}$ we can choose $v_{1}^{n},v_{2}^{n}$ of Euclidean length less than $\frac{\delta_{0}}{\sinh R_{n}}$.

The point $p_{n}$ lifts to some point $\tilde{p}_{n}$ whose coordinates can be assumed to be $(0,0,r_{n})$ where $r_{n}\in(r_{0},R_{n}-\frac{1}{2}\ln n)$. We can assume that either $r_{n}$ converges to some $\bar{r}$ or to $\infty$. In the first case the ambient space around $(0,0,\bar{r})$ is $M_{\infty}=\mathbb{R}^{2}\times(0,+\infty)$ with the metric (1). If $r_{n}\to\infty$, we make the following change of coordinates $a=e^{r_{n}}z$, $b=e^{r_{n}}\theta$ and $\rho=r-r_{n}$. So the ambient space is now $\mathbb{R}^{2}\times(r_{0}-\eta_{1}-r_{n},R_{n}-r_{n})$ with the metric

[TABLE]

As $n$ goes to $+\infty$, these metrics converge smoothly to $\frac{e^{2\rho}}{4}(da^{2}+db^{2})+d\rho^{2}$ on $\mathbb{R}^{3}$. In this model, the vectors $v_{1}^{n},v_{2}^{n}$ become $e^{r_{n}}v_{1}^{n}$ and $e^{r_{n}}v_{2}^{n}$ whose lengths are less that $\frac{\delta_{0}e^{r_{n}}}{\sinh R_{n}}=O(e^{r_{n}-R_{n}})=O(e^{-\frac{1}{2}\ln n})\to 0$.

Actually, the cases $r_{n}\to\bar{r}$ and $r_{n}\to+\infty$ are very similar. Let us look first at the case $r_{n}\to\bar{r}$. We notice that the metric satisfies the hypotheses of Lemma 26 (Appendix A.1) for some parameter $A$ and for $r\in[\bar{r}-\eta_{1},R_{n}]$: we have $x_{1}=z$, $x_{2}=\theta$, $x_{3}=r$ and $h=\sinh$. So there is a $C$ and a function $u_{n}$ defined on the Euclidean disk $\{(z,\theta)\in\mathbb{R}^{2}|z^{2}+\theta^{2}\leq 2C^{2}/\sinh^{2}r_{n}\}$ such that $(z,\theta)\mapsto(z,u_{n}(z,\theta),\theta)$ is a parametrization of a neighborhood of $\tilde{p}_{n}$ in $\widetilde{\Sigma}_{n}$. Moreover we have $u_{n}(0,0)=r_{n}$, $\nabla u_{n}(0,0)=0$ and the estimates

[TABLE]

Here $\nabla$ denote the Euclidean gradient operator.

So the sequence $u_{n}$ is uniformly controlled in the $C^{2}$ topology and moreover $u_{n}$ solves the minimal surface equation (2). Thus, after considering a subsequence, $u_{n}$ converges to some $u$ defined on $D_{\bar{r}}=\{(t,\theta)\in\mathbb{R}^{2}|t^{2}+\theta^{2}\leq\frac{2C^{2}}{\sinh^{2}\bar{r}}\}$ which solves the minimal surface equation.

If $r_{n}\to+\infty$, we apply the change of variables $a=e^{r_{n}}t$, $b=e^{r_{n}}\theta$ and $\rho=r-r_{n}$. So we get a new function $w_{n}(a,b)=u_{n}(e^{-r_{n}}a,e^{-r_{n}}b)-r_{n}$ defined on $\{(a,b)\in\mathbb{R}^{2}|a^{2}+b^{2}\leq\frac{2C^{2}e^{2r_{n}}}{\sinh^{2}r_{n}}\}$. As above $w_{n}$ satisfies the estimates

[TABLE]

and solves a minimal surface equation (2). So we can assume it converges to some function $w$ defined on $\Delta=\{(a,b)\in\mathbb{R}^{2}|a^{2}+b^{2}\leq 4C^{2}\}$.

Let us denote the surface $\{r=R\}$ by $P_{R}$. The surface $\widetilde{\Sigma}_{n}$ is doubly periodic so it is tangent to $P_{r_{n}}$ at any point of the form $(0,0,r_{n})+kv_{1}^{n}+lv_{2}^{n}$ for $(k,l)\in\mathbb{Z}^{2}$. Moreover, around these points, it is parametrized locally on $D_{r_{n}}+kv_{1}^{n}+lv_{2}^{n}$ by $(z,\theta)\mapsto(z,u_{n,k,l}(t,\theta),\theta)$ where $u_{n,k,l}(z,\theta)=u_{n}((z,\theta)-kv_{1}^{n}-lv_{2}^{n})$. The surface $\widetilde{\Sigma}_{n}$ is embedded, this implies that $u_{n}\leq u_{n,k,l}$ or $u_{n}\geq u_{n,k,l}$ on $D_{n}\cap(D_{n}+kv_{1}^{n}+lv_{2}^{n})$ if it is non empty (notice that we can have $u_{n}\equiv u_{n,k,l}$ on the intersection).

If $r_{n}\to\bar{r}$, let $v_{0}$ be a vector in $D_{\bar{r}}$. Since $v_{i}^{n}\to 0$, there are sequences $(k_{n})_{n}$ and $(l_{n})_{n}$ such that $k_{n}v_{1}^{n}+l_{n}v_{2}^{n}\to v_{0}$. As $n\to\infty$, the sequence of functions $u_{n,k_{n},l_{n}}$ then converges to $u_{v_{0}}$ on $D_{\bar{r}}+v_{0}$ where $u_{v_{0}}(\cdot)=u(\cdot-v_{0})$. Because of $u_{n}\leq u_{n,k,l}$ or $u_{n}\geq u_{n,k,l}$, we get $u\leq u_{v_{0}}$ or $u\geq u_{v_{0}}$ on $D_{\bar{r}}\cap(D_{\bar{r}}+v_{0})$.

If $r_{n}\to\infty$, we can do the same with the change of coordinates since $e^{r_{n}}v_{i}^{n}\to 0$. So for any $v_{0}\in\Delta$, we have $w\leq w_{v_{0}}$ or $w\geq w_{v_{0}}$ on $\Delta\cap(\Delta+v_{0})$ where $w_{v_{0}}(\cdot)=w(\cdot+v_{0})$.

We now consider the case $r_{n}\to\bar{r}$ (the second one is similar). Let $G$ be the totally geodesic surface in $M_{\infty}$ tangent to $P_{\bar{r}}$ at $(0,\bar{r},0)$. As $\widetilde{\Sigma}$, $G$ can be described as the graph of a function $h$ over $D_{\bar{r}}$. We have $h(0)=\bar{r}$ and there is some $\alpha>0$ such that, over $D_{\bar{r}}$, $h(z,\theta)\geq\bar{r}+\alpha(z^{2}+\theta^{2})$. This second property comes from the fact that the principal curvatures of $P_{\bar{r}}$ with respect to $\partial_{r}$ are $-\tanh\bar{r}<0$ and $-\coth\bar{r}<0$. The functions $u$ and $h$ are two solutions of the minimal surface equation (2) with the same value and the same gradient at the origin. So by Bers theorem, the function $u-h$ looks like a harmonic polynomial of degree at least $2$.

If the degree of the polynomial is $2$, on can find $v_{0}\in D_{\bar{r}}\setminus\{(0,0)\}$ such that $(u-h)(v_{0})>0$ and $(u-h)(-v_{0})>0$. Then we have

[TABLE]

So this contradicts $u\leq u_{v_{0}}$ or $u\geq u_{v_{0}}$ on the whole $D_{\bar{r}}\cap(D_{\bar{r}}+v_{0})$

If the degree is at least $3$, the growth at the origin of $h$ implies that $u\geq\bar{r}$ on a smaller disk $D^{\prime}\subset D_{\bar{r}}$ and $u>\bar{r}$ on $D^{\prime}\setminus\{(0,0)\}$. So if $v_{0}\in D^{\prime}\setminus\{(0,0)\}$ we have

[TABLE]

Once again, this contradicts $u\leq u_{v_{0}}$ or $u\geq u_{v_{0}}$ on the whole $D_{\bar{r}}\cap(D_{\bar{r}}+v_{0})$

If $r_{n}\to\infty$, the same argument can be done with a totally geodesic surface tangent to the horosphere. ∎

3.3. A first area estimate

The preceding result allows us to estimate the area of a minimal surface with bounded curvature in a tubular end.

Corollary 7**.**

Let $\delta_{0}$ and $k_{0}$ be positive, then there is $\ell_{0}$ and $\overline{R}$ such that the following is true. Let $\ell\leq\ell_{0}$ and $N_{R_{\ell}}$ be the hyperbolic tubular neighborhood of a geodesic loop $\gamma$ of length $\ell$ and such that the diameter of $S_{R_{\ell}}$ is less than $\delta_{0}$. Let $0<R\leq R_{\ell}-\overline{R}$ and $\Sigma$ be a compact embedded minimal surface in $N_{R+1}$ whose curvature is bounded by $k_{0}$ and $\partial\Sigma\subset S_{R+1}$. Then one of the following possibilities occurs

1. (1)

$\Sigma\cap N_{R}=\emptyset$** 2. (2)

$\Sigma\cap N_{R}$* is a finite union of minimal disks. Each of these disks has boundary curve homotopic to a parallel of $S_{R}=\partial N_{R}$ and $|\Sigma\cap N_{R}|\geq 2\pi(\cosh R-1)$.*

A parallel of $S_{R}$ is a curve $\{z=const.\}$ in the tubular coordinates.

Proof.

Let $\ell_{0}$ and $\overline{R}$ be given by Proposition 6 for $\delta_{0}$, $k_{0}$ and $\varepsilon_{0}=1$. Let $\Sigma$ be as in the statement of the corollary and assume $\Sigma\cap N_{R}\neq\emptyset$. By Proposition 6, $\Sigma$ is transverse to the foliation $(S_{r})_{r}$ of $N_{R}$. So any connected component of $\Sigma\cap N_{R}$ intersects the geodesic loop $\gamma$ transversely. This implies that in $N_{\varepsilon}$ for $\varepsilon$ small each connected component of $\Sigma\cap N_{\varepsilon}$ is a disk whose boundary is homotopic to a parallel. Thus this description extends by transversality to $\Sigma\cap N_{R}$. Let $\Pi$ be the geodesic projection from $N_{R}$ to a geodesic parallel disk $\Delta$ (i.e. the map $(z,\theta,r)\mapsto(z_{0},\theta,r)$ for some $z_{0}$). This map is a contraction mapping and it is surjective on any disk component of $\Sigma\cap N_{R}$ since the boundary of such a disk is homotopic to a parallel. As a consequence the area of such a disk component is at least that of $\Delta$, i.e. $2\pi(\cosh(R)-1)$. ∎

4. A maximum principle

One aim of this section is to study some aspect of the behavior of minimal surfaces in a tubular end. Actually we need to study this in a more general setting. So we consider the ambient space $C=T\times[a,b]$ endowed with some reference metric $\overline{g}=h^{2}(x_{3})d\bar{\sigma}^{2}+dx_{3}^{2}$ where $d\bar{\sigma}^{2}$ is a flat metric on the torus $T$. We consider orthonormal coordinates $(x_{1},x_{2})$ on $T$ associated to $d\bar{\sigma}^{2}$; so $\bar{g}=h^{2}(x_{3})(dx_{1}^{2}+dx_{2}^{2})+dx_{3}^{2}$.

On $C$, we also consider a second metric $g=a_{kl}(x_{1},x_{2},x_{3})dx_{k}dx_{l}$. For $s\in[a,b]$, we denote $C_{s}=T\times[s,b]$ and $T_{s}=T\times\{s\}$. We are going to make several hypotheses on the metrics $\bar{g}$ and $g$. In order to formulate them, we need the following notation: for $k_{1},k_{2},k_{3},k_{4},k_{5}\in\{1,2,3\}$ and $p\leq 5$, we define

[TABLE]

The hypotheses on $\bar{g}$ and $g$ are: there is $A\geq 1$ such that

• H1

$\frac{1}{A^{2}}\bar{g}\leq g\leq A^{2}\bar{g}$

• H2

$\frac{|h^{\prime}|}{h}\leq A$, $\frac{|h^{\prime\prime}|}{h}\leq A$ and $\frac{|h^{\prime\prime\prime}|}{h}\leq A$.

• H3

$|a_{kl}|\leq Ah^{n_{2}(k,l)}(x_{3})$, $|\partial_{i}a_{kl}|\leq Ah^{n_{3}(k,l,i)}(x_{3})$, $|\partial_{i}\partial_{j}a_{kl}|\leq Ah^{n_{4}(k,l,i,j)}(x_{3})$ and $|\partial_{i}\partial_{j}\partial_{m}a_{kl}|\leq Ah^{n_{5}(k,l,i,j,m)}(x_{3})$.

• H4

$h^{\prime}\leq 0$ and the mean curvature vector of $T_{s}$ with respect to $g$ points in the $\partial_{x_{3}}$ direction (this is also true for the metric $\bar{g}$ since $h^{\prime}\leq 0$)

One consequence of H1 and H2 is that the sectional curvatures of $\bar{g}$ are uniformly bounded. Actually by $H1$ and $H3$ the sectional curvatures of $g$ are also uniformly bounded. We also notice that these hypotheses does not depend on the choice of the orthonormal coordinates on $(T,d\bar{\sigma}^{2})$.

4.1. The maximum principle

We have the following maximum principle for embedded minimal surfaces in $C$ endowed with the metric $g$.

Proposition 8**.**

Let $i_{0}\in\mathbb{N}$, then there is $h_{0}$ such the following is true. Assume that $h(a)\leq h_{0}$ and let $\Sigma$ be an embedded minimal surface in $(C,g)$ whose non empty boundary is inside $T_{a}$ and its index is less than $i_{0}$. Then $\Sigma\cap C_{a+1/2}=\emptyset$.

We notice that $h_{0}$ will depend on $i_{0}$, $A$ and the metric $d\bar{\sigma}^{2}$. We also notice that this control on $h$ is actually a control on the size of the torus $T_{a}$.

Proof.

If the proposition is not true there is a sequence of function $h_{n}$ with $h_{n}(a)\to 0$ and a minimal surface in $S_{n}\in(C,g_{n})$ ($g_{n}=a_{n,kl}(x_{1},x_{2},x_{3})dx_{k}dx_{l}$) such that $\partial S_{n}\subset T_{a}$, its index is less than $i_{0}$ and $\Sigma\cap C_{a+1/2}\neq\emptyset$.

Let $s_{n}$ be the maximum of the $x_{3}$ coordinate on $S_{n}$, $x_{3}\geq a+1/2$. Let us define $\lambda_{n}=(h_{n}(s_{n}))^{-1}$. Then we change the coordinates by $y_{1}=x_{1}$, $y_{2}=x_{2}$ and $y_{3}=\lambda_{n}(x_{3}-s_{n})$ and blow up the metric by a factor $\lambda_{n}$. This gives us a minimal surface $\Sigma_{n}$ in $T\times[\lambda_{n}(a-s_{n}),0]$ that touches $T_{0}$ and with boundary in $T_{\lambda_{n}(a-s_{n})}$ (we notice that $\lambda_{n}(a-s_{n})\to-\infty$). The ambient metric is then $\tilde{g}_{n}=b_{n,kl}(y_{1},y_{2},y_{3})dy_{k}dy_{l}$ where

[TABLE]

The reference metric becomes

[TABLE]

Because of the hypothesis $H2$, considering a subsequence, this metric converges to the flat metric $d\bar{\sigma}^{2}+dy_{3}^{2}$ in $C^{2,\alpha}$ topology. Because of H3, considering a new subsequence, the metrics $\tilde{g}_{n}$ converges to a flat metric $\bar{h}=\bar{b}_{kl}dy_{k}dy_{l}$ in $C^{2,\alpha}$ topology. For example we have

[TABLE]

So

[TABLE]

Once this is known, the arguments in order to conclude use the fact that $\Sigma_{n}$ converges to a minimal lamination in $T^{2}\times\mathbb{R}_{-}$ endowed with the flat metric $\bar{h}$ : the precise argument can be found in the proof of Proposition 1 in [4]. ∎

Remark 3*.*

We notice that $h_{0}$ can be chosen uniformly if $d\bar{\sigma}^{2}$ lies in a compact subset of flat metrics on $T$.

4.2. Some applications

In this section, we will see some consequences of the above result.

The case of cusp ends $E=T\times\mathbb{R}_{+}$ endowed with $\bar{g}=e^{-2x_{3}}d\bar{\sigma}^{2}+dx_{3}^{2}$ is the simplest one. Indeed in this case the metric $g$ is the reference metric $\bar{g}$. Let $L$ be fixed and replace $x_{3}\mapsto e^{-x_{3}}$ by some non increasing function $h$ such that $h(x_{3})=e^{-x_{3}}$ on $[0,L]$ and $h(x_{3})=e^{-(L+1/2)}$ for $x_{3}>L+1$. Then all the above hypotheses $H_{2},H_{4}$ are satisfied (the constant $A$ can be chosen independently of $L$). So Proposition 8 yields: if $\partial E$ has small diameter, then no compact embedded minimal surface with boundary inside $\partial E$ and index less than $1$ can enter in $E_{1/2}$. As a consequence, in a cusp manifold $M$, there is $\varepsilon>0$ such that any compact embedded minimal surface in $M$ less than $1$ is contained in $M_{[\varepsilon,\infty)}$.

The second case of interest concerns the tubular ends.

Proposition 9**.**

Let $K$ be a compact set of flat tori $T$. Then there are $\ell_{0}$ and $\overline{R}$ such the following is true. Let $\ell\leq\ell_{0}$ and $N_{R_{\ell}}$ be a hyperbolic tubular neighborhood of a geodesic loop of length $\ell$ such that $S_{R_{\ell}}$ belongs in $K$. Let $0<R\leq R_{\ell}-\overline{R}$ and $\Sigma$ be a compact embedded minimal surface in $N_{R+1}$ with $\partial\Sigma\in S_{R+1}$ with index less than $1$. Then one of the following possibilities occurs

1. (1)

$\Sigma\cap N_{R}=\emptyset$** 2. (2)

$\Sigma\cap N_{1}\neq\emptyset$**

Moreover there is a universal constant $\kappa$ such that, in the second case and for any $3\leq R\leq R_{\ell}-\overline{R}$, $|\Sigma\cap N_{R}|\geq\kappa\mathfrak{s}_{0}e^{R-R_{\ell}}$ where $\mathfrak{s}_{0}\leq 1$ is a lower bound on the systole of $T_{R_{\ell}}$.

Proof.

We first prove that $\Sigma\cap N_{R}=\emptyset$ or $\Sigma\cap N_{1}\neq\emptyset$. We have seen in Section 2.1 that we can consider, on $N_{R_{\ell}}$, a coordinate system $C=T\times[0,R_{\ell})$ endowed with the metric $g=\sinh^{2}(R_{\ell}-x_{3})dx_{1}^{2}+\cosh^{2}(R_{\ell}-x_{3})dx_{2}^{2}+dx_{3}^{2}$ ($(x_{1},x_{2})$ are orthonormal coordinates on $T$).

In order to fit with the notations of the preceding section we should introduce the coordinates $y_{i}=\sinh(R_{\ell})x_{i}$ $i=1,2$ and $y_{3}=x_{3}$. The first two are orthonormal coordinates on $(T,d\bar{\sigma}^{2})=\sinh(R_{\ell})^{2}(dx_{1}^{2}+dx_{2}^{2})$. So we define $\bar{g}=h^{2}(x_{3})(dy_{1}^{2}+dy_{2}^{2})+dy_{3}^{2}$ with $h(x)=\sinh(R_{\ell}-x)/\sinh(R_{\ell})$ and in these coordinates the metric $g$ can be written

[TABLE]

There is $A>0$ (that does not depend on $\ell$) such that $g$ and $\bar{g}$ satisfy hypotheses H1, H2, H3 and H4 on $T\times[0,R_{\ell}-\frac{1}{2})$. Moreover we notice that, since $S_{R_{\ell}}$ belongs to $K$, $(T,d\bar{\sigma}^{2})$ belongs to a compact set of flat tori.

Let $h_{0}$ be given by Proposition 8 and let $\overline{R}$ be such that $\sinh(R_{\ell}-\overline{R})\leq h_{0}\sinh(R_{\ell})$. Consider $0<R\leq R_{\ell}-\overline{R}$ and let $\Sigma$ be an embedded minimal surface in $N_{R+1}\setminus N_{\frac{1}{2}}$ with $\partial\Sigma\in S_{R+1}$ and index less than $1$. If $\Sigma\cap N_{1}=\emptyset$, $\Sigma$ can be seen as a minimal surface in $(C,g)$ with boundary in $T_{s}$ where $s=R_{\ell}-(R+1)$. Since $h(s)\leq h_{0}$, Proposition 8 gives $\Sigma\cap C_{s+1/2}=\emptyset$. So in the tubular coordinates, we have $\Sigma\cap N_{R}=\emptyset$.

In the second case we now prove the area estimate. For this we use the tubular coordinates. We notice that $\Sigma$ must meet all the tori $S_{r}$ for $1\leq r\leq R+1$.

Since $g\leq\cosh^{2}r(dz^{2}+d\theta^{2})+dr^{2}$ and the systole of $T_{R_{L}}$ is at least $\mathfrak{s}_{0}$, the disk $\{(z+z_{0},\theta+\theta_{0},R_{\ell})|\,z^{2}+\theta^{2}\leq\frac{\mathfrak{s}_{0}^{2}}{4\cosh^{2}R_{\ell}}\}$ is embedded in $S_{R_{\ell}}$ for any $t_{0},\theta_{0}$. For any $\rho\in[3/2,R_{\ell}]$, let us define $a=\frac{\sinh\rho}{\cosh R_{\ell}}\frac{\mathfrak{s}_{0}}{4}\leq\frac{1}{4}$. The cylinder $Y_{\rho}=\{(z+z_{0},\theta+\theta_{0},r)|r\in[\rho-2a,\rho]\textrm{ and }z^{2}+\theta^{2}\leq\frac{\mathfrak{s}_{0}^{2}}{4\cosh^{2}R_{\ell}}\}$ is embedded in $N_{\rho}$. $Y_{\rho}$ contains the geodesic ball of center $(z_{0},\theta_{0},\rho-a)$ and radius $a$ which is then embedded in $N_{\rho}$. Indeed, in the cylinder, we have

[TABLE]

So the geodesic ball is contained in $\{(z+z_{0},\theta+\theta_{0},r+\rho-a)|\frac{1}{4}\sinh^{2}\rho(z^{2}+\theta^{2})+r^{2}\leq a^{2}\}$ which is a subset of $Y_{\rho}$.

Since $\Sigma$ meets any $S_{r}$ for $r\geq 1$, for any $\rho$ we can select $z_{0},\theta_{0}$ such that $(z_{0},\theta_{0},\rho-a)\subset\Sigma$. So by the monotonicity formula in $\mathbb{H}^{3}$, $|\Sigma\cap Y_{\rho}|\geq\pi a^{2}$. We are going to sum over all these contributions to estimate the area of $\Sigma$.

Let $c(s)=(z(s),\theta(s),R_{\ell})$ be a parametrization of a systole of $S_{R_{\ell}}$ and consider the surface $S$ in $N_{R_{\ell}}$ parametrized by $X:(s,r)\in\mathbb{S}^{1}\times[1,R_{\ell}]\mapsto(z(s),\theta(s),r)$. So, for $\rho_{1}<\rho_{2}$, we can estimate

[TABLE]

So in $N_{\rho}\setminus N_{\rho-2a}$

[TABLE]

for some universal constant $\kappa$ and $\kappa^{\prime}$.

So considering a disjoint union of $N_{\rho}\setminus N_{\rho-2a}$ in $N_{R}\setminus N_{1}$ that covers $N_{R}\setminus N_{3/2}$, we obtain

[TABLE]

for any $R\geq 3$ and some universal constant $\kappa^{\prime\prime\prime}$. ∎

5. The min-max theory

In this section we recall some definitions and results of the min-max theory for minimal surfaces. There are basically two settings for this theory: the discrete and the continuous one. We recall the main points that we will use in the sequel.

5.1. The discrete setting

The discrete setting for the min-max theory was developed by Almgren and Pitts (see [1, 12]).

Let $M$ be a compact orientable $3$-manifold with no boundary. The Almgren-Pitts min-max theory deals with discrete families of elements in $\mathcal{Z}_{2}(M)$ i.e. integral rectifiable $2$-currents in $M$ with no boundary.

If $I=[0,1]$, we define some cell complex structure on $I$ and $I^{2}$.

Definition 10**.**

Let $j$ be an integer. $I(1,j)$ is the cell complex on $I$ whose [math]-cells are points $[\frac{i}{3^{j}}]$ and $1$-cells are intervals $[\frac{i}{3^{j}},\frac{i+1}{3^{j}}]$.

The cell complex $I(2,j)$ on $I^{2}$ is $I(2,j)=I(1,j)\otimes I(1,j)$.

For these cell complexes we can associate some notations

• •

$I(m,j)_{0}$ denotes the set of [math]-cells of $I(m,j)$.

• •

$I_{0}(1,j)$ denotes the set of [math]-cells ${[0],[1]}$.

• •

The distance between two elements of $I(m,j)_{0}$ is

[TABLE]

• •

The projection map $n(i,j):I(m,i)_{0}\to I(m,j)_{0}$ is defined by $n(i,j)(x)$ is the unique element in $I(m,j)_{0}$ such that

[TABLE]

Let $\varphi:I(m,j)_{0}\to\mathcal{Z}_{2}(M)$ be a map. The fineness of $\varphi$ is defined by

[TABLE]

where $\mathbf{M}$ is the mass of a current.

We write $\varphi:I(1,j)_{0}\to(\mathcal{Z}_{2}(M),\{0\})$ to mean $\varphi(I(1,j)_{0})\subset\mathcal{Z}_{2}(M)$ and $\varphi(I_{0}(1,j))=\{0\}$.

Definition 11**.**

Let $\delta$ be positive and $\varphi_{i}:I(1,k_{i})_{0}\to(\mathcal{Z}_{2}(M),\{0\})$ for $i=1,2$. $\varphi_{1}$ and $\varphi_{2}$ are $1$-homotopic in $(\mathcal{Z}_{2}(M),\{0\})$ with fineness $\delta$ if there is $k_{3}\in\mathbb{N}$, $\max(k_{1},k_{2})\leq k_{3}$ and a map

[TABLE]

such that

• •

$f(\psi)\leq\delta$;

• •

$\psi([i-1],x)=\varphi_{i}(n(k_{3},k_{i})(x))$* for $x\in I(1,k_{3})_{0}$;*

• •

$\psi(I(1,k_{3})_{0}\times\{[0],[1]\})=0$.

The main objects in the discrete min-max theory are the $(1,\mathbf{M})$-homotopy sequences.

Definition 12**.**

A $(1,\mathbf{M})$-homotopy sequence of maps into $(\mathcal{Z}_{2}(M),\{0\})$ is a sequence of maps $\{\varphi_{i}\}_{i\in\mathbb{N}}$,

[TABLE]

such that $\varphi_{i}$ is $1$-homotopic to $\varphi_{i+1}$ in $(\mathcal{Z}_{2}(M),\{0\})$ with fineness $\delta_{i}$ and

• •

$\lim_{i\to\infty}\delta_{i}=0$;

• •

$\sup_{i}\{\mathbf{M}(\varphi_{i}(x)),x\in I(1,k_{i})_{0}\}<+\infty$.

Moreover we have a notion of discrete homotopy between $(1,\mathbf{M})$-homotopy sequences

Definition 13**.**

Let $S_{j}=\{\varphi_{i}^{j}\}_{i\in\mathbb{N}}$ ($j=1,2$) be two $(1,\mathbf{M})$-homotopy sequences of maps into $(\mathcal{Z}_{2}(M),\{0\})$. $S_{1}$ is homotopic to $S_{2}$ if there is a sequence $\{\delta_{i}\}_{i\in\mathbb{N}}$ such that

• •

$\lim\delta_{i}=0$;

• •

$\varphi_{i}^{1}$* is $1$-homotopic to $\varphi_{i}^{2}$ in $(\mathcal{Z}_{2}(M),0)$ with fineness $\delta_{i}$.*

This notion defines an equivalence relation between $(1,\mathbf{M})$-homotopy sequences. The set of equivalence classes is denoted by $\pi_{1}^{\#}(\mathcal{Z}_{2}(M),\mathbf{M},\{0\})$. The Almgren-Pitts theory says that $\pi_{1}^{\#}(\mathcal{Z}_{2}(M),\mathbf{M},\{0\})$ is naturally isomorphic to the homology group $H_{3}(M,\mathbb{Z})$ (see Theorem 4.6 in [12] and [1]). We denote by $\Pi_{M}$ the element of $\pi_{1}^{\#}(\mathcal{Z}_{2}(M),\mathbf{M},\{0\})$ that corresponds to the fundamental class in $H_{3}(M)$. If $S\in\Pi_{M}$ we say that $S$ is a discrete sweep-out of $M$.

For $S=\{\varphi_{i}\}_{i}$ a $(1,\mathbf{M})$-homotopy sequence we define

[TABLE]

If $\Pi\in\pi_{1}^{\#}(\mathcal{Z}_{2}(M),\mathbf{M},\{0\})$ is an equivalence class, we define the width associated to $\Pi$ by

[TABLE]

For $\Pi=\Pi_{M}$, the number $W_{M}=W(\Pi_{M})$ is call the width of the manifold $M$. The Almgren-Pitts theory says that this number is positive and is $\mathbf{L}(S)$ for some particular $S\in\Pi_{M}$. If $S=\{\varphi_{i}\}_{i}$ we say that ${\varphi_{i_{j}}(x_{j})}_{j}$ ($x_{j}\in I(1,k_{i_{j}})$) is a min-max sequence if $\mathbf{M}(\varphi_{i_{j}}(x_{j}))\to W_{M}$.

Theorem 14** (Pitts [12]).**

Let $M$ be a closed $3$-manifold, then there is $S=\{\varphi_{i}\}_{i}\in\Pi_{M}$ with $\mathbf{L}(S)=W_{M}$ and a min-max sequence $\{\varphi_{i_{j}}(x_{j})\}_{j}$ that converges (in the varifold sense) to an integral varifold whose support is a finite collection of embedded connected disjoint minimal surfaces $\{S_{i}\}_{i}$ of $M$. So there are positive numbers $\{n_{i}\}_{i}$ such that

[TABLE]

A consequence of this result is that there is always a minimal surface $S$ in $M$ such that $|S|\leq W_{M}$. Actually, Zhou [16] proved that, if $S_{i}$ is a non orientable minimal surface produced by the above theorem, then $n_{i}$ is even.

5.2. The continuous setting

The continuous setting was developed by Colding and De Lellis [3]. Here we present it with the modifications made by Song in [15].

Let $M$ be Riemannian $3$-manifold and consider $N\subset M$ a bounded open subset whose boundary $\partial N$, when non empty, is a rectifiable surface of finite $\mathcal{H}^{2}$-measure. Moreover when $\partial N\neq\emptyset$, we assume that each connected component $C$ of $\partial N$ separates $M$.

If $a<b\in\mathbb{R}$, we then have the following definitions.

Definition 15**.**

A family of $\mathcal{H}^{2}$-measurable closed subsets $\{\Gamma_{t}\}_{t\in[a,b]}$ in $N\cup\partial N$ with finite $\mathcal{H}^{2}$-measure is called a generalized smooth family if

• •

for each $t$ there is a finite set $P_{t}\in N$ such that $\Gamma_{t}\cap N$ is a smooth surface in $N\setminus P_{t}$ or the empty set;

• •

$\mathcal{H}^{2}(\Gamma_{t})$* depends continuously in $t$ and $t\mapsto\Gamma_{t}$ is continuous in the Hausdorff sense;*

• •

on any $U\subset\subset N\setminus P_{t_{0}}$, $\Gamma_{t}\xrightarrow[]{t\to t_{0}}\Gamma_{t_{0}}$ smoothly in $U$.

We notice the smoothness hypothesis is only made on $\Gamma_{t}\cap N$ so this allows $\partial N$ to be non smooth. We now define the notion of continuous sweep-out in this setting.

Definition 16**.**

If $\partial N=\emptyset$, a generalized smooth family $\{\Gamma_{t}\}_{t\in[a,b]}$ is called a continuous sweep-out of $N$ if there exists a family of open subsets $\{\Omega_{t}\}_{t\in[a,b]}$ of $N$ such that

• (sw1)

$(\Gamma_{t}\setminus\partial\Omega_{t})\subset P_{t}$* for any $t$;*

• (sw2)

$\mathcal{H}^{3}(\Omega_{t}\triangle\Omega_{s})\to 0$* as $t\to s$ (where $\triangle$ denotes the symmetric difference of subsets).*

• (sw3)

$\Omega_{a}=\emptyset$* and $\Omega_{b}=N$;*

If $\partial N\neq\emptyset$, for a open subset $\Omega\subset N$ we denote $\partial_{*}\Omega=\partial\Omega\cap N$. A continuous sweep-out of $N$ is then required to satisfy (sw1) and (sw2) above except that $\partial$ is replaced by $\partial_{*}$ and $t>a$ in (sw1). Moreover (sw3) is replaced by

• (sw3’)

$\Omega_{a}=\emptyset$, $\Omega_{b}=N$, $\Sigma_{a}=\partial N$ and $\Sigma_{t}\subset N$ for $t>a$.

For a continous sweep-out as above $\{\Gamma_{t}\}_{t\in[a,b]}$, we define the quantity $\mathbf{L}(\{\Gamma_{t}\})=\max_{t\in[a,b]}\mathcal{H}^{2}(\Gamma_{t})$. When $\partial N$ is a smooth surface, constructing a continuous sweep-out can be done in the following way. Let $f:N\to[0,1]$ be a Morse function such that $\{f^{-1}(0)\}=\partial N$. Then if $\Gamma_{t}=f^{-1}(t)$ for $t\in[0,1]$, $\{\Gamma_{t}\}_{t\in[0,1]}$ is a sweep-out of $N$.

Two continuous sweep-outs $\{\Gamma_{t}^{1}\}_{t\in[a,b]}$ and $\{\Gamma_{t}^{2}\}_{t\in[a,b]}$ are said to be homotopic if, informally, they can be continuously deformed one to the other (the precise definition is Definition 8 in [15]). Then a family $\Lambda$ of sweep-outs is called homotopically closed if it contains the homotopy class of each of its elements. For such a family $\Lambda$, we can define the width associated to $\Lambda$ as

[TABLE]

As in the discrete setting this number can be realized as the mass of some varifold supported by smooth disjoint minimal surfaces (see Theorem 12 in [15]).

5.3. From continous to discrete

In order to construct discrete sweep-outs of a closed orientable $3$-manifold $M$, we will use a result obtained by Zhou (see Theorem 5.1 in [17]). We denote by $\mathcal{C}(M)$ the space of subsets in $M$ with finite perimeter. Let $\mathcal{F}$ denote the flat metric on $\mathcal{Z}_{2}(M)$.

Theorem 17**.**

Let $\Phi:[0,1]\to(\mathcal{Z}_{2}(M),\mathcal{F})$ be a continuous map such that

• •

$\Phi(0)=0=\Phi(1)$;

• •

$\Phi(x)=\partial[[\Omega_{x}]]$, $\Omega_{x}\in\mathcal{C}(M)$ for all $x\in[0,1]$;

• •

$M_{0}=\emptyset$* and $M_{1}=M$*

• •

$\sup_{x}\mathbf{M}(\Phi(x))<+\infty$.

Then there exists a discrete sweep-out $S$ such that

[TABLE]

Here $[[\Omega]]$ denotes the element of $\mathcal{Z}_{3}(M)$ corresponding to $\Omega$.

Let us notice that if $\{\Gamma_{t}\}_{t\in[0,1]}$ is a continuous sweep-out of a compact orientable Riemannian $3$-manifold then $\Phi:t\mapsto[[\Gamma_{t}]]\in\mathcal{Z}_{2}(M)$ satisfies the hypotheses of the above theorem (as above $[[\Gamma_{t}]]$ denotes the element of $\mathcal{Z}_{2}(M)$ corresponding to $\Gamma_{t}$).

6. The quantity $\mathcal{A}_{1}(M)$

In this section we recall some results the authors obtained in [10] (see also [4]).

6.1. The quantity $\mathcal{A}_{1}(M)$ for compact $M$

If $M$ is a closed orientable Riemannian $3$-manifold, we denote by $\mathcal{O}$ the collection of all smooth orientable embedded closed minimal surfaces in $M$ and $\mathcal{U}$ the collection of all smooth non-orientable ones. We then define

[TABLE]

One of the results of [10] is the following theorem (Theorem B in [10])

Theorem 18**.**

Let $M$ be an oriented closed Riemannian $3$-manifold. Then $\mathcal{A}_{1}(M)$ is equal to one of the following possibilities.

1. (1)

$|\Sigma|$* where $\Sigma\in\mathcal{O}$ is a min-max surface of $M$ associated to the fundamental class of $H_{3}(M)$, $\Sigma$ has index $1$, is separating and $\mathcal{A}_{1}(M)=W_{M}$.* 2. (2)

$|\Sigma|$* where $\Sigma\in\mathcal{O}$ is stable.* 3. (3)

$2|\Sigma|$* where $\Sigma\in\mathcal{U}$ is stable and its orientable $2$-sheeted cover has index [math].*

Moreover, if $\Sigma\in\mathcal{O}$ satisfies $|\Sigma|=\mathcal{A}_{1}(M)$, then $\Sigma$ is of type $1$ or $2$ and if $\Sigma\in\mathcal{U}$ satisfies $2|\Sigma|=\mathcal{A}_{1}(M)$, then $\Sigma$ is of type $3$.

Actually in [10], the case (3) mentions the possibility for the orientable $2$-sheeted cover to have index [math] or $1$. In fact, the index $1$ case can be ruled out thanks to the work of Ketover, Marques and Neves [9].

If $\mathcal{S}$ denotes the collection of all smooth embedded stable minimal surfaces, we define $\mathcal{A}_{\mathcal{S}}(M)=\inf(\{|\Sigma|,\Sigma\in\mathcal{O}\cap\mathcal{S}\}\cup\{2|\Sigma|,\Sigma\in\mathcal{U}\cap\mathcal{S}\})$. Actually we proved in [10] that $\mathcal{A}_{1}(M)=\min(W_{M},\mathcal{A}_{S}(M))$. In order to simplify some notations, we will denote $a(\Sigma)=|\Sigma|$ if $\Sigma\in\mathcal{O}$ and $a(\Sigma)=2|\Sigma|$ is $\Sigma\in\mathcal{U}$.

When $M$ is not compact, one can still define $\mathcal{O}$ and $\mathcal{U}$ for $M$ by considering only compact embedded minimal surfaces in $M$. Of course these collections could be empty but if its not $\mathcal{A}_{1}(M)$ is well defined. If $M$ is a cusp manifold this can be done.

6.2. The filler

We want to study $\mathcal{A}_{1}(M)$ when $M$ is a cusp manifold. In order to do that the idea is to change $M$ into a compact manifold $D(M)$ that contains all the compact minimal surfaces of $M$. To do this the main tool are the fillers.

Definition 19**.**

Let $(T,d\sigma^{2})$ be a flat torus and $L>10$ be a real number. A filler $F$ associated to $T$ and $L$ is a solid torus endowed with a Riemannian metric $g$ with the following properties.

• $(i$)

Let $T_{t}$ be the set of points at distance $t$ from $\partial F$. For $t\in[0,L+1)$, $T_{t}$ is a smooth flat torus and $T_{L+1}$ is a closed geodesic.

• $(ii)$

The diameter of $T_{t}$ is a decreasing function and the mean curvature vectors points in the $\partial_{t}$ direction.

• $(iii)$

For $t\in[0,1]$, $T_{t}$ has the metric $e^{-2t}d\sigma^{2}$.

• $(iv)$

Any minimal surface $\Sigma$ that meets all the $T_{t}$ for $0\leq t\leq L-1$ has area at least $\kappa L$ where $\kappa$ is a constant depending on the systole of $(T,d\sigma^{2})$.

Proposition 20**.**

Let $(T,d\sigma^{2})$ be a flat torus and $L>10$. There exists a filler associated to $T$ and $L$. Moreover, let $\delta$ and $\mathfrak{s}$ be the diameter and the systole of $(T,d\sigma^{2})$ and $K\leq\mathfrak{s}/\delta$. Then there is $\delta_{0}>0$ that depends only on $K$ such that

• $(v)$

if $\delta$ is less than $\delta_{0}$, then any minimal surface $\Sigma$ with $\partial\Sigma\subset\partial F$ and index at most $1$ satisfies either $\Sigma\cap T_{1}=\emptyset$ or $\Sigma\cap T_{t}\neq\emptyset$ for any $t\leq L-1$.

Proof.

We construct $F$ as $T\times[L+1]$ with a Riemannian metric which is singular on $T_{L+1}=T\times\{L+1\}$ in order for $T_{L+1}$ to be a geodesic. We use the notation $F_{t}=T\times[t,L+1]$.

Let $f:[0,L+1]\to\mathbb{R}$ be a function satisfying the following properties

• •

$f(t)=t$ on $[0,1]$,

• •

$f^{\prime}>0$ on $[0,L+2/3)$ and $f^{\prime}=0$ on $[L+2/3,L+1]$.

• •

$f\leq 3$.

• •

$f^{\prime}$ and $f^{\prime\prime}$ are bounded independently of $L$.

On $F\setminus F_{L}$, we define the metric $g=e^{-2f(t)}d\sigma^{2}+dt^{2}$. Since $f(t)=t$ on $[0,1]$, $(iii)$ is satisfied.

In order to define the metric on $F_{L}$, we consider a well oriented orthonormal coordinate system on $(T,d\sigma^{2})$ such that $T$ is the quotient of $\mathbb{R}^{2}$ by the translations by $(\alpha,0)$ and $(\beta,\ell)$.

Let $\eta:[0,1]\to[0,1]$ be a non increasing function such that $\eta$ is decreasing on $[1/2,1]$, $\eta=1$ near [math], and $\eta(x)=(1-x)\frac{2\pi}{\alpha}e^{f(L+1)}$ near $1$. On $F_{L}$ we extend the definition of the metric $g$ by

[TABLE]

Since $\eta(1)=0$, the metric is singular at $t=L+1$. Let $D$ be the unit disk and consider the solid torus $\mathbb{T}$ constructed as the quotient of $D\times[0,\ell]$ by the relation $(p,0)\sim(R_{\beta}(p),\ell)$ where $R_{\beta}$ is the rotation of angle $\beta$. If $(\rho,\theta)$ are the polar coordinates on $D$ and $h:\mathbb{T}\to F_{L}$ is the map $(\rho,\theta,z)\mapsto(\frac{\alpha}{2\pi}\theta,v,L+1-\rho)$ the metric $h^{*}g$ is given by

[TABLE]

so, near $\rho=0$ (i.e. $t=L+1$), it is equal to $h^{*}g=\rho^{2}d\theta^{2}+e^{-2f(L+1)}dz^{2}+d\rho^{2}$ which is a smooth metric on $\mathbb{T}$ near the core circle $\{\rho=0\}$. So $F$ is a smooth solid torus with a smooth metric and $(i)$ is satisfied.

Because of the monotonicity of $f$ and $\eta$, $(ii)$ is satisfied. Moreover the curvature of $g$ is uniformly controlled on $F\setminus F_{L}$. If $\rho_{0}$ is the minimum of $1$ and half the systole of $(T,d\sigma^{2})$, then for any $p\in F_{1}\setminus F_{L-1}$ the geodesic ball of center $\rho$ and radius $e^{-3}\rho_{0}$ is embedded in $F\setminus F_{L}$.

Let $\Sigma$ be a minimal surface that meets all the $T_{t}$ for $t\in[0,L]$. Consider $t_{n}=1+2e^{-3}\rho_{0}n$ and, for any $n\in\{0,\dots,n_{0}\}$ where $t_{n_{0}}\leq L+2\leq t_{n_{0}+1}$, let $p_{n}$ be in $T_{t_{n}}$ with $p_{n}\in T_{t_{n}}\cap\Sigma$. Then by the monotonicity formula, the area of $\Sigma$ in the ball of radius $e^{-3}\rho_{0}$ and center $p_{n}$ is at least $ce^{-6}\rho_{0}^{2}$ for some universal constant $c$. Since these balls are disjoint, the area of $\Sigma$ in $F\setminus F_{L}$ is at least

[TABLE]

if $L\geq 10$ and where $\kappa$ only depends on the systole of $(T,d\sigma^{2})$.

For item $(v)$, we notice that $(T,\frac{1}{\delta^{2}}d\Sigma^{2})$ belongs to a compact subset of flat tori fixed by $K$. So Proposition 8 applies to $(F\setminus F_{L},\delta^{2}e^{-2f(t)}\frac{d\sigma^{2}}{\delta^{2}}+dt^{2})$ to prove that if $\Sigma$ has index at most $1$ and $\Sigma\subset T\times[0,,L-1]$ then $\Sigma\subset T\times[0,1]$. ∎

6.3. The quantity $\mathcal{A}_{1}(M)$ for cusp manifolds

In this section we recall the study of compact minimal surfaces inside orientable cusp manifolds we made in [5, 10].

Let $M$ be a cusp manifold. First we prove that $M$ contains a compact embedded minimal surface. Let $\varepsilon$ be such that the $\varepsilon$-thin part is only made of cusp ends. Since $\partial M_{[\varepsilon,\infty)}$ is smooth there is a homotopically closed family $\Lambda$ of sweep-outs associated to a Morse function on $M_{[\varepsilon,\infty)}$ (we recall that the tori components of $\partial M_{[\varepsilon,\infty)}$ are leaves of the sweep-outs). If $\varepsilon^{\prime}<\varepsilon$, $M_{[\varepsilon^{\prime},\varepsilon]}$ is foliated by tori that can be used to extend any continuous sweep-out in $\Lambda$ into a sweep-out of $M_{[\varepsilon^{\prime},\infty)}$ that belongs to a homotopically closed family $\Lambda^{\prime}$. Since $W(M_{[\varepsilon,\infty)},\partial M_{[\varepsilon,\infty)},\Lambda)\geq|\partial M_{[\varepsilon,\infty)}|$ we obtain

[TABLE]

So there is $W_{0}>0$ such that $W_{0}\geq W(M_{[\varepsilon,\infty)},\partial M_{[\varepsilon,\infty)},\Lambda)$ for any $\varepsilon$. Besides a continuous sweep-out of $M_{[\varepsilon,\infty)}$ must sweep out also a fixed geodesic ball in $M$. So there is $w_{0}$ such that $W(M_{[\varepsilon,\infty)},\partial M_{[\varepsilon,\infty)},\Lambda)\geq w_{0}$ for any $\varepsilon$.

Thus we can choose $\varepsilon$ small such that any flat tori $C$ in $\partial M_{[\varepsilon,\infty)}$ has small diameter and $w_{0}>|\partial M_{[\varepsilon,\infty)}|$. For each $C$, we consider a filler $F^{C}$ associated to the flat torus $C$ and $L$ that will be chosen later. $\varepsilon$ is chosen small enough such that item $(v)$ of Proposition 20 is satisfied. Since there are a finite number of $C$, item $(iv)$ of Definition 19 gives some constant $\kappa>0$ independent of $C$. Then $L$ is chosen such that $\kappa L\geq W_{0}+1$.

We can glue each filler $F^{C}$ along $C$ to obtain a compact manifold without boundary denoted $D(M)$ with some metric. The construction of $D(M)$ depends on two parameters $\varepsilon$ and $L$, so sometimes we will write $D_{\varepsilon,L}(\overline{M})$ (actually it also depends on the choice of some coordinates on $F$). We will use this construction in the following sections. $D(M)$ contains isometrically a $1$ tubular neighborhood of $M_{[\varepsilon,\infty)}$. Let $\{\Gamma_{t}\}_{t\in[0,1]}$ be a continuous sweep-out of $M_{[\varepsilon,\infty)}$ with $\mathbf{L}(\Gamma_{t})\leq W(M_{[\varepsilon,\infty)},\partial M_{[\varepsilon,\infty)},\Lambda)+1/2$. We can extend $\{\Gamma\}_{t\in[0,1]}$ to a continuous sweep-out $\{\widetilde{\Gamma}_{t}\}_{t\in[-L-1,1]}$ of $D(M)$ by considering $\widetilde{\Gamma}_{t}=\cup_{C}\partial F_{-t}^{C}$ for $t\in[-L-1,0]$. Since, for $t<0$, $|\widetilde{\Gamma}_{t}|\leq|\partial M_{[\varepsilon,\infty)}|$ we have $\mathbf{L}(\widetilde{\Gamma}_{t})\leq\mathbf{L}(\Gamma_{t})$. By Theorem 17, the width $W_{D(M)}$ is then less than $W_{0}+1/2$. Thus by Theorem 18, there is a minimal surface $\Sigma$ in $D(M)$ with index at most $1$ such that $a(\Sigma)\leq W_{0}+1/2$.

Because of our choice of $\varepsilon$ and items $(iv)$ and $(v)$, if $\Sigma$ enters in some $F_{1}^{C}$ then $a(\Sigma)\geq|\Sigma\cap F^{C}|\geq\kappa L\geq W_{0}+1$; this is impossible. So $\Sigma$ stays outside of $F_{1}^{C}$ for any $C$ so $\Sigma$ is embedded in the part isometric to the $1$-tubular neighborhood of $M_{[\varepsilon,\infty)}\subset M$: $\Sigma$ is a compact minimal surface in $M$.

Now we know that $M$ contains compact embedded minimal surfaces and we can define the number $\mathcal{A}_{1}(M)$. In order to prove that $\mathcal{A}_{1}(M)$ is realized as in Theorem 18, we have the following argument. Let $S$ be a compact minimal surface in $M$. We construct $D(M)$ as above with an extra hypothesis on $\varepsilon$ which is $M_{[\varepsilon,\infty)}$ contains $S$ and all compact embedded minimal surfaces in $M$ with index at most $1$. The above construction gives $\mathcal{A}_{1}(M)\leq W_{0}+1/2$.

Let $\Sigma$ is a minimal surface that realizes $\mathcal{A}_{1}(D(M))$ (it has index at most $1$), we have $a(\Sigma)\leq a(S)$. If $\Sigma$ enters in into $F^{C}_{1}$ for some $C$, we have

[TABLE]

So $\Sigma$ does not enter into such a filler: $\Sigma$ belongs to $M$. Thus $\mathcal{A}_{1}(D(M))=\mathcal{A}_{1}(M)$ and $\mathcal{A}_{1}(D(M))$ is realized by a minimal surface as in Theorem 18.

The remainder of this paper is devoted to the study of the continuity of the $\mathcal{A}_{1}$ functional over the collection of orientable cusp manifolds. We are going to study the lower and the upper semi-continuity of $\mathcal{A}_{1}$.

7. The upper semi-continuity study

In this section, we consider $(M_{i})_{i}$ a sequence of cusp manifolds that converges to $\overline{M}$ for the geometric convergence. The first step and the main step of the upper semi-continuity study is to prove that the sequence $(\mathcal{A}_{1}(M_{i}))_{i}$ is bounded. The following proposition answers this question.

Proposition 21**.**

Let $M_{i}\to\overline{M}$ be a converging sequence of cusp manifolds. Then for small $\varepsilon$ and large $L$, $\limsup\mathcal{A}_{1}(M_{i})\leq W_{D(\overline{M})}$.

Proof.

The idea of the proof consist in constructing a Riemannian manifold $(N_{i},\tilde{g}_{i})$ which is $\kappa_{i}$-quasi isometric to $D(\overline{M})$ with $\kappa_{i}\to 1$ and such that a large part $N_{i}^{1}$ of $N_{i}$ is isometric to a large part $M_{i}$. Moreover $N_{i}$ is such that any minimal surface with index at most $1$ that gets out of $N_{i}^{1}$ has area at least $W_{D(\overline{M})}+1/2$. As a consequence, a minimal surface $S_{i}$ in $N_{i}$ realizing $\mathcal{A}_{1}(N_{i})$ satisfies $a(S_{i})\leq\kappa_{i}^{2}W_{D(\overline{M})}<W_{D(\overline{M})}+1/2$ for large $i$ and is contained in $N_{i}^{1}$. Thus $\limsup\mathcal{A}_{1}(M_{i})\leq\limsup a(S_{i})\leq\limsup\kappa_{i}^{2}W_{D(\overline{M})}=W_{D(\overline{M})}$.

We choose $\varepsilon$ small such that the $\varepsilon$-thin part of $M$ is made only of cusp ends. The convergence $M_{i}\to\overline{M}$ gives us $\varphi_{i}:\overline{M}_{[\varepsilon,\infty)}\to M_{i}$ as in Subsection 2.2. From Section 6.3, we know that there is $W_{0}>0$ independent of $\varepsilon$ and $L$ such that $W_{D_{\varepsilon,L}(\overline{M})}\leq W_{0}+1/2$.

Let $C$ be one boundary component of $\overline{M}_{[\varepsilon,\infty)}$ and $\overline{A}$ the part of the $2$-tubular neighborhood of $C$ inside $\overline{M}_{[\varepsilon,\infty)}$ (the rest of the proof is written as there is only one $C$ in $\partial\overline{M}_{[\varepsilon,\infty)}$, actually we need to repeat the argument for each $C$). $\overline{A}$ can be parametrized by $T\times[-2,0]$ with the metric $\overline{g}=e^{-2x_{3}}(dx_{1}^{2}+dx_{2}^{2})+dx_{3}^{2}$ where $(x_{1},x_{2})\in T$ are orthonormal coordinates on $C$.

By Subsection 2.2, $\varphi_{i}(\overline{A})$ is a one sided neighborhood of $\varphi_{i}(C)$ in $\varphi_{i}(\overline{M})\subset M_{i}$. On $\varphi_{i}(\overline{A})$ we have the coordinates $T\times[-2,0]$ with the metric $g_{i}=a_{i,kl}dx_{k}dx_{l}$ that $C^{\infty}$ converges to $\bar{g}$.

Let $N_{i}^{1}$ be $\varphi_{i}(\overline{M}_{[\varepsilon,\infty)}^{1})$ with the metric $g_{i}$ where $\overline{M}_{[\varepsilon,\infty)}^{1}$ is the set of points in $M_{[\varepsilon,\infty)}$ at distance at least $1$ from $\partial\overline{M}_{[\varepsilon,\infty)}$. We notice that $N_{i}^{1}$ contains the part of $\varphi_{i}(\overline{A})$ parametrized by $T\times[-2,-1]$. We are going to modify the metric $g_{i}$ on $T\times[-1,0]$ in order to define a new metric $\tilde{g}_{i}$ on $\varphi_{i}(\overline{M}_{[\varepsilon,\infty)})$ which will be the Riemannian manifold $N_{i}^{2}$.

Let $\chi:[-1,0]\to\mathbb{R}$ be $C^{\infty}$ such that $0\leq\chi\leq 1$, $\chi=1$ near $-1$ and $\chi=0$ near [math]. We then define $\tilde{g}_{i}=\chi(x_{3})g_{i}+(1-\chi(x_{3}))\bar{g}$. Since $g_{i}$ and $\bar{g}$ are $C^{\infty}$ close. $\tilde{g}_{i}$ is also $C^{\infty}$ close to $\bar{g}$. As explained above, $\tilde{g}_{i}$ turns $\varphi_{i}(\overline{M}_{[\varepsilon,\infty)})$ into a new Riemannian manifold $(N_{i}^{2},\tilde{g}_{i})$. The map $\varphi_{i}:\overline{M}_{[\varepsilon,\infty)}\to N_{i}^{2}$ is still well defined and since the metrics $\tilde{g}_{i}$ converge in the $C^{\infty}$ topology to $\bar{g}$. $\varphi_{i}$ is a $\kappa_{i}^{\prime}$ quasi-isometry where $\kappa_{i}^{\prime}\to 1$. Moreover $\varphi_{i}$ is an isometry close to $\partial\overline{M}_{[\varepsilon,\infty)}$.

Let $L$ be large and consider a filler $F$ associated to $T$ and $L$. Since $N_{i}^{2}$ and $\overline{M}_{[\varepsilon,\infty)}$ are isometric close to their boundary we can glue to all of them the filler $F$ to produce $(D_{\varepsilon,L}(\overline{M}),\tilde{g})$ and $(N_{i},\tilde{g}_{i})$ and extend the definition of $\varphi_{i}$ to a map $D(\overline{M})\to N_{i}$ which is the identity on the filler. As a consequence $\varphi_{i}:D(\overline{M})\to N_{i}$ is a $\kappa_{i}^{\prime}$ quasi-isometry.

Let us estimate the area of a minimal surface $S\subset N_{i}$ with index at most $1$ that is not contained in $N_{i}^{1}$. Thus $S$ must enter in some part of $N_{i}$ which is isometric to $T\times[-2,L]$ endowed with the metric $\tilde{g}_{i}=\tilde{a}_{i,kl}dx_{k}dx_{l}$ which is $C^{\infty}$ close to $\tilde{g}=\bar{g}$ on $T\times[-2,0]$ and is equal to $\tilde{g}=e^{-2f(x_{3})}(dx_{1}^{2}+dx_{2}^{2})+dx_{3}^{2}$ on $T\times[0,L]$ ($f$ is introduced in Section 6.2). Because of our choice of $f$ function, the metrics $\tilde{g}_{i}$ and $\tilde{g}$ satisfy the hypotheses H1, H2, H3 and H4 of Section 4 for a uniform constant $A$.

If $S$ does not meet all the tori $T_{s}$ for $s\in[-2,L]$ then, by Proposition 8, $S$ must stay outside of $T\times[-3/2,L+1]$, so $S\subset N_{i}^{1}$. Since we assume $S_{i}\not\subset N_{i}^{1}$, it must meet all the tori $T_{s}$ for $s\in[0,L]$. Then by Proposition 20, $|S|\geq\kappa L$ for some $\kappa$ that only depends on the injectivity radius of $T_{0}$. Now, we choose $L$ large enough such that $\kappa L>W_{0}+1$. We obtain $|S|\geq\kappa L>W_{0}+1\geq W_{D(\overline{M})}+1/2$. This finishes the construction of $N_{i}$ and then $\limsup\mathcal{A}_{1}(M_{i})\leq W_{D(\overline{M})}$. ∎

We know that for $\varepsilon$ small and $L$ large we have $\mathcal{A}_{1}(\overline{M})=\mathcal{A}_{1}(D(\overline{M}))=\min(\mathcal{A}_{\mathcal{S}}(\overline{M}),W_{D(\overline{M})})$. So the above result gives us a first upper semi-continuity property.

Proposition 22**.**

Let $M_{i}\to\overline{M}$ be a converging sequence of cusp manifolds. If one of the following hypotheses is satisfied then $\limsup\mathcal{A}_{1}(M_{i})\leq\mathcal{A}_{1}(\overline{M})$.

• •

$\mathcal{A}_{1}(\overline{M})=W_{D(\overline{M})}$**

• •

$\mathcal{A}_{1}(\overline{M})$* is realized by a stable non separating minimal surface $\Sigma$*

• •

$\mathcal{A}_{1}(\overline{M})$* is realized by a stable non degenerate minimal surface $\Sigma$*

Proof.

The first case comes directly from the above proposition.

Let $\Sigma$ be a non separating minimal surface that realizes $\mathcal{A}_{1}(\overline{M})$. Let $\varepsilon$ be small such that $\Sigma$ is contained in the $\varepsilon$-thick part of $\overline{M}$. Let $\varphi_{i}:\overline{M}_{[\varepsilon,\infty)}\to M_{i}$ be the $\kappa_{i}$ quasi-isometry associated to the convergence $M_{i}\to\overline{M}$. Then $\varphi_{i}(\Sigma)$ is a surface in $M_{i}$ with $a(\varphi_{i}(\Sigma))\leq\kappa_{i}^{2}a(\Sigma)$. Because of the topology of the $\varepsilon$-thin part (cusps or solid tori), $\varphi_{i}(\Sigma)$ is non separating in $M_{i}$. So taking a small $\varepsilon_{i}$ and a large $L_{i}$ we can see $\varphi_{i}(\Sigma)$ as a non separating surface in $D_{\varepsilon_{i},L_{i}}(M_{i})$. So minimizing the area in the non vanishing homology class of $\Sigma$ there is a minimal surface $S_{i}$ in $D(M_{i})$ with $a(S_{i})\leq a(\varphi_{i}(\Sigma))\leq\kappa_{i}^{2}a(\Sigma)=\kappa_{i}^{2}\mathcal{A}_{1}(\overline{M})$. Thus

[TABLE]

and this gives the result.

Concerning the last case, as above, let $\varepsilon$ be small such that $\Sigma$ is contained in the $\varepsilon$-thick part of $\overline{M}$ and $\varphi_{i}:\overline{M}_{[\varepsilon,\infty)}\to M_{i}$. Let $h_{i}=\varphi_{i}^{*}g_{i}$. Since $M_{i}\to\overline{M}$, the metrics $h_{i}$ converge in the $C^{\infty}$ topology to $\overline{g}$. Since $\Sigma$ is a non degenerate surface, for large $i$, $\Sigma$ can be deformed to a minimal surface $S_{i}$ in $(\overline{M}_{[\varepsilon,\infty)},h_{i})$. So $\varphi_{i}(S_{i})$ is a minimal surface in $M_{i}$ and

[TABLE]

∎

Remark 4*.*

We notice that the hypothesis $\mathcal{A}_{1}(\overline{M})=W_{D(\overline{M})}$ is satisfied if $\mathcal{A}_{1}(\overline{M})$ is realized by an index $1$ minimal surface.

The second case is realized if $\mathcal{A}_{1}(\overline{M})$ is realized by a non orientable minimal surface.

8. The lower semi-continuity study

In this section we are going to prove that the $\mathcal{A}_{1}$ functional is lower semi-continuous.

8.1. An exclusion property

Let $S$ be a two-sided embedded surface. Let $\nu$ be a choice of a unit normal vectorfield along $S$ and $f:S\to\mathbb{R}$ be a smooth function. Then we can define

[TABLE]

If $f$ is sufficiently small, $\exp_{S,f}(S)$ is an embedded surface which inherits from $S$ a natural unit normal vector still denoted by $\nu$. The lemma below is inspired by Lemma 16 in [15]

Lemma 23**.**

Let $S$ be a two-sided embedded surface and $U$ be a subset of $S$ such that the mean curvature of $S$ vanishes on $U$. If $S\setminus U$ has non empty interior, there is a positive function $f$ and $\tau>0$ such that $\exp_{S,sf}(S)$ has positive mean curvature on $\exp_{S,sf}(U)$ with respect to the naturally induced unit normal vector for $0<s<\tau$.

Proof.

Let $S$ and $U$ be as in the statement. Let $q$ be a function on $S$ such that $q=Ric(\nu,\nu)+\|A\|^{2}$ on $U$ and $\mathcal{L}=-\Delta-q$ has negative first eigenvalue on $S$. It is enough to assume $q$ is large enough somewhere in $\Sigma\setminus U$ to ensure that the first eigenvalue $\lambda_{1}$ is negative. Let $f>0$ be the first eigenfunction of $\mathcal{L}$. Consider $S_{t}=\exp_{S,tf}(S)$ and $H_{t}(p)$ be the mean curvature of $S_{t}$ at $\exp_{S,tf}(p)$. It is known that $2\partial_{t}{H_{t}}_{|t=0}=\Delta f+(Ric(\nu,\nu)+\|A\|^{2})f=-\lambda_{1}f+(Ric(\nu,\nu)+\|A\|^{2}-q)f>0$ on $U$. Thus, there is $\tau>0$ such that $H_{t}(p)>0$ for any $t\in(0,\tau]$ and $p\in U$. ∎

Using the above lemma we can prove the following result.

Proposition 24**.**

Let $A_{0}$, $\delta_{0}$ and $\mathfrak{s}_{0}\leq 1$ be positive. Then there is $\ell_{0}$ and $\overline{R}$ such the following is true. Let $\ell\leq\ell_{0}$ and $M$ be a cusp manifold such that $\mathcal{A}_{1}(M)\leq A_{0}$ and $M$ contains a tubular end $N_{R_{\ell}}$ of a geodesic loop of length $\ell$ and such that $S_{R_{\ell}}$ has diameter less than $\delta_{0}$ and systole larger than $\mathfrak{s}_{0}$. Let $\Sigma$ be an embedded minimal surface that realizes $\mathcal{A}_{1}(M)$ then $\Sigma\cap N_{R_{\ell}-\overline{R}}$ is empty.

Proof.

We first assume that $\Sigma$ is stable. This implies that there is $k_{0}$ such that $\Sigma$ has curvature bounded by $k_{0}$. So by Corollary 7, there is $\ell_{0}$ and $\overline{R}$ such that, if $\ell\leq\ell_{0}$, either $\Sigma\cap N_{R_{\ell}-\overline{R}}$ is empty or $\Sigma\cap N_{R_{\ell}-\overline{R}}$ has area at least $2\pi(\cosh(R_{\ell}-\overline{R})-1)$. Actually, if $\ell_{0}$ is chosen such that $2\pi(\cosh(R_{\ell_{0}}-\overline{R})-1)\geq A_{0}$, the second case can not occur and $\Sigma\cap N_{R_{\ell}-\overline{R}}$ is empty.

So we can assume that $\Sigma$ is separating and has index $1$. By Proposition 9, there is $\overline{R}$, $\ell_{0}$ and $\kappa$ such that $\Sigma\cap N_{R_{\ell}-\overline{R}}=\emptyset$ or $|\Sigma\cap N_{R}|\geq\kappa\mathfrak{s}_{0}e^{R-R_{\ell}}$ for any $R\in[3,R_{\ell}-\overline{R}]$. Moreover $\ell_{0}$ and $\overline{R}$ can be chosen such that the preceding paragraph is still true.

Let us assume that $\Sigma\cap N_{R_{\ell}-\overline{R}}$ is not empty. Let us notice that $|S_{R}|=\pi\ell\sinh(2R)\leq\sqrt{3}\frac{\sinh(2R)}{\sinh(2R_{\ell})}\leq\sqrt{3}e^{2(R-R_{\ell})}$. So choosing $R_{*}\geq\overline{R}$ such that $e^{-R_{*}}\leq\frac{\kappa\mathfrak{s}_{0}}{4\sqrt{3}}$ we obtain

[TABLE]

for any $3\leq R\leq R_{\ell}-R_{*}$.

Let $\varepsilon$ be small such that the $\varepsilon$-thin part of $M$ contains only cusp ends. For $L$ large we consider $D_{\varepsilon,L}(M)=D(M)$ such that $\mathcal{A}_{1}(M)=\mathcal{A}_{1}(D(M))$. So $\Sigma$ is a separating index $1$ minimal surface in $D(M)$ that realizes $\mathcal{A}_{1}(D(M))$. The idea is now to construct a discrete sweep-out $S$ of $D(M)$ such that $\mathbf{L}(S)<|\Sigma|$ which contradicts $\Sigma$ realizes $\mathcal{A}_{1}(M)$. We notice that $N_{R_{\ell}}$ is still isometrically included in $D(M)$.

$\Sigma$ separates $M$ into two connected components $\Omega_{1}$ and $\Omega_{2}$. Let $R\in[R_{\ell}-R_{*}-1,R_{\ell}-R_{*}]$ such that $S_{R}$ is transverse to $\Sigma$. We define $\Gamma=\Sigma\cap S_{R}$. The subset $\Omega_{i}\cap N_{R}$ has mean convex boundary made of pieces of $S_{R}$ and $\Sigma$. We can find a least area minimal surface $\Sigma_{i}\subset\Omega_{i}\cap N_{R}$ with $\partial\Sigma_{i}=\Gamma$ and homologous to $S_{R}\cap\Omega_{i}$. Since $S_{R}\cap\Omega_{i}$ is a surface with boundary $\Gamma$, $|\Sigma_{i}|\leq|S_{R}\cap\Omega_{i}|\leq|S_{R}|$. Besides $\Sigma_{i}\cup(S_{R}\cap\Omega_{i})$ bounds a subset $D_{i}$ of $N_{R}\cap\Omega_{i}$. By boundary regularity of solutions to the Plateau problem [6], $\Sigma_{i}$ is a smooth surface up to its boundary $\Gamma$ and, by the maximum principle, $\Sigma_{i}$ is transverse to $\Sigma$ along $\Gamma$. We notice that since $\Sigma_{i}$ is smooth up to its boundary we can slightly extend $\Sigma_{i}$ across $\Gamma$ to $\Sigma_{i}^{\prime}$. $\Sigma_{i}^{\prime}$ is not assumed to be minimal outside of $\Sigma_{i}$ and $\partial\Sigma_{i}^{\prime}$ is assumed to be outside $\Omega_{i}$.

Let us fix $i\in\{1,2\}$ and consider $\nu$ the unit normal along $\Sigma$ pointing into $\Omega_{i}$. Since $\Sigma$ has index $1$, there is $\tau>0$ and $f_{i}>0$ on $\Sigma$ such that $\exp_{\Sigma,sf_{i}}(\Sigma)$ is an embedded surface with positive mean curvature for any $s\in(0,\tau]$. Moreover, we assume $f_{i}>1$. If $\nu_{i}$ denote the unit normal along $\Sigma_{i}^{\prime}$ pointing into $D_{i}$ along $\Sigma_{i}$, by Lemma 23, there is $g_{i}>0$ on $\Sigma_{i}^{\prime}$ such that $\exp_{\Sigma_{i}^{\prime},sg_{i}}(\Sigma_{i}^{\prime})$ is an embedded surface with positive mean curvature on $\exp_{\Sigma_{i}^{\prime},sg_{i}}(\Sigma_{i})$ for any $s\in(0,\tau]$. Moreover, we assume $g_{i}<1$ and $\exp_{\Sigma_{i}^{\prime},sg_{i}}(\partial\Sigma_{i}^{\prime})\not\subset\Omega_{i}$ for any $s\in[0,\tau]$.

Let us define $U_{i,0}=D_{i}\cup(\Omega_{i}\setminus N_{R})$. For $s\leq\tau$ we define

[TABLE]

We postpone the precise description of $\partial U_{i,s}$ to the end of the proof. Actually we are going to prove that there is a smaller $\tau$ such that, for $s\in[0,\tau]$, $\partial U_{i,s}$ is $\exp_{\Sigma,sf_{i}}(A_{i,s})\cup\exp_{\Sigma_{i}^{\prime},sg_{i}}(B_{i,s})$ where $A_{i,s}$ is a smooth subdomain in $\Sigma$ and $B_{i,s}$ is a smooth subdomain of $\Sigma_{i}^{\prime}$. Moreover both components of $\partial U_{i,s}$ are transverse. We also have $B_{i,s}\subset\Sigma_{i}$ and $A_{i,s}\subset\Sigma\setminus N_{R-1}$. $s\mapsto\partial U_{i,s}$ is then a continuous map with values in $\mathcal{Z}_{2}(M)$. Moreover $\mathbf{M}(\partial U_{i,s})\leq\mathbf{M}(\partial U_{i,0})$. We then have

[TABLE]

Besides we notice that, since $A_{i,s}\subset\Sigma\setminus N_{R-1}$ and $B_{i,s}\subset\Sigma_{i}^{\prime}$, $\partial U_{i,s}$ is piecewise smooth mean convex in the sense of Definition 10 in [15].

Using the work of Song in [15], we can adapt the work of the authors in [10] to the case $\partial U_{i,\tau}$ is not smooth and prove the following statement.

Claim 1**.**

There is a continuous sweep-out $\{\partial U_{i,s}\}_{s\in[\tau,1]}$ of $U_{i,\tau}$ such that

[TABLE]

Proof.

Because of the Appendix in [15], we know that there exists a homotopically closed family $\Lambda$ of sweepouts in $U_{i,\tau}$. Let us assume that $W(U_{i,\tau},\partial U_{i,\tau},\Lambda)\geq|\Sigma|-5/2|S_{R}|>|\partial U_{i,\tau}|$.

Thus by Theorem 12 in [15], there is a closed minimal surface $S$ in $U_{i,\tau}$. As in the proof of Lemma 20 in [15], $N=U_{i,\tau}\setminus S$ is then a mean convex subset such that $\partial U_{i,\tau}$ has non vanishing homology class in $N$. Thus we can minimize the area in this homology class in order to get $S^{\prime}$ a stable minimal surface such that $|S^{\prime}|\leq|\partial U_{i,\tau}|$. But this implies $\mathcal{A}_{\mathcal{S}}(D(M))\leq|\partial U_{i,\tau}|\leq|\Sigma|-3|T_{R}|<\mathcal{A}_{1}(D(M))\leq\mathcal{A}_{\mathcal{S}}(D(M))$ which is contradictory. So $W(U_{i,\tau},\partial U_{i,\tau},\Lambda)\leq|\Sigma|-5/2|S_{R}|$ and there is $\{\partial U_{i,s}\}_{s\in[\tau,1]}$ with

[TABLE]

∎

Using these two sweep-outs, we can define a family $G_{s}$ (see Figure 4) of open subsets of $M$ by

[TABLE]

The family $G_{s}$ satisfies $\mathcal{H}^{3}(G_{s}\triangle G_{t})\to 0$ as $s\to t$. Moreover, $\partial G_{s}$ is rectifiable so $\Phi:s\mapsto\partial G_{s}$ is a continuous path in $\mathcal{Z}_{2}(M)$ for the flat topology. Moreover $G_{0}=\emptyset$ and $G_{2R+2}=M$. Let us now study $\mathbf{M}(\Phi(s))$. For $s\in[0,1]$ we have $\mathbf{M}(\Phi(s))=\mathbf{M}(\partial U_{1,1-s})\leq|\Sigma|-2|S_{R}|$. For $s\in[1,R+1]$, $\partial G_{s}$ is contained in $\partial U_{1,0}\cup S_{s-1}$ so $\mathbf{M}(\Phi(s))\leq\mathbf{M}(\partial U_{1,0})+|S_{s-1}|\leq|\Sigma|-2|S_{R}|$. If $s\in[R+1,2R+1]$, $\partial U_{s}$ is contained in $\partial U_{2,0}\cup S_{2R+1-s}$ so $\mathbf{M}(\Phi(s))\leq\mathbf{M}(\partial U_{2,0})+|S_{2R+1-s}|\leq|\Sigma|-2|S_{R}|$. Finally for $s\in[2R+1,2R+2]$, $\mathbf{M}(\Phi(s))=\mathbf{M}(\partial U_{2,s-2R-1})\leq|\Sigma|-2|S_{R}|$.

After a reparametrization, we have then constructed a continuous map $\Phi:[0,1]\to\mathcal{Z}_{2}((D(M)),\mathcal{F})$ satisfying all the hypotheses of Theorem 17 with $\sup\mathbf{M}(\Phi)\leq|\Sigma|-2|T_{R}|<|\Sigma|$. So by Theorem 17, we have $\mathcal{A}_{1}(M)=\mathcal{A}_{1}(D(M))\leq W_{D(M)}\leq\sup\mathbf{M}(\Phi)\leq|\Sigma|-2|T_{R}|<|\Sigma|=\mathcal{A}_{1}(M)$. This gives a contradiction with $\Sigma\cap N_{R_{L}-\overline{R}}\neq\emptyset$ and finishes the proof.

Let us come back to the study of $\partial U_{i,s}$ for small $s$ and check the properties we announced. Clearly this boundary is contained in $\Sigma_{i,s}^{\prime}=\exp_{\Sigma_{i}^{\prime},sg_{i}}(\Sigma_{i}^{\prime})$ and $\Sigma_{s}=\exp_{\Sigma,sf_{i}}(\Sigma)$. We need to understand the intersection of these two surfaces when $s$ is small.

We define $F_{i}:(p,s)\in\Sigma\times(-\tau,\tau)\mapsto\exp_{\Sigma,sf_{i}}(p)\in M$ and $G_{i}:(p,s)\in\Sigma_{i}^{\prime}\times(-\tau,\tau)\mapsto\exp_{\Sigma_{i}^{\prime},sg_{i}}(p)\in M$ for small $\tau$. The map $F_{i}$ defines a smooth coordinate system in a neighborhood $N$ of $\Sigma$. Let us write $F_{i}^{-1}=(P,T):N\to\Sigma\times(-\tau,\tau)$. Let us remark that at a point $p\in\Sigma$, ${DF_{i}^{-1}}_{|p}:X\in T_{p}M\mapsto(\pi_{p}(X),(X,\nu)/f_{i}(p))\in T_{p}\Sigma\times\mathbb{R}$ where $\pi_{p}$ is the normal projection $T_{p}M\to T_{p}\Sigma$.

Let $V$ be a neighborhood of $\Gamma$ inside $\Sigma_{i}^{\prime}$ contained in $N$. There is $\tau^{\prime}$ such that $G_{i}(V\times(-\tau^{\prime},\tau^{\prime}))\subset N$. Let $\eta_{i}$ be the conormal to $\Gamma$ in $\Sigma_{i}^{\prime}$ pointing to $\Sigma_{i}$. So neighboring points to $\Gamma$ in $\Sigma_{i}^{\prime}$ can be parametrized by $(p,t)\in\Gamma\times(-\varepsilon,\varepsilon)\mapsto\exp_{p}^{i}(t\eta_{i}(p))$ where $\exp^{i}$ is the exponential map in $\Sigma_{i}^{\prime}$. Thus such a point has image by $G_{i}(\cdot,s)$ in the intersection $F_{i}(\Sigma,s)\cap G_{i}(\Sigma_{i}^{\prime},s)$ ($s$ small) if

[TABLE]

Solving $t$ as a function of $(p,s)\in\Gamma\times\mathbb{R}$ can be done near $\Gamma\times\{0\}$ using the implicit function theorem since $L_{i}(p,0,0)=0$. Indeed we have $\partial_{t}L_{i}(p,0,0)=(\nu(p),\eta_{i}(p))/f_{i}(p)>0$ since both $\nu$ and $\eta_{i}$ point to $\Omega_{i}$. So $t_{i}(p,s)$ can be defined near $\Gamma\times\{0\}$. At $(p,0)$ we also have

[TABLE]

Thus $\partial_{s}t_{i}=\frac{f_{i}}{(\nu,\eta_{i})}(1-\frac{g_{i}(\nu,\nu_{i})}{f_{i}})>0$ since $g_{i}/f_{i}<1$. The curve $\gamma_{i,s}(p)=\exp_{p}^{i}(t_{i}(p,s)\eta(p))$ is sent by $G_{i}(\cdot,s)$ on the intersection $F_{i}(\Sigma,s)\cap G_{i}(\Sigma_{i}^{\prime},s)$: $\beta_{i}(\cdot,s)=G_{i}(\gamma_{i,s}(\cdot),s)$ is a parametrization of the intersection. Moreover $\gamma_{i,s}$ bounds a subdomain $B_{i,s}$ in $\Sigma_{i}^{\prime}$ whose image by $G_{i}(\cdot,s)$ is the piece of $\partial U_{i,s}$ contained in $G_{i}(\Sigma_{i}^{\prime},s)$. Since $\partial_{s}t_{i}>0$, we have $B_{i,s}\subset\Sigma_{i}$. At $(p,0)$, we also have

[TABLE]

The curve $\gamma_{s}(\cdot)=P(\beta_{i}(\cdot,s))$ on $\Sigma$ is such that $F_{i}(\gamma_{s}(\cdot),s)$ is also a parametrization of the intersection. $\gamma_{s}$ bounds a subdomain $A_{i,s}$ in $\Sigma$ such that $\partial U_{i,s}$ is $F_{i}(A_{i,s},s)\cup G_{i}(B_{i,s},s)$. We notice that, at $s=0$,

[TABLE]

We notice that since $B_{i,s}\subset\Sigma_{i}$, the mean curvature of $G_{i}(B_{i,s},s)$ is positive ($s>0$). The same is true for the mean curvature of $F_{i}(A_{i,s},s)$. Moreover both surfaces intersect at an angle less than $\pi$. Finally using the first variation formula and $\Sigma$ and $\Sigma_{i}$ are minimal, we have at $s=0$

[TABLE]

where $\eta$ is the unit conormal to $\Gamma$ in $\Sigma$ that points outside $N(R)$. We notice that for $s>0$, $\beta_{s}$ is inside $U_{i,0}$ so $\partial_{s}\beta_{s}$ points to $U_{i,0}$ and is orthogonal to the tangent space to $\Gamma$. $\eta+\eta_{i}$ is a vector that bisects the wedge corresponding to $U_{i,0}$ and contained in the orthogonal to the tangent space to $\Gamma$. So $(\partial_{s}\beta_{i},\eta+\eta_{i})>0$ along $\Gamma$ and $\partial_{s}(|F_{i}(A_{i,s},s)|+|G_{i}(B_{i,s},s)|<0$. This implies that $|\partial U_{i,s}|<|\partial U_{i,0}|$ for $s>0$ small. So all the stated properties are satisfied. ∎

8.2. The lower semi-continuity

We have the following result.

Proposition 25**.**

Let $M_{i}\to\overline{M}$ be a converging sequence of cusp manifolds. Then $\liminf\mathcal{A}_{1}(M_{i})\geq\mathcal{A}_{1}(\overline{M})$.

Proof.

Let us consider a minimal surface $\Sigma_{i}$ in $M_{i}$ such that $a(\Sigma_{i})=\mathcal{A}(M_{i})$. By Proposition 21, we know that there is $A_{0}$ such that $|\Sigma_{i}|<A_{0}$. Moreover, by Corollary 4, there is $\ell_{0}$, $\delta_{0}$ and $\mathfrak{s}_{0}$ such that if $M_{i}$ contains a geodesic loop of length $\ell\leq\ell_{0}$ then its tubular neighborhood $N_{R_{\ell}}$ satisfies $S_{R_{\ell}}=\partial N_{R_{\ell}}$ has diameter less than $\delta_{0}$ and systole larger than $\mathfrak{s}_{0}$. Thus there is $\overline{R}$ such that $\Sigma_{i}\cap N_{R_{\ell}-\overline{R}}$ is empty by Proposition 24.

This implies that there is $\varepsilon>0$ such that $\Sigma_{i}\subset{M_{i}}_{[\varepsilon,\infty)}$ for any $i$. Since $M_{i}\to\overline{M}$ there is $\varphi_{i}:\overline{M}_{[\varepsilon/2,\infty)}\to M_{i}$ which is a $\kappa_{i}$ quasi-isometry where $\kappa_{i}\to 1$. Moreover we have ${M_{i}}_{[\varepsilon,\infty)}\subset\varphi_{i}(\overline{M}_{[\varepsilon/2,\infty)})$. Let $\tilde{g}_{i}=\varphi_{i}^{*}g_{i}$ and $\widetilde{\Sigma}_{i}=\varphi_{i}^{-1}(\Sigma_{i})\subset\overline{M}_{[\varepsilon/2,\infty)}$ which is a minimal surface for the metric $\tilde{g}_{i}$. Since $M_{i}\to\overline{M}$ we have $\tilde{g}_{i}\to\bar{g}$ in the $C^{\infty}$ topology. Moreover $a_{\tilde{g}_{i}}(\widetilde{\Sigma}_{i})\leq A_{0}$ and the index of $\Sigma_{i}$ is [math] or $1$.

Thus we can apply the compactness result of Sharp (Theorem A.6 in [14]). It implies that there is a closed connected embedded minimal surface $\overline{\Sigma}$ in $(\overline{M},\bar{g})$ such that $(\widetilde{\Sigma}_{i})$ converges in the varifold sense to $\overline{\Sigma}$ with some multiplicity. Moreover, the convergence is smooth outside a finite number of points. If $\overline{\Sigma}$ is orientable, then

[TABLE]

If $\overline{\Sigma}$ is non-orientable, then either $\widetilde{\Sigma}_{i}$ is non orientable or $\widetilde{\Sigma}_{i}$ is orientable and the convergence must be with multiplicity at least $2$. In both cases, we have

[TABLE]

So the proposition is proved. ∎

Appendix A

A.1. A uniform graph lemma

Let us consider $\mathbb{R}^{3}$ endowed with the metric $\bar{g}=h^{2}(x_{3})(dx_{1}^{2}+dx_{2}^{2})+dx_{3}^{2}$. For $k_{1},k_{2},k_{3},k_{4}\in\{1,2,3\}$ and $p\leq 4$, we recall that

[TABLE]

We consider a second metric $g=a_{kl}(x_{1},x_{2},x_{3})dx_{k}dx_{l}$. We assume that there is some $A$ such that the following hypotheses occurs

• H1

$\frac{1}{A^{2}}\bar{g}\leq g\leq A^{2}\bar{g}$

• H2

$\frac{|h^{\prime}|}{h}\leq A$ and $\frac{|h^{\prime\prime}|}{h}\leq A$.

• H3

$|a_{kl}|\leq Ah^{n_{2}(k,l)}(x_{3})$, $|\partial_{i}a_{kl}|\leq Ah^{n_{3}(k,l,i)}(x_{3})$ and $|\partial_{i}\partial_{j}a_{kl}|\leq Ah^{n_{4}(k,l,i,j)}(x_{3})$.

We notice that the metric $\bar{g}$ satisfies also the hypotheses of the last item.

Lemma 26**.**

Let $\bar{g}$ and $g$ as above and consider $\varepsilon_{0}$, $k_{0}$, then there is $C>0$ such that the following is true. Let $\Sigma$ be a surface in $\mathbb{R}^{2}\times[a,b]$ endowed with the metric $g$ which is tangent to $\mathbb{R}^{2}\times\{\bar{t}\}$ at $\bar{p}=(0,0,\bar{t})$ such that $d_{\Sigma}(\bar{p},\partial\Sigma)\geq\varepsilon_{0}$ and $|A_{\Sigma}|\leq k_{0}$.

Then there is a function $u$ defined on the disk $\{(x_{1},x_{2})\in\mathbb{R}^{2}|x_{1}^{2}+x_{2}^{2}\leq 2C^{2}/h^{2}(\bar{t})\}$ such that $(x_{1},x_{2})\mapsto(x_{1},x_{2},\bar{t}+u(x_{1},x_{2}))$ is a parametrization of a neighborhood of $\bar{p}$ in $\Sigma$. Moreover $u$ satisfies

[TABLE]

Proof.

First we replace $\Sigma$ by the geodesic disk of center $\bar{p}$ and $\varepsilon_{0}$. Since $a_{33}\geq\frac{1}{A^{2}}$, the distance between $\{x_{3}=\bar{t}\}$ and $\{x_{3}=\bar{t}\pm t\}$ is at least $t/A$. So $\Sigma$ is contained in $\mathbb{R}^{2}\times[\bar{t}-A\varepsilon_{0},\bar{t}+A\varepsilon_{0}]$. Let us also remark that since $\frac{|h^{\prime}|}{h}\leq A$ we have $e^{-A|x_{3}-\bar{t}|}h(\bar{t})\leq h(x_{3})\leq e^{A|x_{3}-\bar{t}|}h(\bar{t})$. Then $e^{-A^{2}\varepsilon_{0}}h(\bar{t})\leq h(x_{3})\leq e^{A^{2}\varepsilon_{0}}h(\bar{t})$ on $[\bar{t}-A\varepsilon_{0},\bar{t}+A\varepsilon_{0}]$.

Let us consider $\Psi:\mathbb{R}^{2}\times[\bar{t}-A\varepsilon_{0},\bar{t}+A\varepsilon_{0}]\to\mathbb{R}^{2}\times[\bar{t}-A\varepsilon_{0},\bar{t}+A\varepsilon_{0}]:(y_{1},y_{2},y_{3})\mapsto(\frac{1}{h(\bar{t})}y_{1},\frac{1}{h(\bar{t})}y_{2},y_{3})$. Then the metric $g^{*}=\Psi^{*}g$ can be written $b_{kl}(y_{1},y_{2},y_{3})dy_{k}dy_{l}$ where $b_{kl}=h(\bar{t})^{-n_{2}(k,l)}a_{kl}\circ\Psi$. Thus

[TABLE]

So there is a constant $B$ such that $|b_{kl}|\leq B$. A similar computation proves that $|\nabla b_{kl}|\leq B$ and $|\operatorname{Hess}b_{kl}|\leq B$. Using Hypothesis H1, we also have $\frac{1}{A^{2}}\Psi^{*}\bar{g}\leq g^{*}\leq A^{2}\Psi^{*}\bar{g}$ where $\Psi^{*}\bar{g}=\frac{h^{2}(y_{3})}{h^{2}(\bar{t})}(dy_{1}^{2}+dy_{2}^{2})+dy_{3}^{2}$. This implies that $\det g^{*}$ is far from [math] and $\infty$. So the coefficients $b^{kl}$ of the inverse of $g^{*}$ satisfy $|b^{kl}|\leq B$ and, for any $k\in\{1,2,3\}$, $\frac{1}{B}\leq b_{kk}\leq B$ and $\frac{1}{B}\leq b^{kk}\leq B$.

Let us define $\Sigma^{*}=\Psi^{-1}(\Sigma)$, $\Sigma^{*}\subset(\mathbb{R}^{3},g^{*})$ is a geodesic disk of radius $\varepsilon_{0}$ and curvature bounded by $k_{0}$. Let us consider $g_{e}=dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2}$ the Euclidean metric. Because of the the control we have on $g^{*}$, there is $\varepsilon_{1}$ that depends only on $\varepsilon_{0}$, $A$ and $B$ and $k_{1}$ that depends only on $k_{0}$, $A$ and $B$ such that $(\Sigma^{*},g_{e})$ has curvature bounded by $k_{1}$ and $d_{\Sigma^{*},g_{e}}(\bar{p},\partial\Sigma^{*})\geq\varepsilon_{1}$ (the proof of this result can be found in the Appendix of [13] more precisely see the proof of Propositions 4.1 and 4.3).

So we have a surface in the Euclidean space $\mathbb{R}^{3}$ with curvature bounded by $k_{1}$, $d_{\Sigma^{*},g_{e}}(\bar{p},\partial\Sigma^{*})\geq\varepsilon_{1}$ and that is tangent to $\mathbb{R}^{2}\times\{\bar{t}\}$ at $\bar{p}$. Then a classical uniform graph lemma (see Proposition 2.3 in [13]) implies that there is $C$ that depends only on $k_{1}$ and $\varepsilon_{1}$ such the following is true. There is a function $u$ defined on the Euclidean disk of radius $\sqrt{2}C$ centered at the origin such that $(y_{1},y_{2})\mapsto(y_{1},y_{2},\bar{t}+u(y_{1},y_{2}))$ is a parametrization of a neighborhood of $\bar{p}$ in $\Sigma^{*}$. Moreover

[TABLE]

In order to come back to the original coordinate system we define the function $v(x_{1},x_{2})=u(h(\bar{t})x_{1},h(\bar{t})x_{2})$ which is defined on $\{(x_{1},x_{2})\in\mathbb{R}^{2}|x_{1}^{2}+x_{2}^{2}\leq 2\delta^{2}/h^{2}(\bar{t})\}$ and satisfies

[TABLE]

We notice that, since $\Sigma\subset\mathbb{R}^{2}\times[\bar{t}-A\varepsilon_{0},\bar{t}+A\varepsilon_{0}]$, we have $|v|\leq A\varepsilon_{0}$. ∎

A.2. The minimal surface equation

Several times we consider graphs that are minimal surfaces; let us write the equation solved by these graphs.

On $\mathbb{R}^{3}$ we consider the metric $g=a_{1}^{2}(x_{3})dx_{1}^{2}+a_{2}^{2}(x_{3})dx_{2}^{2}+dx_{3}^{2}$ which is a model for the metric in cusp or tubular ends. Let $u$ be a function in a domain of $\mathbb{R}^{2}$ and consider the graph parametrized by $X(x_{1},x_{2})=(x_{1},x_{2},u(x_{1},x_{2}))$. The induced metric is

[TABLE]

So the area element is $Wdx_{1}dx_{2}=(a_{1}^{2}(u)a_{2}^{2}(u)+a_{2}^{2}(u)u_{x_{1}}^{2}+a_{1}^{2}(u)u_{x_{2}}^{2})dx_{1}dx_{2}$.

So if $v$ is an other function with zero boundary values and $A(t)$ is the area of the graph of $u+tv$, the derivative of $A$ at $t=0$ is

[TABLE]

Thus the graph is minimal if $u$ satisfies

[TABLE]