ON ALMOST-FUCHSIAN SUBMANIFOLDS OF
HADAMARD SPACES AND THE ASYMPTOTIC
PLATEAU PROBLEM

Samuel Bronstein

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Samuel Bronstein. ON ALMOST-FUCHSIAN SUBMANIFOLDS OF HADAMARD SPACES AND THE ASYMPTOTIC PLATEAU PROBLEM. 2024. hal-04549710

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# ON ALMOST-FUCHSIAN SUBMANIFOLDS OF HADAMARD SPACES AND THE ASYMPTOTIC PLATEAU PROBLEM

## SAMUEL BRONSTEIN

**ABSTRACT.** We consider minimal submanifolds of negatively curved spaces with small curvature. We show that in a Hadamard space with negatively pinched curvature $-C \le K \le -1$, complete minimal submanifolds with second fundamental form less than 1 everywhere bound a class of spheres at infinity for which the asymptotic Plateau problem is uniquely solvable.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. The convex hull barrier</td><td>2</td></tr><tr><td>3. Almost-fuchsian submanifolds</td><td>3</td></tr><tr><td>4. The asymptotic Plateau problem</td><td>7</td></tr><tr><td>References</td><td>10</td></tr></table>

## 1. INTRODUCTION

In 1983, Uhlenbeck [Uhl83] introduced the notion of almost-fuchsian representation. A representation of a surface group in PSL(2, $\mathbb{C}$) is almost-fuchsian if it is discrete, faithful and there is an equivariant minimal disc in $\mathbb{H}^3$ whose principal values lie in a compact set of $(-1, 1)$. Since then, almost-fuchsian representations have been a very fruitful field of study, see for instance [Eps86, KS07, HW13, Sep16, San17, Jia21, EES22, BS23] We consider here a generalization of almost-fuchsian submanifolds in negatively curved spaces: Let $X$ be a complete simply connected riemannian manifold whose sectional curvature is less or equal than $-1$.

**Definition 1.1.** An immersion $f: Y \to X$ is almost-fuchsian if it is minimal, proper, and it satisfies

$$ (1.1) \qquad \sup_{y \in Y} |\mathbb{I}_f| < 1 $$

Remark that $Y$ can always be endowed with the induced metric which will be complete by the properness assumption. By extension, a submanifold $Y \subset X$ will be said to be almost-fuchsian if the inclusion map is almost-fuchsian.

In this paper, we prove the following embedding theorem:

**Theorem 1.2.** Let $f: Y \to X$ be an almost-fuchsian immersion. Then $Y$ is contractible. Denote by $NY$ the normal bundle to $Y$. The exponential map $\exp: NY \to X$ is a diffeomorphism. Also, $f$ extends at infinity to an embedding from a sphere $S \to \partial_{\infty} X$.

Date: 2023.
# ON ALMOST-FUCHSIAN SUBMANIFOLDS OF HADAMARD SPACES AND THE ASYMPTOTIC PLATEAU PROBLEM

## SAMUEL BRONSTEIN

**ABSTRACT.** We consider minimal submanifolds of negatively curved spaces with small curvature. We show that in a Hadamard space with negatively pinched curvature $-C \le K \le -1$, complete minimal submanifolds with second fundamental form less than 1 everywhere bound a class of spheres at infinity for which the asymptotic Plateau problem is uniquely solvable.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. The convex hull barrier</td><td>2</td></tr><tr><td>3. Almost-fuchsian submanifolds</td><td>3</td></tr><tr><td>4. The asymptotic Plateau problem</td><td>7</td></tr><tr><td>References</td><td>10</td></tr></table>

## 1. INTRODUCTION

In 1983, Uhlenbeck [Uhl83] introduced the notion of almost-fuchsian representation. A representation of a surface group in PSL(2, $\mathbb{C}$) is almost-fuchsian if it is discrete, faithful and there is an equivariant minimal disc in $\mathbb{H}^3$ whose principal values lie in a compact set of $(-1, 1)$. Since then, almost-fuchsian representations have been a very fruitful field of study, see for instance [Eps86, KS07, HW13, Sep16, San17, Jia21, EES22, BS23] We consider here a generalization of almost-fuchsian submanifolds in negatively curved spaces: Let $\mathcal{X}$ be a complete simply connected riemannian manifold whose sectional curvature is less or equal than $-1$.

**Definition 1.1.** An immersion $f: Y \to \mathcal{X}$ is almost-fuchsian if it is minimal, proper, and it satisfies

$$ (1.1) \qquad \sup_{y \in Y} |\mathbb{I}_f| < 1 $$

Remark that $Y$ can always be endowed with the induced metric which will be complete by the properness assumption. By extension, a submanifold $Y \subset \mathcal{X}$ will be said to be almost-fuchsian if the inclusion map is almost-fuchsian.

In this paper, we prove the following embedding theorem:

**Theorem 1.2.** Let $f: Y \to \mathcal{X}$ be an almost-fuchsian immersion. Then $Y$ is contractible. Denote by $NY$ the normal bundle to $Y$. The exponential map $\exp: NY \to \mathcal{X}$ is a diffeomorphism. Also, $f$ extends at infinity to an embedding from a sphere $S \to \partial_{\infty}\mathcal{X}$.

Date: 2023.
We also consider the following asymptotic Plateau problem: given a sphere $S \subset \partial_{\infty}\mathcal{X}$, can we count the complete minimal submanifolds of $\mathcal{X}$ asymptotically bounding $S$? Jiang [Jia21] and Huang-Lowe-Seppi [HLS23] considered this problem for spheres in $\partial_{\infty}\mathbb{H}^n$ bounding respectively an almost-fuchsian disc or an almost-fuchsian hypersurface. They proved that in that case, the unique minimal submanifold of the given dimension bounding $S$ is the almost-fuchsian one, which is unique.

Here we prove that this statement holds in a broader generality, provided that $\mathcal{X}$ has negatively pinched curvature between $-C$ and $-1$.

**Theorem 1.3.** Let $\mathcal{X}$ be an $n$-dimensional negatively pinched Hadamard space with sectional curvature less or equal than $-1$. Let $Y \subset \mathcal{X}$ be a $k$-dimensional almost-fuchsian submanifold bounding a $(k-1)$-dimensional sphere $S \subset \partial_{\infty}\mathcal{X}$. Then $Y$ is the unique $k$-dimensional complete minimal submanifold of $\mathcal{X}$ asymptotically bounding $S$.

We don't know whether this statement holds when the sectional curvature of $\mathcal{X}$ has no lower bound, because we lack of convex sets to work with. Note that Huang-Lowe-Seppi [HLS23] also cover the case of weakly-almost-fuchsian discs in $\mathbb{H}^3$, that is when the supremum of the principal values is allowed to be one.

This paper is divided in three parts. The first part is about a very standard fact, that minimal submanifolds of a space $\mathcal{X}$ with negatively pinched sectional curvature remain in the convex hull of their asymptotic boundary. Surprisingly, we couldn't prove it without a lower bound on the curvature of $\mathcal{X}$. The second part is devoted to the proof of the embedding theorem, with some explicit estimates on the geometry of an almost-fuchsian submanifold. Finally, the third part is devoted to the proof of the uniqueness to the asymptotic Plateau problem for spheres bounding an almost-fuchsian submanifold.

The author thanks Andrea Seppi, Graham Smith and Nicolas Tholozan for their help, insights and discussions on these topics.

## 2. THE CONVEX HULL BARRIER

This section is devoted to the proof of the following theorem:

**Theorem 2.1.** Let $\mathcal{X}$ be a Hadamard space with negatively pinched curvature. Let $Y$ be a minimal submanifold of $\mathcal{X}$, bounding a subset $F \subset \partial_{\infty}\mathcal{X}$. Denote $C$ the convex hull of $F$. Then $Y \subset C$.

While this theorem is well known when $\mathcal{X} = \mathbb{H}^n$, we didn't find a general proof of this fact in negative curvature. A very elegant proof in $\mathbb{H}^n$ uses totally geodesic hypersurfaces as barriers, as the convex hull is the intersection of all half spaces containing $C$ [Jia21]. While this proof can be word for word done in a symmetric space of a semisimple Lie group of rank 1, here we need to be more careful, as totally geodesic hypersurfaces might not exist.

In all this section $\mathcal{X}$ will denote a negatively pinched Hadamard space, its boundary $\partial_{\infty}\mathcal{X}$ will be its ideal boundary which equals its Gromov boundary in that case [BGS85, Gro78, EO73]. The main idea is to use the following property:

**Proposition 2.2.** Let $F$ be a closed set of $\partial_{\infty}\mathcal{X}$, and denote by $C$ its convex hull. Let $p \in X - C$. Then there is a closed domain $D \subset \mathcal{X} \cup \partial_{\infty}\mathcal{X}$ disjoint from $C \cup F$ such that $p \in D$, and $D$ is foliated by smooth strictly convex hypersurfaces.

*Proof.* We use a theorem of Bowditch [Bow94]. As $\mathcal{X}$ has negatively pinched curvature, there is a metric on $\mathcal{X} \cup \partial_{\infty}\mathcal{X}$ compatible with the topologies such that the map $F \mapsto CH(F)$ is
We also consider the following asymptotic Plateau problem: given a sphere $S \subset \partial_{\infty}\mathcal{X}$, can we count the complete minimal submanifolds of $\mathcal{X}$ asymptotically bounding $S$? Jiang [Jia21] and Huang-Lowe-Seppi [HLS23] considered this problem for spheres in $\partial_{\infty}\mathbb{H}^n$ bounding respectively an almost-fuchsian disc or an almost-fuchsian hypersurface. They proved that in that case, the unique minimal submanifold of the given dimension bounding $S$ is the almost-fuchsian one, which is unique.

Here we prove that this statement holds in a broader generality, provided that $\mathcal{X}$ has negatively pinched curvature between $-C$ and $-1$.

**Theorem 1.3.** Let $\mathcal{X}$ be an $n$-dimensional negatively pinched Hadamard space with sectional curvature less or equal than $-1$. Let $Y \subset \mathcal{X}$ be a $k$-dimensional almost-fuchsian submanifold bounding a $(k-1)$-dimensional sphere $S \subset \partial_{\infty}\mathcal{X}$. Then $Y$ is the unique $k$-dimensional complete minimal submanifold of $\mathcal{X}$ asymptotically bounding $S$.

We don't know whether this statement holds when the sectional curvature of $\mathcal{X}$ has no lower bound, because we lack of convex sets to work with. Note that Huang-Lowe-Seppi [HLS23] also cover the case of weakly-almost-fuchsian discs in $\mathbb{H}^3$, that is when the supremum of the principal values is allowed to be one.

This paper is divided in three parts. The first part is about a very standard fact, that minimal submanifolds of a space $\mathcal{X}$ with negatively pinched sectional curvature remain in the convex hull of their asymptotic boundary. Surprisingly, we couldn't prove it without a lower bound on the curvature of $\mathcal{X}$. The second part is devoted to the proof of the embedding theorem, with some explicit estimates on the geometry of an almost-fuchsian submanifold. Finally, the third part is devoted to the proof of the uniqueness to the asymptotic Plateau problem for spheres bounding an almost-fuchsian submanifold.

The author thanks Andrea Seppi, Graham Smith and Nicolas Tholozan for their help, insights and discussions on these topics.

## 2. THE CONVEX HULL BARRIER

This section is devoted to the proof of the following theorem:

**Theorem 2.1.** Let $\mathcal{X}$ be a Hadamard space with negatively pinched curvature. Let $Y$ be a minimal submanifold of $\mathcal{X}$, bounding a subset $F \subset \partial_{\infty}\mathcal{X}$. Denote $C$ the convex hull of $F$. Then $Y \subset C$.

While this theorem is well known when $\mathcal{X} = \mathbb{H}^n$, we didn't find a general proof of this fact in negative curvature. A very elegant proof in $\mathbb{H}^n$ uses totally geodesic hypersurfaces as barriers, as the convex hull is the intersection of all half spaces containing $C$ [Jia21]. While this proof can be word for word done in a symmetric space of a semisimple Lie group of rank 1, here we need to be more careful, as totally geodesic hypersurfaces might not exist.

In all this section $\mathcal{X}$ will denote a negatively pinched Hadamard space, its boundary $\partial_{\infty}\mathcal{X}$ will be its ideal boundary which equals its Gromov boundary in that case [BGS85, Gro78, EO73]. The main idea is to use the following property:

**Proposition 2.2.** Let $F$ be a closed set of $\partial_{\infty}\mathcal{X}$, and denote by $C$ its convex hull. Let $p \in X - C$. Then there is a closed domain $D \subset \mathcal{X} \cup \partial_{\infty}\mathcal{X}$ disjoint from $C \cup F$ such that $p \in D$, and $D$ is foliated by smooth strictly convex hypersurfaces.

*Proof.* We use a theorem of Bowditch [Bow94]. As $\mathcal{X}$ has negatively pinched curvature, there is a metric on $\mathcal{X} \cup \partial_{\infty}\mathcal{X}$ compatible with the topologies such that the map $F \mapsto CH(F)$ is
continuous on the closed sets of $X \cup \partial_{\infty}X$ equipped with the Hausdorff distance. Therefore, there is a neighborhood $V$ of $F$ such that $\tilde{C}$ the convex hull of $V$ does not contain $p$. The complement of $\tilde{C}$ is then foliated by the equidistant to $\tilde{C}$, which are strictly convex.

Now a priori the convex $\tilde{C}$ has no smooth boundary, and so our foliation is only $C^{1,1}$. However, Parkkonen-Paulin [PP12] (Proposition 6) proved that a $\epsilon$-neighborhood of $\tilde{C}$ contains a smooth convex set, so up to replacing $\tilde{C}$ by this smooth convex set we assume the foliation of the complement to be by smooth convex hypersurfaces. Denote $D$ the closure of the complement of $\tilde{C}$. $\square$

*Proof of theorem 2.1.* Let $Y$ be a minimal submanifold and let $C$ denote the convex hull of $\partial_{\infty}Y$. Let $p \in X - C$ and $D$ denote the domain provided by proposition 2.2.

Suppose that $Y$ intersects $D$. Because $D$ does not bound $\partial_{\infty}Y$, the intersection $Y \cap D$ is compact. Then there must some point $q$ at which $Y$ is tangent to one of the strictly convex hypersurfaces foliating $D$. This contradicts the minimality of $Y$. Hence $Y \cap D$ is empty. Repeating this argument for any $p \in X - C$ proves that $Y \subset C$. $\square$

*Remark 2.1.* The lower bound assumption on the curvature of $X$ is crucial here, otherwise there might not be enough convex sets to pursue our proof. See for instance Ancona [Anc94], there are spaces with curvature less than $-1$ such that the convex hull of any nontrivial open set at infinity is the full space $X$.

### 3. ALMOST-FUCHSIAN SUBMANIFOLDS

This part is about the notion of almost-fuchsian submanifold. First introduced for surface group representations in PSL(2, $\mathbb{C}$) by Uhlenbeck [Uhl83], almost-fuchsian discs and hypersurfaces have a rich literature [Eps86, KS93, Sep16, San17, Jia21, EES22, BS23, HLS23].

Throughout this section $X$ will be a Hadamard space with sectional curvature less or equal than $-1$. Note that we do not require its curvature to admit a lower bound. We consider an immersion $f: Y \to X$, where $Y$ is connected. We assume $f$ to be complete, i.e. the induced metric on $Y$ is complete.

We will prove the following theorem.

**Theorem 3.1.** Assume $f: Y \to X$ is minimal and satisfies:

$$ (3.1) \qquad \sup_{y \in Y} |\mathcal{I}_f| < 1 $$

*Then $Y$ is contractible. Denote by $NY$ the normal bundle to $Y$. The exponential map $\exp: NY \to X$ is a diffeomorphism. Also, $f$ extends at infinity to an embedding from a sphere $S \to \partial_{\infty}X$*

The proof is divided in several steps. First, we show that the induced metric by $\exp$ is nondegenerate on $NY$. This will imply that $\exp$ is a covering map to $X$, and so it is a diffeomorphism. Finally, we will see that $f$ is a quasi-isometric embedding from $Y$ to $X$, so it extends at infinity to an embedding $S \to \partial_{\infty}X$.

First note that because of the condition on $f$, the induced metric has negatively pinched curvature on $Y$, and it makes sense to talk about its boundary $\partial_{\infty}Y$, which is constructed in the works of Gromov [Gro78], Eberlein-O'NEill [EO73] and Ballmann-Gromov-Schroeder [BGS85].
continuous on the closed sets of $X \cup \partial_{\infty}X$ equipped with the Hausdorff distance. Therefore, there is a neighborhood $V$ of $F$ such that $\tilde{C}$ the convex hull of $V$ does not contain $p$. The complement of $\tilde{C}$ is then foliated by the equidistant to $\tilde{C}$, which are strictly convex.

Now a priori the convex $\tilde{C}$ has no smooth boundary, and so our foliation is only $C^{1,1}$. However, Parkkonen-Paulin [PP12] (Proposition 6) proved that a $\epsilon$-neighborhood of $\tilde{C}$ contains a smooth convex set, so up to replacing $\tilde{C}$ by this smooth convex set we assume the foliation of the complement to be by smooth convex hypersurfaces. Denote $D$ the closure of the complement of $\tilde{C}$. $\square$

*Proof of theorem 2.1.* Let $Y$ be a minimal submanifold and let $C$ denote the convex hull of $\partial_{\infty}Y$. Let $p \in X - C$ and $D$ denote the domain provided by proposition 2.2.

Suppose that $Y$ intersects $D$. Because $D$ does not bound $\partial_{\infty}Y$, the intersection $Y \cap D$ is compact. Then there must some point $q$ at which $Y$ is tangent to one of the strictly convex hypersurfaces foliating $D$. This contradicts the minimality of $Y$. Hence $Y \cap D$ is empty. Repeating this argument for any $p \in X - C$ proves that $Y \subset C$. $\square$

*Remark 2.1.* The lower bound assumption on the curvature of $X$ is crucial here, otherwise there might not be enough convex sets to pursue our proof. See for instance Ancona [Anc94], there are spaces with curvature less than $-1$ such that the convex hull of any nontrivial open set at infinity is the full space $X$.

### 3. ALMOST-FUCHSIAN SUBMANIFOLDS

This part is about the notion of almost-fuchsian submanifold. First introduced for surface group representations in PSL(2, $\mathbb{C}$) by Uhlenbeck [Uhl83], almost-fuchsian discs and hypersurfaces have a rich literature [Eps86, KS93, Sep16, San17, Jia21, EES22, BS23, HLS23].

Throughout this section $X$ will be a Hadamard space with sectional curvature less or equal than $-1$. Note that we do not require its curvature to admit a lower bound. We consider an immersion $f: Y \to X$, where $Y$ is connected. We assume $f$ to be complete, i.e. the induced metric on $Y$ is complete.

We will prove the following theorem.

**Theorem 3.1.** Assume $f: Y \to X$ is minimal and satisfies:

$$ (3.1) \qquad \sup_{y \in Y} |\mathcal{I}_f| < 1 $$

*Then $Y$ is contractible. Denote by $NY$ the normal bundle to $Y$. The exponential map $\exp: NY \to X$ is a diffeomorphism. Also, $f$ extends at infinity to an embedding from a sphere $S \to \partial_{\infty}X$*

The proof is divided in several steps. First, we show that the induced metric by $\exp$ is nondegenerate on $NY$. This will imply that $\exp$ is a covering map to $X$, and so it is a diffeomorphism. Finally, we will see that $f$ is a quasi-isometric embedding from $Y$ to $X$, so it extends at infinity to an embedding $S \to \partial_{\infty}X$.

First note that because of the condition on $f$, the induced metric has negatively pinched curvature on $Y$, and it makes sense to talk about its boundary $\partial_{\infty}Y$, which is constructed in the works of Gromov [Gro78], Eberlein-O'NEill [EO73] and Ballmann-Gromov-Schroeder [BGS85].
**Lemma 3.2.** Let $f: Y \to \mathbb{X}$ be an almost-fuchsian immersion. Let $\varepsilon > 0$ such that

$$ (3.2) \qquad \sup_{y \in Y} |\mathbb{I}_f| \le 1 - \varepsilon. $$

Endow $Y$ with the induced metric by $f$. Then its sectional curvature is less than $-\varepsilon(2 - \varepsilon)$. If the sectional curvature of $\mathbb{X}$ is lower bounded by $-C$, then the sectional curvature of $f^*g_{\mathbb{X}}$ is bigger than $-C - 2(1 - \varepsilon)^2$.

**Remark 3.1.** When $Y$ is a surface, the minimality condition ensures that we have more precise bounds: The sectional curvature of an almost-fuchsian surface is then less than $-1$ and bigger than $-C - (1 - \varepsilon)^2$.

**Proof.** This is a consequence of the Gauss condition: For $u, v$ an orthonormal basis of a tangent plane to $Y$, By the Gauss' equation, the curvature of $f^*g_{\mathbb{X}}$ is then

$$ (3.3) \qquad R_{f^*g_{\mathbb{X}}}(u, v, v, u) = R_{g_{\mathbb{X}}}(u, v, v, u) + g_{\mathbb{X}}(\mathbb{I}_f(u, u), \mathbb{I}_f(v, v)) - g_{\mathbb{X}}(\mathbb{I}_f(u, v), \mathbb{I}_f(u, v)) $$

As $R_{g_{\mathbb{X}}}(u, v, v, u) \le -1$ and $|\mathbb{I}_f(u, u)| \le 1 - \varepsilon$, we directly get

$$ (3.4) \qquad R_{f^*g_{\mathbb{X}}}(u, v, v, u) \le -1 + (1 - \varepsilon)^2 = -\varepsilon(2 - \varepsilon). $$

as desired. Moreover, if $R_{g_{\mathbb{X}}}(u, v, v, u) \ge -C$, we directly get that

$$ (3.5) \qquad R_{f^*g_{\mathbb{X}}}(u, v, v, u) \ge -C - 2(1 - \varepsilon)^2, $$

as claimed.

□

**Proposition 3.3.** Let $f: Y \to \mathbb{X}$ be an almost-fuchsian immersion. Denote by $NY$ the normal bundle to $Y$, and note $G$ the induced metric by the exponential map on $NY$. For $t > 0$, denote by $N_tY$ the tangent bundle to the distance $t$ hypersurface to $Y$. Let $(x, tv) \in NY$ with $v$ a unit normal vector. Let $B_t$ be the parallel transport map from $T_x\mathbb{X} \cap (v)^\perp$ to $T_{\exp tv}(N_tY)$. Let $w \in T_v(UY)$ and let $f$ be the function:

$$ (3.6) \qquad f(t) = |B_t w|^2 $$

If $\mathbb{X}$ is the hyperbolic space, $f$ satisfies the following controls:

$$ (3.7) \qquad (\sqrt{f})'' = \sqrt{f} $$

If $\mathbb{X}$ has sectional curvature strictly less than $-1$, then

$$ (3.8) \qquad (\sqrt{f})'' > \sqrt{f} $$

Also, in both cases

$$ (3.9) \qquad (\ln f)'' + \frac{(\ln f)'^2}{2} \ge 2 $$

with the same equality and inequality conditions.

**Proof.** With the introduced notations, fix $\gamma: I \to UY$ a smooth path such that $\gamma(0) = v$ and $\gamma'(0) = w$. The family $c_s(t)$ is a smooth variation of geodesics, hence $J(t) = \partial_s c_s(t)|_{s=0}$ is a Jacobi field along $c_0$. Evaluate the characteristic equation of Jacobi fields against $J$:

$$ (3.10) \qquad g_{\mathbb{X}}(\ddot{J}, J) + R_{\mathbb{X}}(J, \partial_t c, \partial_t c, J) = 0 $$

In the second term, we recognize the sectional curvature of the tangent plane spanned by $J$ and $\partial_t c$. Choose a local orthonormal chart of $T_x\mathbb{X}$ such that $B_0 = I_k \oplus O_{n-k-1}$ and
**Lemma 3.2.** Let $f: Y \to \mathbb{X}$ be an almost-fuchsian immersion. Let $\varepsilon > 0$ such that

$$ (3.2) \qquad \sup_{y \in Y} |\mathbb{I}_f| \le 1 - \varepsilon. $$

Endow $Y$ with the induced metric by $f$. Then its sectional curvature is less than $-\varepsilon(2 - \varepsilon)$. If the sectional curvature of $\mathbb{X}$ is lower bounded by $-C$, then the sectional curvature of $f^*g_{\mathbb{X}}$ is bigger than $-C - 2(1 - \varepsilon)^2$.

**Remark 3.1.** When $Y$ is a surface, the minimality condition ensures that we have more precise bounds: The sectional curvature of an almost-fuchsian surface is then less than $-1$ and bigger than $-C - (1 - \varepsilon)^2$.

*Proof.* This is a consequence of the Gauss condition: For $u, v$ an orthonormal basis of a tangent plane to $Y$, By the Gauss' equation, the curvature of $f^*g_{\mathbb{X}}$ is then

$$ (3.3) \qquad R_{f^*g_{\mathbb{X}}}(u, v, v, u) = R_{g_{\mathbb{X}}}(u, v, v, u) + g_{\mathbb{X}}(\mathbb{I}_f(u, u), \mathbb{I}_f(v, v)) - g_{\mathbb{X}}(\mathbb{I}_f(u, v), \mathbb{I}_f(u, v)) $$

As $R_{g_{\mathbb{X}}}(u, v, v, u) \le -1$ and $|\mathbb{I}_f(u, u)| \le 1 - \varepsilon$, we directly get

$$ (3.4) \qquad R_{f^*g_{\mathbb{X}}}(u, v, v, u) \le -1 + (1 - \varepsilon)^2 = -\varepsilon(2 - \varepsilon). $$

as desired. Moreover, if $R_{g_{\mathbb{X}}}(u, v, v, u) \ge -C$, we directly get that

$$ (3.5) \qquad R_{f^*g_{\mathbb{X}}}(u, v, v, u) \ge -C - 2(1 - \varepsilon)^2, $$

as claimed.

□

**Proposition 3.3.** Let $f: Y \to \mathbb{X}$ be an almost-fuchsian immersion. Denote by $NY$ the normal bundle to $Y$, and note $G$ the induced metric by the exponential map on $NY$. For $t > 0$, denote by $N_tY$ the tangent bundle to the distance $t$ hypersurface to $Y$. Let $(x, tv) \in NY$ with $v$ a unit normal vector. Let $B_t$ be the parallel transport map from $T_x\mathbb{X} \cap (v)^\perp$ to $T_{\exp tv}(N_tY)$. Let $w \in T_v(UY)$ and let $f$ be the function:

$$ (3.6) \qquad f(t) = |B_t w|^2 $$

If $\mathbb{X}$ is the hyperbolic space, $f$ satisfies the following controls:

$$ (3.7) \qquad (\sqrt{f})'' = \sqrt{f} $$

If $\mathbb{X}$ has sectional curvature strictly less than $-1$, then

$$ (3.8) \qquad (\sqrt{f})'' > \sqrt{f} $$

Also, in both cases

$$ (3.9) \qquad (\ln f)'' + \frac{(\ln f)'^2}{2} \ge 2 $$

with the same equality and inequality conditions.

*Proof.* With the introduced notations, fix $\gamma: I \to UY$ a smooth path such that $\gamma(0) = v$ and $\gamma'(0) = w$. The family $c_s(t)$ is a smooth variation of geodesics, hence $J(t) = \partial_s c_s(t)|_{s=0}$ is a Jacobi field along $c_0$. Evaluate the characteristic equation of Jacobi fields against $J$:

$$ (3.10) \qquad g_{\mathbb{X}}(\ddot{J}, J) + R_{\mathbb{X}}(J, \partial_t c, \partial_t c, J) = 0 $$

In the second term, we recognize the sectional curvature of the tangent plane spanned by $J$ and $\partial_t c$. Choose a local orthonormal chart of $T_x\mathbb{X}$ such that $B_0 = I_k \oplus O_{n-k-1}$ and
path lifting property, so it is a covering map. But $Y$ is connected and $X$ is contractible, so $\exp$ is a global diffeomorphism, $Y$ is embedded and contractible too.

It remains to prove that $f$ is a quasi-isometric embedding. In order to prove we introduce a metric on $NY$ for which the inverse map of the exponential map will be Lipschitz, and such that $Y$ sits in $NY$ totally geodesically with the induced metric by $f$:

Consider $(x, tv) \in NY$ and decompose its tangent space into $T_{(x,tv)}NY = T_t(x, \mathbb{R}v) \oplus T_xY \oplus T_v\mathbb{S}^{n-k-1}$.

Introduce the metric $h$ on $NY$ which in that decomposition is:

$$ (3.20) \qquad h_{(x,tv)} = 1 \oplus \cosh^2(t)f^*g_{\mathbb{X},x} \oplus \sinh^2(t)g_{\mathbb{S}^{n-k-1}} $$

By construction, $Y$ sits inside $NY$ totally geodesically with the metric induced by $f$, that we denote $g_Y$.

The equation on $f$ 3.19 ensures that:

$$ (3.21) \qquad \exp^* g_{\mathbb{X}} \ge 1 \oplus (\cosh(t)I_k + B(v)\sinh(t))^2 g_Y \oplus \sinh^2(t)g_{\mathbb{S}^{n-k-1}} $$

Denote $\delta = 1 - \sup |\Pi_f| > 0$. Then

$$ (3.22) \qquad \exp^* g_{\mathbb{X}} \ge \delta^2(1 \oplus \cosh^2(t)g_Y \oplus \sinh^2(t)g_{\mathbb{S}^{n-k-1}}) = \delta^2 h $$

Hence the inverse map of the exponential map is $\frac{1}{\delta}$-Lipschitz.

As a consequence, $f$ is a quasi-isometric embedding, as it satisfies:

$$ (3.23) \qquad d_{\mathbb{X}}(f(x_1), f(x_2)) \le d_Y(x_1, x_2) \le \frac{1}{\delta} d_{\mathbb{X}}(f(x_1), f(x_2)) $$

Combined with the fact that $Y$ has negatively pinched sectional curvature proven in lemma 3.2, it ensures that $f$ extends at infinity to an embedding from a sphere $\mathbb{S}^{k-1} \to \partial_{\infty}\mathbb{X}$. $\square$

As a corollary of the computations we made, we have some explicit estimates of the geometry of $NY$, endowed with the induced metric by $\exp$.

**Corollary 3.4.** Let $f: Y \to X$ be a complete almost-fuchsian immersion, with $X$ a Hadamard space. Denote by $N_tY$ the distance t hypersurface to $Y$. At a point $\exp_x(tv) \in N_tY$, consider $\lambda_1 \le \dots \le \lambda_k$ the eigenvalues of $B(v)$ the shape operator of $Y$ evaluated at $(x, v)$.

Then the second fundamental form $\Pi_t$ of $N_tY$ has eigenvalues $\lambda_1^t \le \dots \le \lambda_{n-1}^t$ which satisfy

$$ (3.24) \qquad \forall i \le k, \quad \sum_{j=1}^{i} \lambda_j^t \ge \sum_{j=1}^{i} \frac{\lambda_j + \tanh(t)}{1 + \lambda_j \tanh(t)} $$

$$ (3.25) \qquad \forall i > k, \quad \sum_{j=1}^{i} \lambda_j^t \ge \sum_{j=1}^{k} \frac{\lambda_j + \tanh(t)}{1 + \lambda_j \tanh(t)} + (i - k) \frac{1}{\tanh(t)} $$

In particular, $N_tY$ is $k$-convex in the sense of Sha [Sha86].

*Proof.* This is a consequence of the control on $f/f$ gotten in 3.19. The trace of $\Pi_t$ on a $k$-plane $P$ is the sum of $\frac{1}{2}\frac{f'(t)}{f(t)}$ for $w$ spanning a basis of the plane $P$.

The almost-fuchsian condition ensures that the sum of the $\lambda_i$ is 0, and that they live in a compact set of $(-1, 1)$, which ensures the $k$-convexity of $\Pi_t$. $\square$
path lifting property, so it is a covering map. But $Y$ is connected and $X$ is contractible, so $\exp$ is a global diffeomorphism, $Y$ is embedded and contractible too.

It remains to prove that $f$ is a quasi-isometric embedding. In order to prove we introduce a metric on $NY$ for which the inverse map of the exponential map will be Lipschitz, and such that $Y$ sits in $NY$ totally geodesically with the induced metric by $f$:

Consider $(x, tv) \in NY$ and decompose its tangent space into $T_{(x,tv)}NY = T_t(x, \mathbb{R}v) \oplus T_xY \oplus T_v\mathbb{S}^{n-k-1}$.

Introduce the metric $h$ on $NY$ which in that decomposition is:

$$ (3.20) \qquad h_{(x,tv)} = 1 \oplus \cosh^2(t)f^*g_{\mathbb{X},x} \oplus \sinh^2(t)g_{\mathbb{S}^{n-k-1}} $$

By construction, $Y$ sits inside $NY$ totally geodesically with the metric induced by $f$, that we denote $g_Y$.

The equation on $f$ 3.19 ensures that:

$$ (3.21) \qquad \exp^* g_{\mathbb{X}} \ge 1 \oplus (\cosh(t)I_k + B(v)\sinh(t))^2 g_Y \oplus \sinh^2(t)g_{\mathbb{S}^{n-k-1}} $$

Denote $\delta = 1 - \sup |\Pi_f| > 0$. Then

$$ (3.22) \qquad \exp^* g_{\mathbb{X}} \ge \delta^2(1 \oplus \cosh^2(t)g_Y \oplus \sinh^2(t)g_{\mathbb{S}^{n-k-1}}) = \delta^2 h $$

Hence the inverse map of the exponential map is $\frac{1}{\delta}$-Lipschitz.

As a consequence, $f$ is a quasi-isometric embedding, as it satisfies:

$$ (3.23) \qquad d_{\mathbb{X}}(f(x_1), f(x_2)) \le d_Y(x_1, x_2) \le \frac{1}{\delta} d_{\mathbb{X}}(f(x_1), f(x_2)) $$

Combined with the fact that $Y$ has negatively pinched sectional curvature proven in lemma 3.2, it ensures that $f$ extends at infinity to an embedding from a sphere $\mathbb{S}^{k-1} \to \partial_{\infty}\mathbb{X}$. $\square$

As a corollary of the computations we made, we have some explicit estimates of the geometry of $NY$, endowed with the induced metric by $\exp$.

**Corollary 3.4.** Let $f: Y \to X$ be a complete almost-fuchsian immersion, with $X$ a Hadamard space. Denote by $N_tY$ the distance t hypersurface to $Y$. At a point $\exp_x(tv) \in N_tY$, consider $\lambda_1 \le \dots \le \lambda_k$ the eigenvalues of $B(v)$ the shape operator of $Y$ evaluated at $(x, v)$.

Then the second fundamental form $\Pi_t$ of $N_tY$ has eigenvalues $\lambda_1^t \le \dots \le \lambda_{n-1}^t$ which satisfy

$$ (3.24) \qquad \forall i \le k, \quad \sum_{j=1}^{i} \lambda_j^t \ge \sum_{j=1}^{i} \frac{\lambda_j + \tanh(t)}{1 + \lambda_j \tanh(t)} $$

$$ (3.25) \qquad \forall i > k, \quad \sum_{j=1}^{i} \lambda_j^t \ge \sum_{j=1}^{k} \frac{\lambda_j + \tanh(t)}{1 + \lambda_j \tanh(t)} + (i - k) \frac{1}{\tanh(t)} $$

In particular, $N_tY$ is $k$-convex in the sense of Sha [Sha86].

*Proof.* This is a consequence of the control on $f/f$ gotten in 3.19. The trace of $\Pi_t$ on a $k$-plane $P$ is the sum of $\frac{1}{2}\frac{f'(t)}{f(t)}$ for $w$ spanning a basis of the plane $P$.

The almost-fuchsian condition ensures that the sum of the $\lambda_i$ is 0, and that they live in a compact set of $(-1, 1)$, which ensures the $k$-convexity of $\Pi_t$. $\square$
# 4. THE ASYMPTOTIC PLATEAU PROBLEM

The Plateau problem was introduced in 1847, originally about a mathematical proof of the existence of soap films bounding a wire frame. It was solved by Douglas [Dou31] and Rado [Rad30]. Here we are interested in a noncompact analogous problem: the asymptotic Plateau problem. Fix $\mathcal{X}$ a Hadamard space of sectional curvature less than $-1$, the asymptotic Plateau problem can be formulated as

**Problem 4.1 (Asymptotic Plateau Problem).** Let $S$ be a $(k-1)$-sphere embedded in $\partial_{\infty}\mathcal{X}$. Is there a minimal $k$-dimensional submanifold $M$ of $\mathcal{X}$ asymptotically bounding $S$? Is $M$ unique?

The asymptotic Plateau problem has been studied by Anderson [And83]. The problem has since had a rich history, see the survey of Cozkunuzer [Cos09].

Recently, Huang-Lowe-Seppi [HLS23] considered the asymptotic Plateau problem for a class of almost-fuchsian submanifolds of $\mathbb{H}^n$. If a sphere in $\partial_{\infty}\mathbb{H}^n$ bounds an almost-fuchsian hypersurface, then it is the unique minimal hypersurface bounding it. Jiang [Jia21] also solved the asymptotic Plateau problem for Jordan curves bounding an almost-fuchsian disc in $\partial_{\infty}\mathbb{H}^n$. In this section, we generalize these results to any sphere bounding an almost-fuchsian submanifold in a space $\mathcal{X}$, under the assumption that $\mathcal{X}$ has negatively pinched curvature.

**Theorem 4.2 (Asymptotic Plateau Problem for almost-fuchsian spheres).** Let $\mathcal{X}$ be a negatively pinched Hadamard space with sectional curvature less or equal than $-1$. Let $Y \subset \mathcal{X}$ be an almost-fuchsian submanifold of dimension $k$, bounding at infinity a $(k-1)$-dimensional sphere $S \subset \partial_{\infty}\mathcal{X}$. Then $Y$ is the unique complete $k$-dimensional minimal submanifold of $\mathcal{X}$ bounding $S$.

In all the remainder of the article, $\mathcal{X}$ is assumed to have pinched sectional curvature between $-C$ and $-1$. $Y \subset \mathcal{X}$ is a complete almost-fuchsian $k$-dimensional submanifold, and $S$ is its asymptotic boundary in $\partial_{\infty}\mathcal{X}$.

The first step in the proof is that eventual other minimal submanifolds bounding $S$ remain at bounded distance from $Y$:

**Proposition 4.3.** Let $Z$ be a minimal submanifold of $\mathcal{X}$ bounding $S$. Then $Z \subset \mathcal{N}_rY$, where $\mathcal{N}_rY$ is the $r$-uniform neighborhood of $Y$, and $r = \tanh^{-1}(1 - \sup |\mathbb{I}_f|)$.

*Proof.* Thanks to theorem 2.1, we know that $Z$ is included in the convex hull of $S$. But the explicit bounds shown in corollary 3.4 show that for $r \ge \tanh^{-1}(1 - \sup |\mathbb{I}_f|)$, the uniform neighborhoods $\mathcal{N}_r(Y)$ are convex, and contain $Y$ so bound $S$ too. As a consequence, $\mathcal{N}_r(Y)$ contains the convex hull of $Y$ which contains $Z$. $\square$

*Remark 4.1.* The same corollary 3.4 shows that all uniform neighborhoods are $k$-convex, which directly implies that the function $z \in Z \mapsto d(z, Y)$ cannot have a local maximum. When the induced metric of $Z$ has lower bounded sectional curvature, we could apply the Omori-Yau [Omo67, Yau75] maximum principle to conclude that $Z = Y$. Here we don't prove that $Z$ has lower bounded sectional curvature, but we prove that the maximum principle is still applicable. This property is called *stochastical completeness*, cf Pigola-Rigoli-Setti [PRS05].

We want to apply a maximum principle to the distance function to $Y$ restricted on a minimal submanifold $Z$, so we first prove it satisfies a strong subharmonicity condition.
# 4. THE ASYMPTOTIC PLATEAU PROBLEM

The Plateau problem was introduced in 1847, originally about a mathematical proof of the existence of soap films bounding a wire frame. It was solved by Douglas [Dou31] and Rado [Rad30]. Here we are interested in a noncompact analogous problem: the asymptotic Plateau problem. Fix $\mathcal{X}$ a Hadamard space of sectional curvature less than $-1$, the asymptotic Plateau problem can be formulated as

**Problem 4.1 (Asymptotic Plateau Problem).** Let $S$ be a $(k-1)$-sphere embedded in $\partial_{\infty}\mathcal{X}$. Is there a minimal $k$-dimensional submanifold $M$ of $\mathcal{X}$ asymptotically bounding $S$? Is $M$ unique?

The asymptotic Plateau problem has been studied by Anderson [And83]. The problem has since had a rich history, see the survey of Cozkunuzer [Cos09].

Recently, Huang-Lowe-Seppi [HLS23] considered the asymptotic Plateau problem for a class of almost-fuchsian submanifolds of $\mathbb{H}^n$. If a sphere in $\partial_{\infty}\mathbb{H}^n$ bounds an almost-fuchsian hypersurface, then it is the unique minimal hypersurface bounding it. Jiang [Jia21] also solved the asymptotic Plateau problem for Jordan curves bounding an almost-fuchsian disc in $\partial_{\infty}\mathbb{H}^n$. In this section, we generalize these results to any sphere bounding an almost-fuchsian submanifold in a space $\mathcal{X}$, under the assumption that $\mathcal{X}$ has negatively pinched curvature.

**Theorem 4.2 (Asymptotic Plateau Problem for almost-fuchsian spheres).** Let $\mathcal{X}$ be a negatively pinched Hadamard space with sectional curvature less or equal than $-1$. Let $Y \subset \mathcal{X}$ be an almost-fuchsian submanifold of dimension $k$, bounding at infinity a $(k-1)$-dimensional sphere $S \subset \partial_{\infty}\mathcal{X}$. Then $Y$ is the unique complete $k$-dimensional minimal submanifold of $\mathcal{X}$ bounding $S$.

In all the remainder of the article, $\mathcal{X}$ is assumed to have pinched sectional curvature between $-C$ and $-1$. $Y \subset \mathcal{X}$ is a complete almost-fuchsian $k$-dimensional submanifold, and $S$ is its asymptotic boundary in $\partial_{\infty}\mathcal{X}$.

The first step in the proof is that eventual other minimal submanifolds bounding $S$ remain at bounded distance from $Y$:

**Proposition 4.3.** Let $Z$ be a minimal submanifold of $\mathcal{X}$ bounding $S$. Then $Z \subset \mathcal{N}_rY$, where $\mathcal{N}_rY$ is the $r$-uniform neighborhood of $Y$, and $r = \tanh^{-1}(1 - \sup |\mathbb{I}_f|)$.

*Proof.* Thanks to theorem 2.1, we know that $Z$ is included in the convex hull of $S$. But the explicit bounds shown in corollary 3.4 show that for $r \ge \tanh^{-1}(1 - \sup |\mathbb{I}_f|)$, the uniform neighborhoods $\mathcal{N}_r(Y)$ are convex, and contain $Y$ so bound $S$ too. As a consequence, $\mathcal{N}_r(Y)$ contains the convex hull of $Y$ which contains $Z$. $\square$

*Remark 4.1.* The same corollary 3.4 shows that all uniform neighborhoods are $k$-convex, which directly implies that the function $z \in Z \mapsto d(z, Y)$ cannot have a local maximum. When the induced metric of $Z$ has lower bounded sectional curvature, we could apply the Omori-Yau [Omo67, Yau75] maximum principle to conclude that $Z = Y$. Here we don't prove that $Z$ has lower bounded sectional curvature, but we prove that the maximum principle is still applicable. This property is called *stochastical completeness*, cf Pigola-Rigoli-Setti [PRS05].

We want to apply a maximum principle to the distance function to $Y$ restricted on a minimal submanifold $Z$, so we first prove it satisfies a strong subharmonicity condition.
**Proposition 4.4.** Let $Z \subset \mathbb{X}$ be a minimal $k$-dimensional submanifold. Consider the function $u: z \in Z \mapsto d(z, Y)^2$. Then there is $C > 0$ such that

$$ (4.1) \qquad \Delta u \geq Cu $$

*Proof.* Consider $d$ the distance function to $Y$, defined on the whole space $X$. Because $Y$ is almost-fuchsian, for any point of $\mathbb{X} - Y$, the distance is attained at a unique point of $\mathbb{X}$, so the function $d$ is smooth on $X - Y$.

Furthermore, at a point $x$ where $d(x) = t > 0$, we have the formula

$$ (4.2) \qquad \nabla^2 d = 0 \oplus \mathbb{I}_{N_t(Y)} $$

the $0$ is simply because $d$ is linear in the direction of the minimizing geodesic to $Y$. Using the formula

$$ (4.3) \qquad \nabla^2(d^2) = 2\nabla d \cdot \nabla d + 2d\nabla^2 d $$

We deduce that

$$ (4.4) \qquad \nabla^2(u) \geq 2 \oplus 2d \mathbb{I}_{N_t(Y)} $$

Now, because $Z$ is a minimal submanifold of $X$, the laplacian of the restriction of $u$ equals the trace of the restriction of its Hessian to the tangent space of $Z$

$$ (4.5) \qquad \Delta(u|_Z) = \mathrm{Tr}((\nabla^2 u)|_{TZ}) $$

Hence the bounds on the eigenvalues of $\mathbb{I}_{N_t(Y)}$ computed in corollary 3.4 and the control

$$ (4.6) \qquad \frac{\tanh(t) + \lambda_i}{1 + \lambda_i \tanh(t)} \leq 1 $$

allow us to get the control:

$$ (4.7) \qquad \Delta(u|_Z) \geq 2d \inf_{\lambda_1 + \dots + \lambda_k = 0, |\lambda_i| \leq 1 - \epsilon} \sum_{i=1, \dots, k} \frac{\tanh(d) + \lambda_i}{1 + \tanh(d)\lambda_i} = 2d\Phi(d) $$

Remark that $\Phi(d)$ is continuous, positive, vanishes only when $d = 0$, and satisfies

$$ (4.8) \qquad \Phi'(0) \geq k(1 - (1 - \sup |\mathbb{I}_f|)^2) > 0 $$

As $d$ is bounded by a constant $r = \tanh^{-1}(1 - \sup |\mathbb{I}_f|)$, there is a constant $C > 0$ such that on $[0, r]$, $\Phi(d) \geq Cd$. We deduce that the restriction of $u$ to $Z$ satisfies

$$ (4.9) \qquad \Delta u \geq 2Cu $$

, as claimed.

□

Now we want to prove that we can apply the maximum principle to $u$. In order to do so, we will use the Khas'minskii test [Km60], as stated in Theorem 3.1 and Proposition 3.2. of [PRS05].

**Theorem 4.5.** Let $M, g$ be a Riemannian manifold, and assume that $M$ supports a $C^2$ function $\gamma$, which tends to infinity at infinity, and satisfies

$$ (4.10) \qquad \Delta\gamma \leq \lambda\gamma \text{ off a compact set} $$

for some $\lambda > 0$. Then $M, g$ is stochastically complete, or equivalently for every $\lambda > 0$ the only non-negative bounded smooth solution $u$ of $\Delta u \geq \lambda u$ on $M$ is zero.
**Proposition 4.4.** Let $Z \subset \mathbb{X}$ be a minimal $k$-dimensional submanifold. Consider the function $u: z \in Z \mapsto d(z, Y)^2$. Then there is $C > 0$ such that

$$ (4.1) \qquad \Delta u \geq Cu $$

*Proof.* Consider $d$ the distance function to $Y$, defined on the whole space $X$. Because $Y$ is almost-fuchsian, for any point of $\mathbb{X} - Y$, the distance is attained at a unique point of $\mathbb{X}$, so the function $d$ is smooth on $X - Y$.

Furthermore, at a point $x$ where $d(x) = t > 0$, we have the formula

$$ (4.2) \qquad \nabla^2 d = 0 \oplus \mathbb{I}_{N_t(Y)} $$

the $0$ is simply because $d$ is linear in the direction of the minimizing geodesic to $Y$. Using the formula

$$ (4.3) \qquad \nabla^2(d^2) = 2\nabla d \cdot \nabla d + 2d\nabla^2 d $$

We deduce that

$$ (4.4) \qquad \nabla^2(u) \geq 2 \oplus 2d \mathbb{I}_{N_t(Y)} $$

Now, because $Z$ is a minimal submanifold of $X$, the laplacian of the restriction of $u$ equals the trace of the restriction of its Hessian to the tangent space of $Z$

$$ (4.5) \qquad \Delta(u|_Z) = \mathrm{Tr}((\nabla^2 u)|_{TZ}) $$

Hence the bounds on the eigenvalues of $\mathbb{I}_{N_t(Y)}$ computed in corollary 3.4 and the control

$$ (4.6) \qquad \frac{\tanh(t) + \lambda_i}{1 + \lambda_i \tanh(t)} \leq 1 $$

allow us to get the control:

$$ (4.7) \qquad \Delta(u|_Z) \geq 2d \inf_{\lambda_1 + \dots + \lambda_k = 0, |\lambda_i| \leq 1 - \epsilon} \sum_{i=1, \dots, k} \frac{\tanh(d) + \lambda_i}{1 + \tanh(d)\lambda_i} = 2d\Phi(d) $$

Remark that $\Phi(d)$ is continuous, positive, vanishes only when $d = 0$, and satisfies

$$ (4.8) \qquad \Phi'(0) \geq k(1 - (1 - \sup |\mathbb{I}_f|)^2) > 0 $$

As $d$ is bounded by a constant $r = \tanh^{-1}(1 - \sup |\mathbb{I}_f|)$, there is a constant $C > 0$ such that on $[0, r]$, $\Phi(d) \geq Cd$. We deduce that the restriction of $u$ to $Z$ satisfies

$$ (4.9) \qquad \Delta u \geq 2Cu $$

, as claimed.

□

Now we want to prove that we can apply the maximum principle to $u$. In order to do so, we will use the Khas'minskii test [Km60], as stated in Theorem 3.1 and Proposition 3.2. of [PRS05].

**Theorem 4.5.** Let $M, g$ be a Riemannian manifold, and assume that $M$ supports a $C^2$ function $\gamma$, which tends to infinity at infinity, and satisfies

$$ (4.10) \qquad \Delta\gamma \leq \lambda\gamma \text{ off a compact set} $$

for some $\lambda > 0$. Then $M, g$ is stochastically complete, or equivalently for every $\lambda > 0$ the only non-negative bounded smooth solution $u$ of $\Delta u \geq \lambda u$ on $M$ is zero.
Note that the equivalence between stochastical completeness and the applicability of the maximum principle is due to Grigor'Yan [Gri99].

The following proposition will ensure that we can apply our maximum principle on Z:

**Proposition 4.6.** Let $Z \subset \mathbb{X}$ be a proper submanifold of $\mathbb{X}$ complete contractible space with sectional curvature pinched between $-b$ and $-a$, $b \ge a \ge 0$. Fix $p \in \mathbb{X}$ and denote $f$ the distance function to the point $p$. Then out of a compact set $K$, there is a constant $C > 0$ such that the restriction of $f$ to $Z$ satisfies

$$ (4.11) \qquad \Delta(f|_Z) \le C $$

*Proof.* As $\mathbb{X}$ is negatively pinched, there are $b \ge a \ge 0$ such that $-b \le K_{\mathbb{X}} \le -a$. By the Hessian comparison Theorem, the Hessian of $f$ is bounded between the hessian of the distance functions in the spaceforms of sectional curvature $-b$ and $-a$. Explicitely, this means

$$ (4.12) \qquad a \coth(af)g_{\mathbb{X}} \le \nabla^2 f \le b \coth(bf)g_{\mathbb{X}} $$

If $a$ is zero, one has to replace $a \coth(af)$ by $\frac{1}{f}$, but it doesn't change anything to the proof. In particular, out of a compact set $K$ containing $p$, the hessian of $f$ is bounded by a constant $C$. Now because $Z$ is proper, the intersection $K \cap Z$ is a compact subset of $Z$. Also, because $Z$ is minimal, the laplacian of the restriction of $f$ satisfies

$$ (4.13) \qquad \Delta(f|_Z) = \mathrm{Tr}(\nabla^2 f|_{TZ}) \le C \dim Z. $$

as desired. □

Applying theorem 4.5, we deduce that the maximum principle is applicable on a proper submanifold of such a space $\mathbb{X}$.

**Corollary 4.7.** Let $\mathbb{X}$ be a complete contractible space whose sectional curvature is bounded between $-b$ and $0$ for some $b > 0$. Let $Z \subset \mathbb{X}$ a proper minimal submanifold. Then for any $u \in C^2(Z)$ such that $\sup u < \infty$, for every $c < \sup u$,

$$ (4.14) \qquad \inf_{z \in Z: u(z) > c} \Delta u \le 0 $$

*Proof.* Thanks to proposition 4.6, the function $f|_Z$ tends to $\infty$ at $\infty$, and out of a compact set $K$, it satisfies

$$ (4.15) \qquad \Delta f \le C \le C' f $$

for some constant $C' > 0$. In particular, we can apply theorem 4.5 to deduce that $Z$ is stochastically complete, which is equivalent to the statement of our corollary by Theorem 3.1. of [PRS05]. □

We now have everything needed to prove our main theorem.

*Proof of theorem 4.2.* Introduce $u: Z \to \mathbb{R}$, such that $u(z) = d(z, Y)^2$. Thanks to proposition 4.3, $u$ is bounded. But thanks to proposition 4.4, there is $C > 0$ such that

$$ (4.16) \qquad \Delta u \ge Cu $$

Now because of corollary 4.7, we can apply the maximum principle on $u$ to get that $u = 0$. Hence $Z \subset Y$, and by completeness $Z = Y$, as desired. □
Note that the equivalence between stochastical completeness and the applicability of the maximum principle is due to Grigor'Yan [Gri99].

The following proposition will ensure that we can apply our maximum principle on Z:

**Proposition 4.6.** Let $Z \subset \mathbb{X}$ be a proper submanifold of $\mathbb{X}$ complete contractible space with sectional curvature pinched between $-b$ and $-a$, $b \ge a \ge 0$. Fix $p \in \mathbb{X}$ and denote $f$ the distance function to the point $p$. Then out of a compact set $K$, there is a constant $C > 0$ such that the restriction of $f$ to $Z$ satisfies

$$ (4.11) \qquad \Delta(f|_Z) \le C $$

*Proof.* As $\mathbb{X}$ is negatively pinched, there are $b \ge a \ge 0$ such that $-b \le K_{\mathbb{X}} \le -a$. By the Hessian comparison Theorem, the Hessian of $f$ is bounded between the hessian of the distance functions in the spaceforms of sectional curvature $-b$ and $-a$. Explicitely, this means

$$ (4.12) \qquad a \coth(af)g_{\mathbb{X}} \le \nabla^2 f \le b \coth(bf)g_{\mathbb{X}} $$

If $a$ is zero, one has to replace $a \coth(af)$ by $\frac{1}{f}$, but it doesn't change anything to the proof. In particular, out of a compact set $K$ containing $p$, the hessian of $f$ is bounded by a constant $C$. Now because $Z$ is proper, the intersection $K \cap Z$ is a compact subset of $Z$. Also, because $Z$ is minimal, the laplacian of the restriction of $f$ satisfies

$$ (4.13) \qquad \Delta(f|_Z) = \mathrm{Tr}(\nabla^2 f|_{TZ}) \le C \dim Z. $$

as desired. □

Applying theorem 4.5, we deduce that the maximum principle is applicable on a proper submanifold of such a space $\mathbb{X}$.

**Corollary 4.7.** Let $\mathbb{X}$ be a complete contractible space whose sectional curvature is bounded between $-b$ and $0$ for some $b > 0$. Let $Z \subset \mathbb{X}$ a proper minimal submanifold. Then for any $u \in C^2(Z)$ such that $\sup u < \infty$, for every $c < \sup u$,

$$ (4.14) \qquad \inf_{z \in Z: u(z) > c} \Delta u \le 0 $$

*Proof.* Thanks to proposition 4.6, the function $f|_Z$ tends to $\infty$ at $\infty$, and out of a compact set $K$, it satisfies

$$ (4.15) \qquad \Delta f \le C \le C'f $$

for some constant $C' > 0$. In particular, we can apply theorem 4.5 to deduce that $Z$ is stochastically complete, which is equivalent to the statement of our corollary by Theorem 3.1. of [PRS05]. □

We now have everything needed to prove our main theorem.

*Proof of theorem 4.2.* Introduce $u: Z \to \mathbb{R}$, such that $u(z) = d(z, Y)^2$. Thanks to proposition 4.3, $u$ is bounded. But thanks to proposition 4.4, there is $C > 0$ such that

$$ (4.16) \qquad \Delta u \ge Cu $$

Now because of corollary 4.7, we can apply the maximum principle on $u$ to get that $u = 0$. Hence $Z \subset Y$, and by completeness $Z = Y$, as desired. □
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[KS07] Kirill Krasnov and Jean-Marc Schlenker. Minimal surfaces and particles in 3-manifolds. *Geometriae Dedicata*, 126(1):187–254, 2007.

[Omo67] Hideki Omori. Isometric immersions of riemannian manifolds. *Journal of the Mathematical Society of Japan*, 19(2):205–214, 1967.

[PP12] Jouni Parkkonen and Frédéric Paulin. On strictly convex subsets in negatively curved manifolds. *Journal of Geometric Analysis*, 22:621–632, 2012.

[PRS05] Stefano Pigola, Marco Rigoli, and Alberto Giulio Setti. *Maximum principles on Riemannian manifolds and applications*. American Mathematical Soc., 2005.

[Rad30] Tibor Radó. On Plateau's problem. *Annals of Mathematics*, pages 457–469, 1930.

[San17] Andrew Sanders. Domains of discontinuity for almost-fuchsian groups. *Transactions of the American Mathematical Society*, 369(2):1291–1308, 2017.

[Sep16] Andrea Seppi. Minimal discs in hyperbolic space bounded by a quasicircle at infinity. *Commentarii Mathematici Helvetici*, 91(4):807–839, 2016.

[Sha86] Ji-Ping Sha. p-convex riemannian manifolds. *Inventiones mathematicae*, 83(3):437–447, 1986.

[Uhl83] Karen K Uhlenbeck. Closed minimal surfaces in hyperbolic 3-manifolds. In *Seminar on minimal submanifolds*, volume 103, pages 147–168, 1983.

[Yau75] Shing-Tung Yau. Harmonic functions on complete riemannian manifolds. *Communications on Pure and Applied Mathematics*, 28(2):201–228, 1975.
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