# Uniqueness and non-uniqueness for the asymptotic Plateau problem in
hyperbolic space

Zheng Huang  Department of Mathematics, The City University of New York,
Staten Island, NY 10314, USA  The Graduate Center, The City University of New
York, 365 Fifth Ave., New York, NY 10016, USA<EMAIL_ADDRESS>,  Ben
Lowe  Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
<EMAIL_ADDRESS>and  Andrea Seppi  Univ. Grenoble Alpes, CNRS, IF, 38000
Grenoble, France<EMAIL_ADDRESS>
###### Abstract.

We prove a number of results on the number of solutions to the asymptotic
Plateau problem in $\mathbb{H}^{3}$. In the direction of non-uniqueness, we
construct an example of a quasicircle that is the asymptotic boundary of
uncountably many pairwise distinct stable minimal disks.

Moreover, we discuss criteria that ensure uniqueness. Given a Jordan curve
$\Lambda$ in the asymptotic boundary of $\mathbb{H}^{3}$, we show that
uniqueness of the minimal surfaces with asymptotic boundary $\Lambda$ is
equivalent to uniqueness in the smaller class of stable minimal disks, and,
when $\Lambda$ is invariant by a Kleinian group, to uniqueness in the even
smaller class of group invariant stable minimal disks.

Finally, we show that if a quasicircle (or more generally, a Jordan curve of
finite width) $\Lambda$ is the asymptotic boundary of a minimal surface
$\Sigma$ with principal curvatures less than or equal to 1 in absolute value,
then uniqueness holds.

###### 2020 Mathematics Subject Classification:

Primary 53C42, 57K32

###### Contents

  1. 1 Introduction
    1. 1.1 Characterizing uniqueness
    2. 1.2 Uniqueness criteria via curvature conditions
    3. 1.3 Strong non-uniqueness
    4. 1.4 Quasiconformal constant
    5. 1.5 Organization of the paper
    6. 1.6 Acknowledgements
  2. 2 Preliminaries
    1. 2.1 Hypersurfaces theory
    2. 2.2 Convex hull and width of a Jordan curve
    3. 2.3 Stable and area-minimizing minimal hypersurfaces
    4. 2.4 Compactness results
  3. 3 Uniqueness among stable minimal disks
    1. 3.1 Outline
    2. 3.2 Adapting the arguments of Anderson
    3. 3.3 Conclusion of the proofs
  4. 4 Small curvatures and uniqueness
    1. 4.1 Small curvature conditions
    2. 4.2 Proof of Theorem 1.3 for $n=2$
    3. 4.3 Alternative argument for Theorem 1.2
  5. 5 Higher dimension and codimension
    1. 5.1 Hypersurfaces in higher dimensions
    2. 5.2 Higher codimension submanifolds
  6. 6 Uncountably many minimal disks
    1. 6.1 Constructing the Jordan curve
    2. 6.2 Topological disks
    3. 6.3 Conclusion of the proof
  7. A Small curvatures

##  1\. Introduction

The classical “asymptotic Plateau problem” asks, given a Jordan curve
$\Lambda$ on ${\mathbb{S}}^{2}_{\infty}=\partial_{\infty}\mathbb{H}^{3}$, how
to count the number of (properly embedded) minimal surfaces $\Sigma$ in
$\mathbb{H}^{3}$, if any, that are asymptotic to $\Lambda$, in the sense that
the closure of $\Sigma$ in ${\mathbb{S}}^{2}_{\infty}\cup\mathbb{H}^{3}$ is
equal to $\Lambda\cup\Sigma$. The existence of minimal disk solutions to the
asymptotic Plateau problem was obtained by Anderson ([And83]). Using geometric
measure theory, Anderson also obtained existence results in higher dimensions.

The uniqueness does not hold in general: as shown in [And83, HW15], taking
advantage of group actions, one can construct a Jordan curve $\Lambda$ in
${\mathbb{S}}^{2}_{\infty}=\partial_{\infty}\mathbb{H}^{3}$ which is the limit
set of some quasi-Fuchsian group such that $\Lambda$ spans multiple minimal
disks (even an arbitrarily large, but finite, number). Anderson ([And83]) even
constructed a curve $\Lambda$ which spans infinitely many minimal surfaces
(the surfaces he constructs have positive genus). On the other hand, when
$\Lambda$ is a round circle, the unique minimal surface it spans is a totally
geodesic disk. To look for unique solutions, it is therefore natural to
consider the class of minimal surfaces that are “close” to totally geodesic,
for which $\Lambda$ is “close” to a round circle. Related questions with
conditions on natural invariants of $\Lambda$ were studied in [Sep16] (for the
quasi-conformal constant of $\Lambda$), and [HW13, San18] (for the Hausdorff
dimension of $\Lambda$). Some properness questions for the asymptotic Plateau
problem solutions for various classes of curves were addressed, for example,
in [GS00, AM10]. For an overview of this active area of research see the
survey paper [Cos14].

Motivated by the above results and by many natural questions arising from the
study of the asymptotic Plateau problem, in this paper we address two basic
questions, which, for the sake of simplicity, we state only in dimension
three, but may naturally be extended to hypersurfaces in $\mathbb{H}^{n+1}$:

  * •

under what conditions does a Jordan curve $\Lambda$ on
${\mathbb{S}}^{2}_{\infty}$ span exactly one minimal surface in
$\mathbb{H}^{3}$?

  * •

does there exist a Jordan curve $\Lambda$ on ${\mathbb{S}}^{2}_{\infty}$ that
spans infinitely many minimal disks in $\mathbb{H}^{3}$, and if yes, which
cardinality may the set of solutions have?

###  1.1. Characterizing uniqueness

Recall that a (hyper)surface is _minimal_ if it is a critical point of the
area under compactly supported variations. It is _stable_ if moreover the
second variation of the area under any compactly supported variation is non-
negative. In this introduction, we will always implicitly consider _properly
embedded_ (hyper)surfaces. Our first theorem shows that it suffices to check
uniqueness in the class of _stable minimal disks_.

######  Theorem 1.1.

Let $\Lambda$ be a Jordan curve on
${\mathbb{S}}^{2}_{\infty}=\partial_{\infty}\mathbb{H}^{3}$. Then $\Lambda$
spans a unique minimal surface if and only if it spans a unique stable minimal
disk.

A statement similar to Theorem 1.1, but in the context of the _finite_ Plateau
problem, was proved in [MY19]. When $\Lambda$ is invariant under the action of
a Kleinian group (i.e. a discrete subgroup of isometries of $\mathbb{H}^{3}$),
which is for instance the case for the limit set of a quasi-Fuchsian group, we
can prove a stronger statement.

######  Theorem 1.2.

Let $\Lambda$ be a Jordan curve on
${\mathbb{S}}^{2}_{\infty}=\partial_{\infty}\mathbb{H}^{3}$, and let $\Gamma$
be any Kleinian group preserving $\Lambda$. Then $\Lambda$ spans a unique
minimal surface if and only if it spans a unique $\Gamma$-invariant stable
minimal disk.

The main idea in the proof of Theorem 1.1 is an adaptation of an argument in
[And83, Theorem 3.1]: we show that if there is a minimal surface, which is not
stable or is not topologically a disk, with asymptotic boundary $\Lambda$,
then we can construct two distinct — actually, disjoint — stable minimal disks
with the same asymptotic boundary $\Lambda$ (Theorem 3.1). The proof of
Theorem 1.2 then relies on a further improvement of the arguments of [And83,
Theorem 3.1], showing that if there is a non-invariant stable minimal disk,
then we can construct two disjoint invariant stable minimal disks (Theorem
3.2).

###  1.2. Uniqueness criteria via curvature conditions

Next, we turn our attention to sufficient conditions for uniqueness. For an
immersed hypersurface in $\mathbb{H}^{n+1}$, or more generally in a hyperbolic
($n+1$)-manifold, we say it has strongly small curvature if its principal
curvatures $\\{\lambda_{i}\\}$ satisfy that

(1.1) $|\lambda_{i}|\leq 1-\epsilon,\ i=1,\ldots,n,\ \ \ \text{for some
small}\ \ \epsilon>0.$

Similarly we say it has small curvature if $|\lambda_{i}|<1$, and it has
weakly small curvature if $|\lambda_{i}|\leq 1$. This definition has some
immediate consequences: for instance, a complete immersion of weakly small
curvature is in fact a properly embedded topological disk ([Eps84, Eps86], see
also [EES22] for a generalization). See Section 4 and Appendix A for more
details.

Surfaces of small curvature are very special in three-manifold theory:
Thurston observed that a closed surface of small curvature in a complete
hyperbolic three-manifold is incompressible ([Thu86, Lei06]); they are
abundant in closed hyperbolic three-manifolds ([KM12]); many results have been
extended to the study of complete noncompact hyperbolic three-manifold of
finite volume ([Rub05, CF19, KW21]). It is often favorable to consider
canonical representatives within a homotopy class of surfaces, and minimal
surfaces are in many ways the most natural choice. Among hyperbolic three-
manifolds, _almost-Fuchsian_ manifolds are quasi-Fuchsian manifolds which
admit a closed minimal surface of small curvature. This notion was introduced
by Uhlenbeck and it played an important role in her study of parametrization
of the moduli space of minimal surfaces in hyperbolic three-manifolds
([Uhl83]). Subsequently many different aspects of this subclass of quasi-
Fuchsian manifold have been studied up to recent years (for instance [KS07,
GHW10, HL21] and many others).

It is known ([Uhl83]) that any almost-Fuchsian manifold admits a unique closed
minimal surface — in other words, identifying the almost-Fuchsian manifold
with a quotient $\mathbb{H}^{3}/\Gamma$, the limit set $\Lambda$ of the group
$\Gamma$ bounds a unique $\Gamma$-invariant minimal disk asymptotic to
$\Lambda$. Inspired by this fact, we prove (Corollary 1.4 below) that if a
Jordan curve $\Lambda$ (not necessarily group equivariant) spans a minimal
disk $\Sigma$ of strongly small curvature in $\mathbb{H}^{3}$, then $\Sigma$
is the unique minimal surface asymptotic to $\Lambda$. Our results, however,
are more general. The main result we prove in this direction is the following:

######  Theorem 1.3.

Let $\Lambda$ be a topologically embedded $n$-sphere on
${\mathbb{S}}^{n}_{\infty}=\partial_{\infty}\mathbb{H}^{n+1}$ of finite width,
and let $\Sigma$ be a minimal hypersurface in $\mathbb{H}^{n+1}$ of weakly
small curvature asymptotic to $\Lambda$. Then $\Sigma$ is the unique minimal
hypersurface in $\mathbb{H}^{n+1}$ asymptotic to $\Lambda$. Moreover, $\Sigma$
is area-minimizing.

Let us explain the terminology of the statement. First, recall that a
hypersurface $\Sigma$ is _area-minimizing_ if any compact codimension-zero
submanifold with boundary has smaller area than any rectifiable hypersurface
with the same boundary in the ambient space. This implies that $\Sigma$ is a
stable minimal hypersurface. Second, the _width_ of a Jordan curve $\Lambda$
in ${\mathbb{S}}^{2}_{\infty}=\partial_{\infty}\mathbb{H}^{3}$ has been
introduced in [BDMS21] — and the definition is immediately extended to higher
dimensions — as the supremum over all points in the convex hull of $\Lambda$
of the sum of the distances from each boundary component of the convex hull.

We include two proofs of Theorem 1.3 as they offer different perspectives. The
first proof only works for $n=2$, but it is based on elementary geometric
arguments, which we sketch here. The first observation is that the finite
width condition implies that every minimal surface $\Sigma^{\prime}$ has,
roughly speaking, finite normal distance from $\Sigma$. Moreover, by Theorem
1.1, it suffices to show uniqueness among _stable_ minimal disks. Now, if the
maximum normal distance between $\Sigma$ and another stable minimal surface is
realized in the interior, the conclusion follows from a maximum principle
argument, taking advantage the properties of the normal flow from $\Sigma$. If
the maximum normal distance is not realized in the interior, we use isometries
to send the run-away sequence of points back to a fixed point and use the
compactness theorems for stable minimal surfaces, so as to reduce essentially
to the previous case. The second proof, which works in any dimension, rests on
an application of a more general maximum principle proved in [Whi10] in the
context of minimal varifolds (see also [JT03]).

We now derive several corollaries of Theorem 1.3. Firstly, observe that if a
properly embedded hypersurface $\Sigma$ has strongly small curvatures, then
its asymptotic boundary has finite width (See Lemma A.4 in Appendix A). Hence
we obtain:

######  Corollary 1.4.

Let $\Lambda$ be a topologically embedded $n$-sphere on
${\mathbb{S}}^{n}_{\infty}=\partial_{\infty}\mathbb{H}^{n+1}$, and let
$\Sigma$ be a minimal hypersurface in $\mathbb{H}^{n+1}$ of strongly small
curvature asymptotic to $\Lambda$. Then $\Sigma$ is the unique minimal
hypersurface in $\mathbb{H}^{n+1}$ asymptotic to $\Lambda$. Moreover, $\Sigma$
is area-minimizing.

Secondly, in dimension $n=2$, quasicircles are an important class of Jordan
curves, which are known to have finite width. We thus obtain immediately the
following.

######  Corollary 1.5.

Let $\Lambda$ be a quasicircle on
${\mathbb{S}}^{2}_{\infty}=\partial_{\infty}\mathbb{H}^{3}$, and let $\Sigma$
be a minimal surface in $\mathbb{H}^{3}$ of weakly small curvature asymptotic
to $\Lambda$. Then $\Sigma$ is the unique minimal surface in $\mathbb{H}^{3}$
asymptotic to $\Lambda$. Moreover, $\Sigma$ is area-minimizing.

We remark that the setting of Theorems 1.1 and 1.2 is more general than
Theorem 1.3 and Corollary 1.5. Indeed, it follows from [HL21, Theorem 5.2]
that there are examples of quasi-Fuchsian groups $\Gamma$ whose limit set
$\Lambda$ bounds a unique $\Gamma$-invariant stable minimal disk $\Sigma$
(hence, by Theorem 1.2, a unique minimal surface), but $\Sigma$ does not have
weakly small curvature.

###  1.3. Strong non-uniqueness

Now we turn to the other extreme case: we aim to construct a Jordan curve on
$\mathbb{S}^{2}_{\infty}$ which spans _a lot_ of minimal disks. To give some
context, Hass-Thurston conjectured that no closed hyperbolic 3-manifold admits
a foliation by minimal surfaces. Anderson even conjectured ([And83]) that no
hyperbolic 3-manifold admits a local 1-parameter family of closed minimal
surfaces, and proved this statement for quasi-Fuchsian hyperbolic 3-manifolds.
It has been a folklore conjecture that no Jordan curve in
${\mathbb{S}}^{2}_{\infty}=\partial_{\infty}\mathbb{H}^{3}$ asymptotically
bounds a 1-parameter family of minimal surfaces. The full extent of these
conjectures remain as major open questions in the field.

What has been proven up to this point tends to support these conjectures.
Huang-Wang [HW19] and Hass [Has15] made progress on the Hass-Thurston
conjecture for certain fibered closed hyperbolic 3-manifolds containing short
geodesics; Wolf-Wu ([WW20]) ruled out so-called geometric local 1-parameter
families of closed minimal surfaces; it follows from the work of Alexakis-
Mazzeo [AM10] that a generic $C^{3,\alpha}$ simple closed curve in the
boundary at infinity of $\mathbb{H}^{3}$ bounds only finitely many surfaces of
any given finite genus; Coskunuzer proved that generic simple closed curves in
$\partial_{\infty}\mathbb{H}^{3}$ bound unique area-minimizing surfaces
[Cos11].

On the other hand what we prove, while compatible with the folklore
conjecture, is in the other direction. Based on the aforementioned results,
one might be tempted to strengthen the folklore conjecture to the statement
that any Jordan curve in
${\mathbb{S}}^{2}_{\infty}=\partial_{\infty}\mathbb{H}^{3}$ bounds at most
countably many minimal surfaces. We show that this stronger statement is
false:

######  Theorem 1.6.

There exists a quasicircle in
${\mathbb{S}}^{2}_{\infty}=\partial_{\infty}\mathbb{H}^{3}$ spanning
uncountably many pairwise distinct stable minimal disks.

Let us emphasize some important features of the construction of this extreme
curve $\Lambda$. In ([And83]), Anderson constructed a Jordan curve which is
the limit set of a quasi-Fuchsian group (hence continuous but almost nowhere
differentiable) such that it spans infinitely many minimal surfaces, one of
which is a minimal disk. In [HW15], for each integer $N>1$, also using the
limit set of a quasi-Fuchsian group, an extreme curve spanning at least
$2^{N}$ distinct minimal disks was constructed. However, Anderson ([And83])
has shown that any quasi-Fuchsian manifold only admits finitely many least
area closed minimal surfaces diffeomorphic to the fiber, which poses a
possible limitation on how much one can improve the aforementioned
constructions to find infinitely many minimal disks if one insists on using
the limit set of some quasi-Fuchsian group as the curve at infinity. The
starting point of our construction is similar to the ideas in [HW15], but the
Jordan curve is constructed in such a way to allow an improvement of the
argument, leading to $2^{\mathbb{N}}$ pairwise distinct minimal disk.
Moreover, since this Jordan curve is not invariant under any quasi-Fuchsian
group, we must adopt a different approach in order to produce the minimal
disks, namely, in a spirit similar to the proofs of Theorems 1.1 and 1.2, we
take the limit of a sequence of solutions of the finite Plateau problems
inside $\mathbb{H}^{3}$.

###  1.4. Quasiconformal constant

We conclude this introduction with an improvement of the curvature estimates
obtained in [Sep16] in terms of quasiconformal constants, by a direct
application of Corollary 1.5. More concretely, [Sep16, Theorem A] showed that
there exist universal constants $C>0$ and $K_{0}>1$ such that any stable
minimal disk in $\mathbb{H}^{3}$ with asymptotic boundary a $K$-quasicircle,
for $K<K_{0}$, has principal curvatures bounded in absolute value by $C\log
K$. This result has been recently applied in several directions, see [Bis19,
Low21, CMN22].

The proof, however, relies on the application of compactness for minimal
surfaces, and therefore requires stability. However, when $K$ is sufficiently
small, the principal curvatures of the area-minimizing (hence stable) disk
whose existence is guaranteed by [And83] are less than $1-\epsilon$ in
absolute value, and therefore, as a consequence of Corollary 1.5, $\Sigma$ is
the unique minimal surface. Up to taking a smaller constant $K_{0}$, we can
therefore remove the stability assumption:

######  Corollary 1.7.

There exist universal constants $C>0$ and $K_{0}>1$ such that the principal
curvatures $\lambda_{i}$ of any minimal surface $\Sigma$ in $\mathbb{H}^{3}$
with asymptotic boundary a $K$-quasicircle with $K\leq K_{0}$ satisfy

$|\lambda_{i}|\leq C\log K,\;i=1,2.$

In particular, we also improve [Sep16, Theorem B] (up to choosing a smaller
constant) by removing the stability assumption.

######  Corollary 1.8.

There exists a universal constant $K^{\prime}_{0}>1$ such that any
$K$-quasicircle with $K\leq K_{0}^{\prime}$ is the asymptotic boundary of a
unique minimal surface, which is an area-minimizing disk of strongly small
curvature.

###  1.5. Organization of the paper

In the preliminary section §2, we collect and prove some facts in preparation
to work towards proofs of our main results introduced above. In §3 we prove
Theorems 1.1 and 1.2. In §4 we work in the case of small curvature conditions
and prove Theorem 1.3 and, using similar methods, we provide an alternative
argument for Theorem 1.2 in a special case; in §5 we address some
generalizations of our results in higher dimensions and codimensions; in §6 we
detail a construction of an extreme Jordan curve which spans uncountably many
minimal disks in $\mathbb{H}^{3}$ and hence prove Theorem 1.6. In Appendix A
we provide the details to extend some well-known arguments for hypersurfaces
of small curvature to the setting of weakly small curvature that is of
interest here.

###  1.6. Acknowledgements

The first-named author wishes to thank Bill Meeks for his insightful
suggestions and Biao Wang for his generous help. Part of this work was done
during a visit of the first-named author at the Institut Fourier (Université
Grenoble Alpes) in the framework of the “Visiting Scientist Campaign 2023”, he
wishes to thank the institute for excellent working environment. The second-
named author was supported by NSF grant DMS-2202830. The third-named author
was partially supported by the ANR JCJC grant GAPR (ANR-22-CE40-0001, Geometry
and Analysis in the Pseudo-Riemannian setting).

##  2\. Preliminaries

In this article, we denote by $\mathbb{H}^{n+1}$ the hyperbolic space of
dimension $n+1$, and by
$\mathbb{S}^{n}_{\infty}=\partial_{\infty}\mathbb{H}^{n+1}$ its visual
boundary.

###  2.1. Hypersurfaces theory

Given an immersed hypersurface $\Sigma$ in $\mathbb{H}^{n+1}$, we recall that
its first fundamental form is the restriction to $T\Sigma$ of the hyperbolic
metric $h$ of $\mathbb{H}^{n+1}$. The second fundamental form is defined as

$A_{\Sigma}(v,w)=h(\nabla^{h}_{v}W,N_{\Sigma})~{},$

where $\nabla^{h}$ is the Levi-Civita connection of $h$, $W$ is a local smooth
extension of $w$, and $N_{\Sigma}:\Sigma\to T\mathbb{H}^{n+1}$ is a continuous
choice of a unit normal vector to the immersion. (Since $N_{\Sigma}$ is
uniquely determined only up to a sign, so too is $A_{\Sigma}$.)

The second fundamental form satisfies the identity

$A_{\Sigma}(v,w)=h(B_{\Sigma}(v),w)$

where $B_{\Sigma}$ is the shape operator, namely the endomorphism of the
tangent bundle of $\Sigma$ given by $B_{\Sigma}(v)=-\nabla^{h}_{v}N_{\Sigma}$.
The principal curvatures of $\Sigma$ are the eigenvalues of $B_{\Sigma}$,
denoted by $\lambda_{1},\ldots,\lambda_{n}$.

The mean curvature of the immersed hypersurface $\Sigma$ is

$H_{\Sigma}=\mathrm{tr}(B_{\Sigma})=\lambda_{1}+\ldots+\lambda_{n}~{},$

and $\Sigma$ is minimal if and only if its mean curvature vanishes
identically. Although $H_{\Sigma}$ depends (but only up to a sign) on the
choice of the normal vector $N_{\Sigma}$, the condition of being minimal does
not. Also, the mean curvature vector, which is defined as $H_{\Sigma}\cdot
N_{\Sigma}$, does not depend on such a choice.

Finally, the norm of the second fundamental form is

$\|A_{\Sigma}\|=\sqrt{\mathrm{tr}(B_{\Sigma}B_{\Sigma}^{T})}=\sqrt{\lambda_{1}^{2}+\ldots+\lambda_{n}^{2}}~{}.$

###  2.2. Convex hull and width of a Jordan curve

Let $\Lambda$ be a topologically embedded $n$-sphere in
$\mathbb{S}^{n}_{\infty}=\partial\mathbb{H}^{n+1}$. We recall that the convex
hull
$\mathcal{C}(\Lambda)\subset\overline{\mathbb{H}^{n+1}}=\mathbb{H}^{n+1}\cup\mathbb{S}^{n}_{\infty}$
of $\Lambda$ is the smallest geodesically convex subset that contains
$\Lambda$. When $\Lambda$ is not the boundary of a totally geodesic hyperplane
in $\mathbb{H}^{n+1}$, $\mathcal{C}(\Lambda)$ is homeomorphic to a ball, and,
by the Jordan-Brouwer separation theorem, its boundary is the union of
$\Lambda$ and two properly embedded disks, denoted by
$\partial^{+}\mathcal{C}(\Lambda)$ and $\partial^{-}\mathcal{C}(\Lambda)$.
When $\Lambda$ is the boundary of a totally geodesic hyperplane $P$,
$\mathcal{C}(\Lambda)$ equals $P\cup\Lambda$. In this case, by an abuse of
notation, we will still use the symbols $\partial^{+}\mathcal{C}(\Lambda)$ and
$\partial^{-}\mathcal{C}(\Lambda)$, meaning that
$P=\partial^{+}\mathcal{C}(\Lambda)=\partial^{-}\mathcal{C}(\Lambda)$.

Following (and extending to all dimensions) the recent work [BDMS21], we now
define the width of $\Lambda$.

######  Definition 2.1.

Given a topologically embedded $n$-sphere $\Lambda$ in
$\mathbb{S}_{\infty}^{n}$, the width of $\Lambda$ is defined as:

$w(\Lambda)=\sup_{x\in\mathcal{C}(\Lambda)}\left(d(x,\partial^{+}\mathcal{C}(\Lambda))+d(x,\partial^{-}\mathcal{C}(\Lambda))\right)\in[0,+\infty]~{}.$

The following lemma, that relates minimal hypersurfaces and the convex hull of
their asymptotic boundaries, is well-known.

######  Lemma 2.2.

Given a topologically embedded $n$-sphere $\Lambda$ in
$\mathbb{S}_{\infty}^{n}$, let $\Sigma$ be any properly embedded minimal
hypersurface such that $\partial_{\infty}\Sigma=\Lambda$. Then $\Sigma$ is
contained in $\mathcal{C}(\Lambda)$. Moreover, if $\Lambda$ is not the
boundary of a totally geodesic hyperplane, then $\Sigma$ is contained in the
interior of $\mathcal{C}(\Lambda)$.

###### Proof.

Let $P$ be any totally geodesic hyperplane disjoint from $\Lambda$. The signed
distance function from $P$, defined in such a way that it goes to $-\infty$ as
it approaches $\Lambda$, cannot have a positive maximum by a standard
application of the geometric maximum principle (see also Corollary 4.4 for a
more general statement). This implies that $\Sigma$ is contained in the half-
space bounded by $P$ whose closure contains $\Lambda$. Since
$\mathcal{C}(\Lambda)$ is the intersection of all such half-spaces, this
concludes the proof of the first assertion.

For the second assertion, suppose that $\Sigma$ contains a point $x$ in the
boundary of $\mathcal{C}(\Lambda)$. Let $P$ be any support hyperplane for
$\mathcal{C}(\Lambda)$ containing $x$. This means that $\Sigma$ is tangent to
$P$ and, by the first assertion, it is contained in a half-space bounded by
$P$. By the strong maximum principle, $\Sigma=P$, and therefore
$\Lambda=\partial_{\infty}P$.  ∎

###  2.3. Stable and area-minimizing minimal hypersurfaces

A minimal hypersurface $\Sigma$ in a Riemannian manifold $M^{n+1}$ is stable
if and only if, for every $u\in C_{0}^{\infty}(\Sigma)$,

$\int_{\Sigma}uL_{\Sigma}(u)\mathrm{dVol}_{\Sigma}\geq 0~{},$

where $L_{\Sigma}$ is the Jacobi operator:

(2.1)
$L_{\Sigma}(u)=-\Delta_{\Sigma}u-\left(\|A_{\Sigma}\|^{2}+\mathrm{Ric}_{M}(N,N)\right)u~{}.$

Equivalently, integrating by parts, $\Sigma$ is stable if and only if

(2.2)
$\int_{\Sigma}\left(\|A_{\Sigma}\|^{2}+\mathrm{Ric}_{M}(N,N)\right)u^{2}\mathrm{dVol}_{\Sigma}\leq\int_{\Sigma}\|\nabla
u\|^{2}\mathrm{dVol}_{\Sigma}$

for every $u\in C_{0}^{\infty}(\Sigma)$.

A compact hypersurface $\Sigma_{c}$ with boundary is a _least area
hypersurface_ if its area is less than or equal to that of any other compact
hypersurface $\Sigma_{c}^{\prime}$ such that
$\partial\Sigma_{c}=\partial\Sigma_{c}^{\prime}$. For the non-compact case, we
say a hypersurface $\Sigma$ is area-minimizing if any compact codimension 0
submanifold with boundary of $\Sigma$ has least area in the sense above.

An area-minimizing hypersurface is stable. Indeed, the area-minimizing
condition implies that $\Sigma$ is a critical point of the area functional
among compactly supported variation (that is, $\Sigma$ is minimal) and the
second derivative of the area is non-negative (which is equivalent to the
definition of stability given above).

When $n=2$, the following theorem provides an important a priori bound on the
curvature of a stable minimal surface. We state the theorem here in its
version for embedded minimal surfaces in $\mathbb{H}^{3}$, but it holds more
generally for immersed CMC surfaces in a complete Riemannian manifold of
bounded sectional curvature.

######  Theorem 2.3.

[RST10, Main Theorem] There exists a constant $c>0$ such that, for any stable
embedded two-sided minimal surface $\Sigma$ in $\mathbb{H}^{3}$,

$\|A_{\Sigma}(p)\|\leq\frac{c}{\min\\{d(p,\partial\Sigma),\pi/2\\}}~{}.$

Applying Theorem 2.3 to a properly embedded minimal surface, which is
automatically two-sided (see [Sam69]), we immediately get the following
corollary. See also [Ros06, Corollary 11].

######  Corollary 2.4.

There exists a constant $c^{\prime}>0$ such that, for any stable properly
embedded minimal surface $\Sigma$ in $\mathbb{H}^{3}$, $\|A_{\Sigma}\|\leq
c^{\prime}$.

###  2.4. Compactness results

We conclude the preliminaries by stating an important compactness theorem for
minimal surfaces. Again, we state the theorem here only for embedded minimal
surfaces in $\mathbb{H}^{3}$, but it holds more generally for immersed
surfaces in a complete Riemannian manifold (see [And85, Section 2]). This
result is well-known and is often stated under the assumption of bounded
Gaussian curvature (in absolute value); however, by the Gauss’ equation for
minimal surfaces when $n=2$, $K_{\Sigma}=-1-\|A_{\Sigma}\|^{2}/2$, hence this
is equivalent to a bound on the norm of the second fundamental form.

######  Theorem 2.5.

Let $B$ be a bounded domain in $\mathbb{H}^{3}$ and let
$\\{\Sigma_{n}\\}_{n\in\mathbb{N}}$ be a sequence of embedded minimal surfaces
in $B$ with $\partial\Sigma_{n}\subset\partial B$. Suppose there exists a
constant $C$ such that $\|A_{\Sigma_{n}}\|\leq C$ and that there exists a
compact subset $K\subset\mathrm{int}(B)$ such that $\Sigma_{n}\cap
K\neq\emptyset$, for all $n$. Then, up to extracting a subsequence,
$\Sigma_{n}$ converges smoothly to an embedded minimal surface
$\Sigma_{\infty}$.

In particular, for properly embedded minimal surfaces, we obtain the following
compactness result.

######  Corollary 2.6.

Let $\\{\Sigma_{n}\\}_{n\in\mathbb{N}}$ be a sequence of properly embedded
minimal surfaces in $\mathbb{H}^{3}$. Suppose there exists a constant $C$ such
that $\|A_{\Sigma_{n}}\|\leq C$ and that there exists a compact subset
$K\subset\mathbb{H}^{3}$ such that $\Sigma_{n}\cap K\neq\emptyset$, for all
$n$. Then, up to extracting a subsequence, $\Sigma_{n}$ converges smoothly on
compact subsets of $\mathbb{H}^{3}$ to a properly embedded minimal surface
$\Sigma_{\infty}$.

###### Proof.

Fix a point $x_{o}\in\mathbb{H}^{3}$. Let $m_{0}\in\mathbb{N}$ be such that
$K\subset B(x_{o},m_{0})$. For any $m\geq m_{0}$, by applying Theorem 2.5 on
the balls $B(x_{o},m)$ of radius $m$ centered at $x_{o}$, we find a
subsequence converging smoothly to a minimal surface in $B(x_{o},m)$. By a
standard diagonal argument, we can then extract a subsequence converging
smoothly on all compact subsets to an embedded minimal surface
$\Sigma_{\infty}$.

It only remains to show that $\Sigma_{\infty}$ is properly embedded. Suppose
by contradiction that $x_{i}\in\Sigma_{\infty}$ is a diverging sequence in
$\Sigma_{\infty}$, which accumulates to $x_{\infty}\in\mathbb{H}^{3}$. Then
for $m$ large, $x_{\infty}$ would be contained in the interior of
$B(x_{o},m)$, and this would contradict the smooth convergence.  ∎

##  3\. Uniqueness among stable minimal disks

In this section, we prove Theorems 1.1 and 1.2.

###  3.1. Outline

The key result to prove Theorem 1.1 is the following:

######  Theorem 3.1.

Let $\Lambda$ be a Jordan curve on
${\mathbb{S}}^{2}_{\infty}=\partial\mathbb{H}^{3}$. Suppose that there exists
a properly embedded minimal surface in $\mathbb{H}^{3}$, which is not a stable
minimal disk, spanning $\Lambda$, then $\Lambda$ spans at least two disjoint
stable minimal disks in $\mathbb{H}^{3}$.

Before proving Theorem 3.1, let us explain how this implies Theorem 1.1. By
the results of [And83], we know that $\Lambda$ spans a stable (actually, area-
minimizing) minimal disk. So uniqueness among minimal surfaces clearly implies
uniqueness among stable minimal disks. For the converse direction, suppose by
contradiction that $\Lambda$ span a unique stable minimal disk, but it spans
several minimal surfaces. Hence it spans a minimal surface $U$ which is not a
stable minimal disk (that is, either it is not stable, or it is not a disk, or
both). By Theorem 3.1, $\Lambda$ spans two distinct stable minimal disks, and
this gives a contradiction.

When $\Lambda$ is invariant by the action of a Kleinian group, in order to
achieve Theorem 1.2, we will first prove the following result in the same
spirit of Theorem 3.1.

######  Theorem 3.2.

Let $\Lambda$ be a Jordan curve on
${\mathbb{S}}^{2}_{\infty}=\partial\mathbb{H}^{3}$ and let $\Gamma$ be a
Kleinian group preserving $\Lambda$. Suppose that there exists a properly
embedded minimal surface in $\mathbb{H}^{3}$, which is not $\Gamma$-invariant,
spanning $\Lambda$. Then $\Lambda$ spans at least two disjoint
$\Gamma$-invariant stable minimal disks in $\mathbb{H}^{3}$.

Theorem 3.2 implies Theorem 1.2 very similarly to above. Indeed, Anderson
proved the existence of a $\Gamma$-invariant stable minimal disk, hence
uniqueness among minimal surfaces implies uniqueness among $\Gamma$-invariant
stable minimal disks. Conversely, suppose that $\Lambda$ spans a minimal
surface which is not a $\Gamma$-invariant stable minimal disk. If it is not a
minimal disk, Theorem 3.1 applies as above. If it is instead not
$\Gamma$-invariant, we apply Theorem 3.2 to produce two distinct
$\Gamma$-invariant stable minimal disks, and this gives a contradiction.

###  3.2. Adapting the arguments of Anderson

To prove Theorems 3.1 and 3.2, we will adapt an argument in [And83]. We first
need the following result.

######  Lemma 3.3.

Let $\Lambda$ be a Jordan curve on
${\mathbb{S}}^{2}_{\infty}=\partial\mathbb{H}^{3}$, let $U$ be a properly
embedded minimal surface in $\mathbb{H}^{3}$ spanning $\Lambda$, and let
$\Omega$ be a connected component of $\mathbb{H}^{3}\setminus U$. Then there
exists a properly embedded stable minimal disk $\Sigma$ in $\Omega\cup U$
spanning $\Lambda$. Moreover, if $\Lambda$ is invariant under a Kleinian group
$\Gamma$, then $\Sigma$ can be taken such that

(3.1) $\Sigma\subset\bigcap_{g\in\Gamma}g(\Omega\cup U)~{}.$

###### Proof.

We can clearly assume that $\Lambda$ is not the boundary of a totally geodesic
plane. Then, since $U$ is contained in the interior of $\mathcal{C}(\Lambda)$
by Lemma 2.2, $\Omega$ contains either $\partial^{+}\mathcal{C}(\Lambda)$ or
$\partial^{-}\mathcal{C}(\Lambda)$. Suppose it contains
$\partial^{+}\mathcal{C}(\Lambda)$, the other case being analogous. Fix a
homeomorphism
$\phi:\overline{\mathbb{D}}\to\Lambda\cup\partial^{+}\mathcal{C}(\Lambda)$ and
let $\gamma_{n}=\phi(\\{|z|=1-\frac{1}{n}\\})$. Then
$\\{\gamma_{n}\\}_{n\in\mathbb{N}}$ is a sequence of nullhomotopic simple
closed curves in $\Omega\cap\mathcal{C}(\Lambda)$ that converge to $\Lambda$
as $n\to+\infty$.

Now, fix a point $x_{o}\in U$. For any $n$, let $R_{n}$ be a sufficiently
large radius so that $\gamma_{n}\subset B(x_{o},R_{n})$. By genericity of the
transversality of intersection of submanifolds, we can furthermore pick
$R_{n}$ so that $U$ and $\partial B(x_{o},R_{n})$ intersect transversely. By
[HS88, Theorem 6.3] applied to $M_{n}:=B(x_{o},R_{n})\cap\Omega$, which (as a
consequence of transversality) is sufficiently convex in the terminology of
[HS88], we can solve the Plateau problem to find a minimal disk $\Sigma_{n}$,
contained in $M_{n}$, with $\partial\Sigma_{n}=\gamma_{n}$, and of least area
(in $M_{n}$), hence in particular stable.

Next we show that the limit of these $\Sigma_{n}$ provides the desired stable
minimal disk $\Sigma$. However, when $\Lambda$ is $\Gamma$-invariant, we first
need an additional adjustement which will ensure (3.1). Since $\Gamma$ is a
discrete subgroup of $\mathrm{Isom}(\mathbb{H}^{3})$, it is countable, and so
too is the coset space $G/\mathrm{Stab}(U)$. Let us enumerate its elements as

$G/\mathrm{Stab}(U)=\\{g_{0}=[\mathrm{id}],g_{1},g_{2},\ldots\\}$

and let us denote $U_{n}:=g_{n}(U)$, which is a properly embedded minimal
surface spanning $\Lambda$. Here, we construct $\Sigma_{n}$ as the solution of
the Plateau problem with $\partial\Sigma_{n}=\gamma_{n}$, but instead of
$M_{n}$, we apply [HS88, Theorem 6.3] to the ambient manifold

$M_{n}^{\prime}:=B(x_{o},R_{n})\cap\Omega\cap g_{1}(\Omega)\ldots\cap
g_{n}(\Omega)~{},$

where $R_{n}$ is chosen so that $\partial B(x_{o},R_{n})$ intersects
transversely all the minimal surfaces
$\Omega,g_{1}(\Omega),\ldots,g_{n}(\Omega)$. Since the points of tangency of
two minimal surfaces are isolated (see [CM11, Section 5.3]), the boundary of
$M_{n}^{\prime}$ is piecewise smooth, meaning that it is smooth (and minimal)
in the complement of an embedded finite graph. Hence $M_{n}^{\prime}$ is is
sufficiently convex in the terminology of [HS88], which allows us to apply
[HS88, Theorem 6.3].

We now claim that, up to extracting a subsequence, $\Sigma_{n}$ converges
smoothly on compact subsets of $\mathbb{H}^{3}$ to a properly embedded minimal
disk $\Sigma$. For this, first observe that there exists $R_{0}>0$ such that
$\Sigma_{n}\cap B(x_{o},R_{0})\neq\emptyset$ for all $n$. Indeed, since
$\partial\Sigma_{n}=\gamma_{n}$ is contained in
$\partial^{+}\mathcal{C}(\Lambda)$, by the maximum principle (as in the first
part of Lemma 2.2) $\Sigma_{n}$ is entirely contained in
$\mathcal{C}(\Lambda)$, and by the Jordan-Brouwer Theorem, $\Sigma_{n}$ must
disconnect $\mathcal{C}(\Lambda)$ in two components, one of which contains
$x_{o}$, and the other contains $\phi(0)\in\partial^{+}\mathcal{C}(\Lambda)$.
Fixing a path $\eta$ connecting $x_{o}$ to $\phi(0)$ which is contained in the
interior of $\mathcal{C}(\Lambda)$ (except for its endpoints), $\Sigma_{n}$
must therefore intersect $\eta$. Hence $\Sigma_{n}$ intersects
$B(x_{o},R_{0})$ for any choice of $R_{0}$ such that $B(x_{o},R_{0})$ contains
$\eta$.

Now, fix any radius $R>0$ and consider the ball $B(x_{o},R)$. For $n=n(R)$
sufficiently large, $\gamma_{n}$ is contained in $\mathbb{H}^{3}\setminus
B(x_{o},R+1)$. Hence the distance from any point of $\Sigma_{n}\cap
B(x_{o},R)$ and $\partial\Sigma_{n}=\gamma_{n}$ is bounded below by $1$.
Theorem 2.3 implies that there is a constant $C$, independent of $n$, such
that $\|A_{\Sigma_{n}}(x)\|\leq C$ for every $x\in\Sigma_{n}\cap B(x_{o},R)$.
Applying Theorem 2.5, we can extract a subsequence $n_{k}$ such that
$\Sigma_{n_{k}}\cap B(x_{o},R)$ converges smoothly to a minimal disk. By a
diagonal argument, we can then find a subsequence of
$\\{\Sigma_{n}\\}_{n\in\mathbb{N}}$ that converges smoothly on compact subsets
to a minimal disk $\Sigma_{\infty}$. Moreover, the limit $\Sigma_{\infty}$ is
stable as a smooth limit of stable minimal surfaces, and is contained in
$\Omega\cup U$ because every $\Sigma_{n}$ is contained in $\Omega$.

It remains to show that $\Sigma_{\infty}$ is properly embedded and
$\partial_{\infty}\Sigma=\Lambda$. To see that it is properly embedded, the
same argument as in the proof of Corollary 2.6 applies. Observe that
$\Sigma_{n}$ is contained in $\mathcal{C}(\Lambda)$ for all $n$, and therefore
so too is $\Sigma_{\infty}$. Since
$\mathcal{C}(\Lambda)\cap\partial_{\infty}\mathbb{H}^{3}=\Lambda$, we have
that $\partial_{\infty}\Sigma=\Lambda$. This concludes the proof.

For the “moreover” part when $\Lambda$ is $\Gamma$-invariant, recall that in
that case we have constructed the $\Sigma_{n}$ to be contained in
$M_{n}^{\prime}$. In particular, for fixed $m$, $\Sigma_{n}$ is contained in
$g_{m}(\Omega)$ for all $n\geq m$. Taking limits, $\Sigma_{\infty}$ is
contained in $g(\Omega\cup U)$ for all $g\in\Gamma$. This concludes the proof.
∎

In the case where $\Lambda$ is $\Gamma$-invariant, we improve Lemma 3.3 to the
following lemma. We include a proof, which follows closely the construction
done by Anderson in [And83, Theorem 3.1], for convenience of the reader.

######  Lemma 3.4.

Let $\Lambda$ be a Jordan curve on
${\mathbb{S}}^{2}_{\infty}=\partial\mathbb{H}^{3}$ and let $\Gamma$ be a
Kleinian group preserving $\Lambda$. Let $U$ be a properly embedded minimal
surface in $\mathbb{H}^{3}$ spanning $\Lambda$, and let $\Omega$ be a
connected component of $\mathbb{H}^{3}\setminus U$. Then there exists a
$\Gamma$-invariant properly embedded stable minimal disk $\Sigma$ in
$\Omega\cup U$ spanning $\Lambda$.

###### Proof.

We will use the following notation. Choose a connected component $D$ of
$\partial_{\infty}\mathbb{H}^{3}\setminus\Lambda$. Given a properly embedded
(minimal) surface $\Sigma$ with $\partial_{\infty}\Sigma=\Lambda$, recalling
that $\Sigma$ disconnects $\mathbb{H}^{3}$ by the Jordan-Brower separation
theorem, we denote $\Omega^{+}(\Sigma)$ to be the connected component of
$\mathbb{H}^{3}\setminus\Sigma$ whose closure in
$\mathbb{H}^{3}\cup\partial_{\infty}\mathbb{H}^{3}$ contains $D$. Finally,
given $\Sigma_{1}$ and $\Sigma_{2}$, we write $\Sigma_{1}\preceq\Sigma_{2}$ if
$\Omega^{+}(\Sigma_{1})\supseteq\Omega^{+}(\Sigma_{2})$.

Now, given $U$, we define a sequence $\\{\Sigma_{n}\\}_{n\in\mathbb{N}}$
inductively as follows. Set $\Sigma_{0}:=U$ and let $\Sigma_{n+1}$, using
Lemma 3.3, to be a properly embedded stable minimal surface such that

(3.2)
$\Sigma_{n+1}\subset\bigcap_{g\in\Gamma}g(\Omega^{+}(\Sigma_{n})\cup\Sigma_{n})~{}.$

As in the proof of Lemma 3.3, there exists a fixed compact set (for example a
compact set containing a path connecting the two boundary components of
$\mathcal{C}(\Lambda)$) that intersects each $\Sigma_{n}$. Moreover, by
Corollary 2.4, $\|A_{\Sigma_{n}}\|\leq C$ for some universal constant $C$.
Hence by Corollary 2.6, $\Sigma_{n}$ converges up to a subsequence to a
properly embedded minimal surface $\Sigma_{\infty}$, which is moreover stable.
Arguing as in the second last paragraph of Lemma 3.3,
$\partial_{\infty}\Sigma_{\infty}=\Lambda$.

The main point of the construction is that $\Sigma_{\infty}$ is
$\Gamma$-invariant. To see this, fix $g\in\Gamma$. By the inductive step,
$g(\Sigma_{n})\preceq\Sigma_{n+1}$ for all $n$. Taking limits along the chosen
subsequence, $g(\Sigma_{\infty})\preceq\Sigma_{\infty}$. Repeating the
argument for $g^{-1}$, this shows that $g(\Sigma_{\infty})=\Sigma_{\infty}$
and concludes the proof.  ∎

###  3.3. Conclusion of the proofs

We now prove Theorem 3.1.

###### Proof of Theorem 3.1.

Suppose $U$ is a properly embedded minimal surface in $\mathbb{H}^{3}$ spanned
by $\Lambda$, which is not a stable minimal disk. By the Jordan-Brouwer
separation theorem, $U$ disconnects $\mathbb{H}^{3}$. Let $\Omega^{\pm}$ be
the two connected components of $\mathbb{H}^{3}\backslash U$. By Lemma 3.3,
there exists a properly embedded stable minimal disk $\Sigma^{\pm}$ contained
in $\Omega^{\pm}\cup U$ with $\partial_{\infty}\Sigma^{\pm}=\Lambda$. We claim
that $\Sigma^{\pm}$ is disjoint from $U$. Indeed, $\Sigma^{\pm}$ is contained
by construction in $\Omega^{\pm}\cup U$, so by the strong maximum principle,
if $\Sigma^{\pm}$ and $U$ intersect, then they are equal. Since $\Sigma^{\pm}$
is a stable minimal disk, and $U$ is not, they must be disjoint. This shows
that $\Sigma^{+}\subset\Omega^{+}$ and $\Sigma^{-}\subset\Omega^{-}$, hence
they are in particular disjoint.  ∎

The proof of Theorem 3.2 is completely analogous, relying on Lemma 3.4 instead
of Lemma 3.3.

###### Proof of Theorem 3.2.

If $U$ is a properly embedded minimal surface in $\mathbb{H}^{3}$ spanned by
$\Lambda$, which is not $\Gamma$-invariant, denoting by $\Omega_{\pm}$ the two
connected components of the complement of $U$, then by Lemma 3.4 there exists
a properly embedded $\Gamma$-invariant stable minimal disk $\Sigma^{\pm}$
contained in $\Omega^{\pm}\cup U$ with
$\partial_{\infty}\Sigma^{\pm}=\Lambda$. Since $\Sigma^{\pm}$ is
$\Gamma$-invariant, and $U$ is not, they are distinct, and by the strong
maximum principle they are disjoint. Hence $\Sigma^{+}$ and $\Sigma^{-}$ are
disjoint.  ∎

##  4\. Small curvatures and uniqueness

In this section, we show that a small curvature condition can imply uniqueness
for solutions to the asymptotic Plateau problem. This will prove Theorem 1.3
in the case of $n=2$.

###  4.1. Small curvature conditions

We now introduce several conditions of having small curvature.

######  Definition 4.1.

Let $\Sigma$ be an immersed hypersurface in $\mathbb{H}^{n+1}$, and let
$\lambda_{i}$ denote its principal curvatures, for $i=1,\ldots,n$. We say
that:

  * •

$\Sigma$ has _weakly small curvature_ if $|\lambda_{i}(x)|\leq 1$ for every
$x$ in $\Sigma$ and every $i=1,\ldots,n$;

  * •

$\Sigma$ has _small curvature_ if $|\lambda_{i}(x)|<1$ for every $x$ in
$\Sigma$ and every $i=1,\ldots,n$;

  * •

$\Sigma$ has _strongly small curvature_ if there exists $\epsilon>0$ such that
$|\lambda_{i}(x)|\leq 1-\epsilon$ for every $x$ in $\Sigma$ and every
$i=1,\ldots,n$.

In this article, will need two results on hypersurfaces of weakly small
curvatures, namely Theorem 4.2 and Theorem 4.3 below. These results are well
known under the small curvatures assumptions, see [Eps84] and [EES22, Section
4]. In Appendix A we will explain how the arguments adapt under the weakly
small curvatures assumption.

######  Theorem 4.2.

Let $\Sigma\subset\mathbb{H}^{n+1}$ be a properly immersed hypersurface of
weakly small curvature. Then $\Sigma$ is properly embedded and diffeomorphic
to $\mathbb{R}^{n}$.

By this result, we can hereafter assume that all the hypersurfaces of weakly
small curvatures are properly embedded.

######  Theorem 4.3.

Let $\Sigma\subset\mathbb{H}^{n+1}$ be a properly embedded hypersurface of
weakly small curvatures. Then the map
$F:\Sigma\times\mathbb{R}\to\mathbb{H}^{n+1}$ defined by

$F_{\Sigma}(x,t)=\exp_{x}(tN(x))$

is a diffeomorphism. For every $t\in\mathbb{R}$,
$\Sigma_{t}:=F_{\Sigma}(\Sigma,t)$ is a properly embedded hypersurface of
weakly small curvature.

If moreover $\Sigma$ is minimal, then the mean curvature of $\Sigma_{t}$,
computed with respect to the unit normal vector field pointing towards the
direction where $t$ is increasing, has the same sign as $-t$.

A consequence of Theorem 4.3 is that, if $\Sigma$ is a properly embedded
hypersurface of weakly small curvature, the signed distance function
$f_{\Sigma}:\mathbb{H}^{n+1}\to\mathbb{R}$ is smooth and has no critical
points. Indeed, $f_{\Sigma}$ is simply the composition

(4.1) $f_{\Sigma}=\pi\circ F_{\Sigma}^{-1}$

where $\pi:\Sigma\times\mathbb{R}\to\mathbb{R}$ is the projection to the
second factor.

Let us focus again on minimal hypersurfaces. The following statement is an
immediate consequence of great importance for this work.

######  Corollary 4.4.

Let $\Sigma\subset\mathbb{H}^{n+1}$ be a properly embedded minimal
hypersurface of weakly small curvature. For any embedded minimal hypersurface
$\Sigma^{\prime}\subset\mathbb{H}^{n+1}$, the restriction of
$d_{\Sigma}:=d(\cdot,\Sigma)$ to $\Sigma^{\prime}$ has no positive local
maximum.

###### Proof.

Assume by contradiction that $(d_{\Sigma})|_{\Sigma^{\prime}}$ has a positive
local maximum. Let $x_{\max}\in\Sigma^{\prime}$ the maximum point and
$t_{\max}=d_{\Sigma}(x_{\max})>0$ be the corresponding maximum value. Up to
switching the sign of $f_{\Sigma}$, $d_{\Sigma}$ coincides with $f_{\Sigma}$
in a neighbourhood of $x_{\max}$, where $f_{\Sigma}$ is the signed distance
function as in (4.1). Hence $(f_{\Sigma})|_{\Sigma^{\prime}}$ also has a
positive local maximum at $x_{\max}$. This implies that $\Sigma^{\prime}$ is,
in an embedded neighbourhood of $x_{\max}$, tangent to $\Sigma_{t_{\max}}$ and
contained in the region $F_{\Sigma}(\Sigma\times\\{t\leq t_{\max}\\})$, which
is strictly mean convex by the last part of Theorem 4.3. This contradicts the
maximum principle for mean curvature.  ∎

Finally, it is immediate to check that minimal hypersurfaces of weakly small
curvatures are stable.

######  Lemma 4.5.

Let $\Sigma$ be a minimal hypersurface in $\mathbb{H}^{n+1}$ of weakly small
curvatures. Then $\Sigma$ is stable.

###### Proof.

If $\Sigma$ has weakly small curvatures, then

$\|A_{\Sigma}\|^{2}=\lambda_{1}^{2}+\ldots+\lambda_{n}^{2}\leq
n=-\mathrm{Ric}_{\mathbb{H}^{n+1}}(N,N)~{}.$

Hence the left hand side of (2.2) is non-positive, and this concludes the
proof.  ∎

###  4.2. Proof of Theorem 1.3 for $n=2$

In this subsection, we prove Theorem 1.3 for surfaces in $\mathbb{H}^{3}$. By
Theorem 1.1, we know that it will be sufficient to prove the uniqueness in the
class of stable minimal disks.

The fundamental result is therefore the following theorem.

######  Theorem 4.6.

Let $\Lambda$ be a Jordan curve on
${\mathbb{S}}^{2}_{\infty}=\partial\mathbb{H}^{3}$ of finite width, and let
$\Sigma$ be a properly embedded minimal surface in $\mathbb{H}^{3}$ of weakly
small curvature asymptotic to $\Lambda$. Then $\Sigma$ is the unique properly
embedded stable minimal surface in $\mathbb{H}^{3}$ asymptotic to $\Lambda$.

Combining Theorem 4.6 with Theorem 1.1, we immediately obtain the first part
Theorem 1.3 in the case $n=2$. The “moreover” part of Theorem 1.3 then follows
from the results of [And83], which show that every Jordan curve bounds at
least one area-minimizing minimal disk.

We begin by an easy lemma.

######  Lemma 4.7.

Let $\Lambda$ be a Jordan curve on
${\mathbb{S}}^{2}_{\infty}=\partial\mathbb{H}^{3}$ of width
$w(\Lambda)<+\infty$, and let $\Sigma$ be any properly embedded surface
contained in $\mathcal{C}(\Lambda)$ with $\partial_{\infty}\Sigma=\Lambda$ .
Then for every $x\in\mathcal{C}(\Lambda)$, $d(x,\Sigma)\leq w(\Lambda)$.

###### Proof.

We may assume that $\Lambda$ is not the boundary of a totally geodesic
hyperplane, as in that case the conclusion holds trivially. Let
$x\in\mathcal{C}(\Lambda)$. By definition of the width, there exist two
geodesic segments joining $x$ to $y_{\pm}\in\mathcal{C}^{\pm}(\Lambda)$, each
of length at most $w(\Lambda)+\epsilon$. Since $\Sigma$ is contained in the
interior of $\mathcal{C}(\Lambda)$ by Lemma 2.2, and disconnects
$\mathcal{C}(\Lambda)$ by the Jordan-Brouwer separation theorem, one of these
two segments must meet $\Sigma$ at a point $y$, which is on the segment
between $x$ and the other endpoint $y_{\pm}$. Hence

$d(x,\Sigma)\leq d(x,y)\leq d(x,y_{\pm})\leq w(\Lambda)+\epsilon~{}.$

Since $\epsilon$ was arbitrary, this concludes the proof.  ∎

###### Proof of Theorem 4.6.

Let $\Sigma^{\prime}$ be a properly embedded stable minimal surface such that
$\partial_{\infty}\Sigma^{\prime}=\Lambda$. Consider the restriction to the
surface $\Sigma^{\prime}$ of the distance function
$d_{\Sigma}:=d(\cdot,\Sigma)$. We want to show that $d_{\Sigma}$ vanishes
identically on $\Sigma^{\prime}$. This will imply
$\Sigma^{\prime}\subseteq\Sigma$ and thus, since $\Sigma^{\prime}$ is also
properly embedded, $\Sigma=\Sigma^{\prime}$.

Suppose by contradiction that

(4.2) $\sup_{\Sigma^{\prime}}d_{\Sigma}>0~{}.$

By Lemma 4.7, the supremum of $d_{\Sigma}$ on $\Sigma^{\prime}$ is bounded
above by $w(\Lambda)$, hence in particular it is finite. Let
$\\{x_{n}^{\prime}\\}_{n\in\mathbb{N}}$ be a maximizing sequence for
$(d_{\Sigma})|_{\Sigma^{\prime}}$, namely

(4.3)
$d(x^{\prime}_{n},\Sigma)\geq\sup_{\Sigma^{\prime}}d_{\Sigma}-\frac{1}{n}~{}.$

Now, fix a point $x_{o}\in\mathbb{H}^{3}$. Let $\phi_{n}$ be isometries of
$\mathbb{H}^{3}$ such that $\phi_{n}(x^{\prime}_{n})=x_{o}$. We denote
$\Sigma_{n}=\phi_{n}(\Sigma)$ and
$\Sigma^{\prime}_{n}=\phi_{n}(\Sigma^{\prime})$. Observe that
$\Sigma_{n}^{\prime}$ contains $x_{o}$, and thus it intersects every ball
centered at $x_{o}$. For $\Sigma_{n}$, we have

$d(x_{o},\Sigma_{n})=d(\phi_{n}(x^{\prime}_{n}),\phi_{n}(\Sigma))=d(x_{n}^{\prime},\Sigma)\leq
w(\Lambda)~{}.$

This shows that $\Sigma_{n}$ intersects every ball centered at $x_{o}$ of
radius $R\geq w(\Lambda)$.

Recall that $\Sigma$ has weakly small curvatures, hence $\|A_{\Sigma}\|\leq
2$, and since this condition is preserved by isometries,
$\|A_{\Sigma_{n}}\|\leq 2$. By Corollary 2.6, $\Sigma_{n}$ converges up to a
subsequence, smoothly on a compact subsets in $\mathbb{H}^{3}$, to a stable
properly embedded minimal surface $\Sigma_{\infty}$. Moreover, since the
weakly small curvatures condition is closed for the topology of smooth
convergence on compact domains, $\Sigma_{\infty}$ has weakly small curvatures.

Since $\Sigma^{\prime}$ is stable by assumption, and stability is preserved by
isometries, $\Sigma^{\prime}_{n}$ is stable for every $n$. However, we do not
have explicit curvature bounds as for $\Sigma_{n}$. In this case, we use
Theorem 2.4 to find that each $\|A_{\Sigma^{\prime}_{n}}\|$ is bounded by a
universal constant. This enables us to apply Corollary 2.6 again to find that
the sequence $\Sigma^{\prime}_{n}$ converges, up to a subsequence, smoothly on
a compact domains in $\mathbb{H}^{3}$ to another properly embedded minimal
surface $\Sigma^{\prime}_{\infty}$.

Observe that, since by construction $x_{o}\in\Sigma_{n}^{\prime}$ for all $n$,
$x_{o}\in\Sigma_{\infty}^{\prime}$. We now claim that

(4.4)
$d(x_{o},\Sigma_{\infty})=\sup_{\Sigma_{\infty}^{\prime}}d_{\Sigma_{\infty}}$

where as usual $d_{\Sigma_{\infty}}=d(\cdot,\Sigma_{\infty})$. Clearly we have
$d(x_{o},\Sigma_{\infty})\leq\sup_{\Sigma_{\infty}^{\prime}}d_{\Sigma_{\infty}}$.
By contradiction, if the inequality were strict, then one would find
$p\in\Sigma_{\infty}^{\prime}$ and $\epsilon>0$ such that
$d(p,\Sigma_{\infty})>d(x_{o},\Sigma_{\infty})+\epsilon$. By smooth
convergence on compact sets, this would imply that, for large $n$,

$d(p_{n},\Sigma_{n})>d(x_{o},\Sigma_{n})+\frac{\epsilon}{2}$

where $p_{n}\in\Sigma_{n}^{\prime}$ and $p_{n}\to p$. But
$d(p_{n},\Sigma_{n})=d(\phi^{-1}(p_{n}),\Sigma)$ and
$d(x_{o},\Sigma_{n})=d(x_{n}^{\prime},\Sigma)$. Hence we would have

$d(\phi^{-1}(p_{n}),\Sigma)>d(x_{n}^{\prime},\Sigma)+\frac{\epsilon}{2}$

thus contradicting (4.3).

Having established the claim, (4.4) implies that $x_{o}$ is a global maximum
for the restriction to $\Sigma^{\prime}$ of the distance function
$d_{\Sigma_{\infty}}$. This contradicts Corollary 4.4 and thus concludes the
proof.  ∎

###  4.3. Alternative argument for Theorem 1.2

We now explain an alternative proof of Theorem 1.2, which however only applies
when the Kleinian group $\Gamma$ is quasi-Fuchsian and isomorphic to the
fundamental group of a closed surface. This is based on the work of Guaraco,
Lima and Vargas-Pallete [GLVP21], which we summarize now. Let $M$ be a quasi-
Fuchsian manifold. One of their main results is that $M$ admits a foliation by
closed surfaces that are either minimal, or have non-vanishing mean curvature
(i.e., are either strictly mean-convex, or strictly mean-concave.) The sign of
the mean curvature switches at the minimal leaves. Their work implies that if
$M$ contains a unique closed stable minimal surface $S$, then $M\setminus S$
has a foliation $\mathcal{F}$ by surfaces that are isotopic to $S$ and have
mean-curvature vector everywhere pointing towards $S$.

###### Proof of Theorem 1.2 for $\Gamma$ quasi-Fuchsian.

Suppose $\Gamma$ admits a unique invariant stable minimal disk $\Sigma$, which
thus gives a unique closed stable minimal surface $\Sigma/\Gamma$ in
$M:=\mathbb{H}^{3}/\Gamma$. Let $F:M\to\mathbb{R}$ be a function whose level
sets $F^{-1}(t)$ are the leafs of the foliation of $M$ provided by [GLVP21],
and such that $F^{-1}(0)=\Sigma/\Gamma$. The foliation $\mathcal{F}$ lifts to
a foliation of $\mathbb{H}^{3}$ by properly embedded disks
$\widetilde{\Sigma}_{t}:=\widetilde{F}^{-1}(t)$ (where $\widetilde{F}$ is the
lift of $F$ to $\mathbb{H}^{3}$, and therefore
$\widetilde{\Sigma}_{0}=\Sigma$) with asymptotic boundary the limit set
$\Lambda$ of $\Gamma$.

By Theorem 1.1, it suffices to show that, if $\Sigma^{\prime}$ is a properly
embedded stable minimal disk in $\mathbb{H}^{3}$ with
$\partial_{\infty}\Sigma^{\prime}=\Lambda$, then $\Sigma^{\prime}=\Sigma$.
Consider the function
$f:=\widetilde{F}|_{\Sigma^{\prime}}:\Sigma^{\prime}\to\mathbb{R}$. We claim
that $f\equiv 0$, which will imply $\Sigma^{\prime}=\Sigma$ since both are
properly embedded. First, observe that $f$ is bounded. Indeed, since $\Gamma$
is quasi-Fuchsian, the convex core $\mathcal{C}(\Lambda)/\Gamma$ of $M$ is
compact, and $F$ is thus bounded on the convex core. Lifting to
$\mathbb{H}^{3}$, this shows that $\widetilde{F}|_{\mathcal{C}(\Lambda)}$ is
bounded, and so too is $f$ by Lemma 2.2.

Now, the proof is similar to the proof of Theorem 4.6, using the foliation
lifted from $\mathcal{F}$ instead of the equidistant foliation. Suppose by
contradiction that $\sup f>0$, the other case being analogous. Let
$x_{n}^{\prime}$ be a sequence such that $f(x_{n}^{\prime})\to\sup f$, let
$\phi_{n}$ be a sequence of isometries sending $x_{n}^{\prime}$ to a fixed
point $x_{o}$, and let $\Sigma_{n}^{\prime}:=\phi_{n}(\Sigma^{\prime})$ and
$\Sigma_{n}:=\phi_{n}(\widetilde{\Sigma}_{\sup f})$. Using Theorem 2.4 and
Corollary 2.6, the sequence $\Sigma^{\prime}_{n}$ converges, up to a
subsequence, smoothly on a compact domains in $\mathbb{H}^{3}$ to another
properly embedded minimal surface $\Sigma^{\prime}_{\infty}$. We claim that
the sequence $\Sigma_{n}$ also converges up to subsequences to a smooth
surface in $\mathbb{H}^{3}$. For this, first observe that the width
$\mathcal{C}(\Lambda)$ is finite because $\Lambda$ is a quasicircle, and, by
cocompactness of the action of $\Gamma$ on $\widetilde{\Sigma}_{\sup f}$, the
distance of $\widetilde{\Sigma}_{\sup f}$ from $\mathcal{C}(\Lambda)$ is
bounded by some constant $c$. It follows that $\Sigma_{n}$ intersects every
ball centered at $x_{o}$ of radius $R\geq w(\Lambda)+c$. Second, by smoothness
of $\widetilde{\Sigma}_{\sup f}$ and cocompactness again, the norm of the
second fundamental form of $\widetilde{\Sigma}_{\sup f}$ and of all its
covariant derivative is bounded. Since $\phi_{n}$ is an isometry, the norm of
the second fundamental form of $\Sigma_{n}$ and of all its covariant
derivative is bounded too. This allows to extract a subsequence of
$\\{\Sigma_{n}\\}_{n\in\mathbb{N}}$ converging smoothly on compact sets to a
surface $\Sigma_{\infty}$ (see for instance [Smi07]). As we are assuming $\sup
f>0$, $\widetilde{\Sigma}_{\sup f}$ has negative mean curvature with respect
to the unit normal pointing towards $\widetilde{F}=+\infty$, hence the same
holds for the limit $\Sigma_{\infty}$.

It follows from the construction that $x_{o}\in\Sigma^{\prime}_{\infty}$. By
cocompactness, the distance between different leaves $\widetilde{\Sigma}_{\sup
f}$ and $\widetilde{\Sigma}_{t}$ goes to zero as $t\to\sup f$. Using this
fact, one can show similarly to the proof of Theorem 4.6 that
$x_{o}\in\Sigma_{\infty}$. Moreover, since $\Sigma^{\prime}$ is contained in
the region $\\{x\,|\,\widetilde{F}(x)\leq\sup f\\}$ whose boundary is
$\widetilde{\Sigma}_{\sup f}$, after taking limits $\Sigma^{\prime}_{\infty}$
is contained in the side of $\Sigma_{\infty}$ corresponding to decreasing
values of $\widetilde{F}$. This contradicts the geometric maximum principle at
the tangency point $x_{o}$, and therefore shows that $\sup f\leq 0$. Repeating
the argument for the infimum of $f$ instead of the supremum shows that
$f\equiv 0$ and concludes the proof.  ∎

##  5\. Higher dimension and codimension

In this section, we outline some generalization of previous results to higher
dimensions and higher codimensions.

###  5.1. Hypersurfaces in higher dimensions

We first provide a proof of Theorem 1.3 in higher dimensions. The argument is
similar to the proof of Theorem 4.6. However, in that case it was sufficient
to work with a _stable_ minimal surface, by Theorem 1.1. The difference here
is that instead of passing to a limit of minimal submanifolds (which we will
be unable to do if they are unstable), we pass to a limit of metric balls in
the ambient space. We are then able to apply a theorem of White [Whi10] in the
setting of minimal varifolds to get a contradiction.

###### Proof of Theorem 1.3.

Recall that, if $d_{\Sigma}=d(\cdot,\Sigma)$ denotes the distance function
from $\Sigma$, by Theorem 4.3 the level hypersurfaces of $d_{\Sigma}$ form a
strictly mean-convex foliation of $\mathbb{H}^{n}\backslash\Sigma$, meaning
that the mean curvature vector does not vanish and always points towards
$\Sigma$. Let $\Sigma^{\prime}$ be another properly embedded minimal
hypersurface with the same asymptotic boundary as $\Sigma$. Since $\Lambda$
has finite width, we know that $d_{\Sigma}$ is bounded on $\Sigma^{\prime}$ by
some constant. (Indeed, the proof of Lemma 4.7 works identically in any
dimension.) If $(d_{\Sigma})|_{\Sigma^{\prime}}$ attains its supremum at an
interior point, then the strong maximum principle implies that
$\Sigma^{\prime}=\Sigma$. If not, we can take a sequence of points $x_{k}$ on
$\Sigma^{\prime}$ with $d_{\Sigma}(x_{k})$ tending to the supremal distance
$D:=\sup_{\Sigma^{\prime}}d_{\Sigma}$. Without loss of generality assume that
all of the $x_{k}$ are in the same connected component $\Omega_{+}$ (say) of
$\mathbb{H}^{n+1}\setminus\Sigma$.

Then we claim that the intersections of the open metric balls of radius $2D$
centered at the $x_{k}$ intersected with
$d_{\Sigma}^{-1}([0,D])\cup\Omega_{-}$ (where $\Omega_{-}$ is the other
connected component of $\mathbb{H}^{n+1}\setminus\Sigma$) are a sequence of
smooth Riemannian manifolds with boundary, that we call $M_{k}$, that
subsequentially converge to a smooth limit Riemannian manifold with strictly
mean convex boundary $M_{\infty}$. Note that since we are intersecting with
open balls, the $M_{k}$ are incomplete and have boundary contained in
$d_{\Sigma}^{-1}(D)\cap\Omega_{+}$. We just need to check smooth convergence
at the boundary.

For every $x_{k}$ there is a point $p_{k}$ on $\Sigma$ such that the ball of
radius $4D$ centered at $p_{k}$ contains $M_{k}$. Let $\phi_{k}$ be an
isometry of $\mathbb{H}^{n+1}$ sending $p_{k}$ to a given point $p$, and fix
any radius $R\geq 4D$. Using that $\|A_{\Sigma}\|\leq n$ by assumption, as in
Corollary 2.6 (which holds in any dimensions) we can pass to a subsequential
limit $\Sigma_{\infty}$ of the minimal hypersurfaces $\phi_{k}(\Sigma)$ inside
every ball $B(p,R)$.

Note that $\Sigma_{\infty}$ has weakly small curvature. By Theorem 4.3, the
set of points at signed distance $D$ from $\Sigma_{\infty}$ is a smooth
strictly mean convex submanifold, and so is its intersection with the interior
of $B(p,4D)$. Since $\phi_{k}(d_{\Sigma}^{-1}(D)\cap B(p_{k},4D))$ converges
smoothly to the set of points in $B(p,4D)$ at signed distance $D$ from
$\Sigma_{\infty}$, it follows that the boundary of $M_{k}$ (which is contained
in $d_{\Sigma}^{-1}(D)$) subsequentially converges smoothly, after applying
$\phi_{k}$. In other words, we have shown that the $M_{k}$ subconverge
smoothly, in the sense of limits of Riemannian manifolds, to a manifold with
strictly mean convex boundary $M_{\infty}$.

Now, since $d_{\Sigma}(x_{n})$ tends to $D$, the set of limit points of
$\Sigma\cap M_{k}$ in $M_{\infty}$ contains a point in the boundary of
$M_{\infty}$. But [Whi10, Theorem 3] says exactly that this is impossible.

For the “moreover” part, Anderson [And85] proved the existence of an area-
minimizing integral current with asymptotic boundary $\Lambda$, which,
however, is a smooth hypersurface only in the complement of a singular set of
Hausdorff dimension at most $n-7$. Nevertheless, calling this integral current
$\Sigma^{\prime}$, the above argument based on the theorem of White [Whi10]
applies to such a $\Sigma^{\prime}$ as well, showing that
$\Sigma^{\prime}=\Sigma$ is, in particular, a smooth hypersurace. Hence
$\Sigma$ is area-minimizing.  ∎

###  5.2. Higher codimension submanifolds

We briefly discuss some generalizations to higher codimension. By following
Uhlenbeck’s approach [Uhl83], Jiang [Jia21, Lemma 2.2] defined a notion of
almost Fuchsian for quotients $M$ of $\mathbb{H}^{n+1}$ by discrete faithful
representations of surface groups in $\text{Isom}(\mathbb{H}^{n+1})$. Such a
quotient $M$ is called almost-Fuchsian if it admits a (two-dimensional)
minimal surface $\Sigma$ with principal curvatures smaller than 1 in
magnitude. Jiang showed that an almost-Fuchsian $M$ with $\Sigma$ removed is
foliated by equidistant surfaces to $\Sigma$ that are mean 2-convex (the sum
of the smallest two principal curvatures is positive), and thus by the maximum
principle that $\Sigma$ is the unique closed minimal surface in $M$. Bronstein
recently showed that in four dimensions $M$ could in fact be homeomorphic to a
nontrivial disk bundle over a surface [Bro23]. The limit set of such an $M$
would then be a self-similar everywhere-non-tame knot in
$\partial_{\infty}\mathbb{H}^{4}\cong S^{3}$ (see [GLT22].)

For any almost-Fuchsian $M$, the foliation by mean 2-convex equidistant
surfaces to $\Sigma$ lifts to a foliation of the complement of a lift
$\widetilde{\Sigma}$ of $\Sigma$ to $\mathbb{H}^{n+1}$. We thus expect that
the same argument by contradiction using [Whi10, Theorem 3] (which applies in
any codimension) can be used, to show that the limit set of $M$ bounds a
unique minimal disk, namely $\widetilde{\Sigma}$. Jiang’s work is for two
dimension surfaces in arbitrary codimension, but we expect that similar
arguments would work for $k$-dimensional minimal surfaces with small principal
curvatures in any dimension. In that case, the same arguments would give a
uniqueness result for asymptotic Plateau problems in that setting as well.

Finally here we mention that Fine has recently developed a theory for counting
minimal surfaces in $\mathbb{H}^{4}$ with asymptotic boundary a given knot
[Fin21]. In view of [Bro23] and the previous discussion, one should be able to
give examples of (wild) knots in $S^{3}$ that bound a unique minimal surface
in $\mathbb{H}^{4}$.

##  6\. Uncountably many minimal disks

The aim of this section is to prove Theorem 1.6.

###  6.1. Constructing the Jordan curve

We begin by constructing the Jordan curve $\Lambda$ in
$\mathbb{S}_{\infty}=\partial_{\infty}\mathbb{H}^{3}$, that will be the
asymptotic boundary of uncountably many stable minimal disks. For this
purpose, we will construct an explicit continuous injective map

$f:\mathbb{R}\mathrm{P}^{1}=\mathbb{R}\cup\\{\infty\\}\to\mathbb{C}\mathrm{P}^{1}=\mathbb{C}\cup\\{\infty\\}~{}.$

We begin by fixing some notation. Let $\epsilon,\delta>0$ be some constants
which will be fixed later. For the moment, we only need to assume that
$\epsilon<1/3$, so that the circle of radius $2^{n}(1+\epsilon)$ is smaller
than the circle of radius $2^{n+1}(1-\epsilon)$, and $\delta<\pi/8$. Let

$\centering\begin{aligned}
p^{n}_{+,+}:=&2^{n}\left(1-{\epsilon}\right)e^{i\left(\frac{\pi}{2}-\delta\right)}\\\
q^{n}_{+,+}:=&2^{n}\left(1+{\epsilon}\right)e^{i\left(\frac{\pi}{2}-\delta\right)}\\\
p^{n}_{-,+}:=&2^{n}\left(1-{\epsilon}\right)e^{i\left(\frac{\pi}{2}+\delta\right)}\\\
q^{n}_{-,+}:=&2^{n}\left(1+{\epsilon}\right)e^{i\left(\frac{\pi}{2}+\delta\right)}\end{aligned}\qquad\begin{aligned}
p^{n}_{+,-}:=&2^{n}\left(1-{\epsilon}\right)e^{i\left(-\frac{\pi}{2}+\delta\right)}\\\
q^{n}_{+,-}:=&2^{n}\left(1+{\epsilon}\right)e^{i\left(-\frac{\pi}{2}+\delta\right)}\\\
p^{n}_{-,-}:=&2^{n}\left(1-{\epsilon}\right)e^{i\left(-\frac{\pi}{2}-\delta\right)}\\\
q^{n}_{-,-}:=&2^{n}\left(1+{\epsilon}\right)e^{i\left(-\frac{\pi}{2}-\delta\right)}\\\
\end{aligned}\@add@centering$

Observe that the points $p^{n}_{\sigma_{1},\sigma_{2}}$ and
$q^{n}_{\sigma_{1},\sigma_{2}}$ lie in the quadrant determined by the two
signs $\sigma_{1}$ and $\sigma_{2}$, and that the Euclidean distance between
$p^{n}_{\sigma_{1},\sigma_{2}}$ and $q^{n}_{\sigma_{1},\sigma_{2}}$ is
$2^{n+1}\epsilon$. Moreover, denoting $\alpha(z)=2z$, we see that
$p^{n}_{\sigma_{1},\sigma_{2}}=2^{n}p^{0}_{\sigma_{1},\sigma_{2}}$ and the
same for $q^{n}_{\sigma_{1},\sigma_{2}}$.

Now, we define the map $f$, which will be injective and piecewise smooth when
restricted to $\mathbb{R}\setminus\\{0\\}$, with junctions exactly at the
points $p^{n}_{\sigma_{1},\sigma_{2}}$ and $q^{n}_{\sigma_{1},\sigma_{2}}$.
The explicit formula for $f$ is the following:

$f(t)=\begin{cases}0&\textrm{ if }t=0\\\
2^{n}\left(1-{\epsilon}\right)e^{i\left((4\pi-8\delta)t-\frac{9\pi}{2}+9\delta\right)}&\textrm{
if }t\in\left[2^{n},2^{n}\left(1+\frac{1}{4}\right)\right]\\\
2^{n}\left(1+\epsilon(-8t+7)/3\right)e^{i\left(\frac{\pi}{2}-\delta\right)}&\textrm{
if
}t\in\left[2^{n}\left(1+\frac{1}{4}\right),2^{n}\left(1+\frac{1}{2}\right)\right]\\\
2^{n}\left(1+{\epsilon}\right)e^{i\left((-4\pi+8\delta)t+\frac{5\pi}{2}-5\delta\right)}&\textrm{
if
}t\in\left[2^{n}\left(1+\frac{1}{2}\right),2^{n}\left(1+\frac{3}{4}\right)\right]\\\
2^{n}\left((-4/3+4\epsilon)t+(10/3-6\epsilon)\right)e^{i\left(-\frac{\pi}{2}+\delta\right)}&\textrm{
if }t\in\left[2^{n}\left(1+\frac{3}{4}\right),2^{n+1}\right]\\\
\infty&\textrm{ if }t=\infty\\\
-2^{n}\left(1-{\epsilon}\right)e^{i\left((4\pi-8\delta)t+\frac{9\pi}{2}-9\delta\right)}&\textrm{
if }t\in\left[-2^{n}\left(1+\frac{1}{4}\right),-2^{n}\right]\\\
-2^{n}\left(1+\epsilon(8t+7)/3\right)e^{i\left(-\frac{\pi}{2}+\delta\right)}&\textrm{
if
}t\in\left[-2^{n}\left(1+\frac{1}{2}\right),-2^{n}\left(1+\frac{1}{4}\right)\right]\\\
-2^{n}\left(1+{\epsilon}\right)e^{i\left((-4\pi+8\delta)t-\frac{5\pi}{2}+5\delta\right)}&\textrm{
if
}t\in\left[-2^{n}\left(1+\frac{3}{4}\right),-2^{n}\left(1+\frac{1}{2}\right))\right]\\\
-2^{n}\left((4/3-4\epsilon)t+(10/3-6\epsilon)\right)e^{i\left(\frac{\pi}{2}-\delta\right)}&\textrm{
if }t\in\left[-2^{n+1},-2^{n}\left(1+\frac{3}{4}\right)\right]\\\ \end{cases}$

Figure 1. The construction of the Jordan curve $\Lambda$.

Let us explain this definition (see also Figure 1). First, observe that the
image of $(0,+\infty)$ lies in the half-space $\\{\Re(z)>0\\}$, while the
image of $(-\infty,0)$ lies in the half-space $\\{\Re(z)<0\\}$. The intervals
$[2^{n},2^{n}(1+1/4)]$ are mapped to the arcs of circles contained in
$\\{\Re(z)>0\\}$ connecting $p^{n}_{+,-}$ to $p^{n}_{+,+}$, and similarly the
intervals $[2^{n}(1+1/2),2^{n}(1+3/4)]$ are mapped to the arcs of circles
contained in $\\{\Re(z)>0\\}$ connecting $q^{n}_{+,+}$ to $q^{n}_{+,-}$,
parameterized proportionally to arclenght. The intervals
$[2^{n}(1+1/4),2^{n}(1+1/2)]$ are mapped to straight lines connecting
$p^{n}_{+,+}$ to $q^{n}_{+,+}$. The intervals $[2^{n}(1+1/4),2^{n}(1+1/2)]$
are mapped to straight lines connecting $q^{n}_{+,-}$ to $p^{n+1}_{+,-}$. For
negative values of $t$, the situation is absolutely analogous up to composing
with a reflection in the imaginary axis. Namely, $f$ satisfies the symmetry:

(6.1) $f(-t)=-\overline{f(t)}~{}.$

Also, $f$ satisfies the equivariance property

(6.2) $f\circ\alpha^{n}=\alpha^{n}\circ f~{},$

where we recall that $\alpha(z)=2z$.

######  Lemma 6.1.

The image of $f$ is a quasicircle in $\mathbb{C}\mathrm{P}^{1}$.

###### Proof.

We show that $f$ is continuous and injective, which shows that the image of
$f$ is a Jordan curve. One immediately checks from the formula that $f$ is
well-defined and continuous (actually, piecewise smooth) when restricted to
$\mathbb{R}\setminus\\{0\\}$. Observe moreover that, if
$|t|\in[2^{n},2^{n+1}]$, then
$|f(t)|\in[2^{n}(1-\epsilon),2^{n+1}(1-\epsilon)]$. This immediately implies
that $f$ is continuous at $t=0$ and $t=\infty$.

To check that $f$ is injective, we have already observed that $f$ maps
$(0,+\infty)$ to the open right half-space and $(-\infty,0)$ to the open left
half-space. Hence, together with (6.1), it suffices to check that $f$ is
injective on $(0,+\infty)$. By the condition on $|f(t)|$ in the previous
paragraph and (6.2) it is actually sufficient to check that $f$ is injective
on $[1,2]$, which is an immediate consequence of the construction.

Let us denote by $\Lambda$ the image of $f$. We now show that $\Lambda$ is a
quasicircle, by using the following characterization of quasicircles given in
[Ahl66, Chapter IV, Theorem 5]. Let $\Lambda$ be a Jordan curve in
$\mathbb{C}\mathrm{P}^{1}=\mathbb{C}\cup\\{\infty\\}$ going through $\infty$.
Then $\Lambda$ is a quasicircle if and only if there exists a constant $C>0$
such that, for every $z_{1},z_{2}\in\Lambda\cap\mathbb{C}$ and every $z_{3}$
in the arc of $\Lambda\cap\mathbb{C}$ with endpoints $z_{1},z_{2}$,

(6.3) $|z_{3}-z_{1}|\leq C|z_{2}-z_{1}|~{}.$

To check that (6.3) holds for the image of $f$, first observe that the
condition (6.3) is scale-invariant. Hence, since $\Lambda$ is invariant by the
transformations $\alpha^{n}(z)=2^{n}z$, we can assume that
$\max\\{|z_{1}|,|z_{2}|\\}\in[1/2,1]$. By definition of $f$, the arc of
$\Lambda\cap\mathbb{C}$ with endpoints $z_{1},z_{2}$ is then contained in the
Euclidean ball $B(0,1)$. It follows that it is enough to check (6.3) for
$|z_{2}-z_{1}|\leq\epsilon_{0}$, for any fixed $\epsilon_{0}>0$. Indeed, if
$|z_{2}-z_{1}|\geq\epsilon_{0}$, since by the triangle inequality
$|z_{3}-z_{1}|\leq 2$, we have
$|z_{3}-z_{1}|\leq(2/\epsilon_{0})|z_{2}-z_{1}|$. In conclusion, it is
sufficient to check that (6.3) holds when either $z_{1}$ or $z_{2}$ has
modulus in $[1/2,1]$ and $|z_{2}-z_{1}|\leq\epsilon_{0}$. This is immediately
verified since $\Lambda\cap\\{1/2-\epsilon_{0}\leq|z|\leq 1+\epsilon_{0}\\}$
is a union of finitely many straight line segments or arcs of circles meeting
orthogonally at the adjacency points.  ∎

######  Remark 6.2.

We remark that $\Lambda$ is not a symmetric quasicircle. Indeed, as a
consequence of the invariance under the scaling $\alpha(z)=2z$, $\Lambda$
cannot satisfy the so-called strong reverse triangle property in a
neighbourhood of $0$, see [GS92, Section 6].

Now, in order to fix the suitable values of $\epsilon$ and $\delta$, we need
to introduce minimal catenoids. A _minimal catenoid_ is a minimal surface in
$\mathbb{H}^{3}$, homeomorphic to an open annulus, whose asymptotic boundary
is the union of two disjoint circles $C$ and $C^{\prime}$ in
$\partial_{\infty}\mathbb{H}^{3}=\mathbb{C}\mathrm{P}^{1}$, and which is
obtained as a surface of revolution with respect to the unique geodesic
meeting orthogonally both totally geodesic planes $P$ and $P^{\prime}$ with
$\partial_{\infty}P=C$ and $\partial_{\infty}P^{\prime}=C^{\prime}$. See
Figure 2.

Figure 2. A minimal catenoid in $\mathbb{H}^{3}$ and its axis of revolution.

We will use the following existence result:

######  Theorem 6.3 ([dMG87]).

There exists a constant $d>0$ such that, if $C$ and $C^{\prime}$ are disjoint
circles in $\partial_{\infty}\mathbb{H}^{3}=\mathbb{C}\mathrm{P}^{1}$ and the
distance between the totally geodesic planes $P$ and $P^{\prime}$ that they
span is less than $d$, then there exists a minimal catenoid with asymptotic
boundary equal to $C\cup C^{\prime}$.

Now, let us define the following circles in
$\mathbb{C}\mathrm{P}^{1}=\mathbb{C}\cup\\{\infty\\}$:

$\displaystyle C_{n,0}$
$\displaystyle=\left\\{z\,:\,\left|z-2^{n}\left(1-{2\epsilon}-\frac{1+\epsilon}{100}\right)\right|=\frac{2^{n}}{100}\right\\}$
$\displaystyle C^{\prime}_{n,0}$
$\displaystyle=\left\\{z\,:\,\left|z-2^{n}\left(1+{2\epsilon}+\frac{1+\epsilon}{100}\right)\right|=\frac{2^{n}}{100}\right\\}$
$\displaystyle C_{n,1}$
$\displaystyle=\\{z\,:\,|z-2^{n}e^{i\left(\frac{\pi}{2}-\delta-(1+\delta)\epsilon\right)}|={\epsilon}\\}$
$\displaystyle C^{\prime}_{n,1}$
$\displaystyle=\\{z\,:\,|z-2^{n}e^{i\left(\frac{\pi}{2}+\delta+(1+\delta)\epsilon\right)}|=\epsilon\\}$

See Figure 3. Observe that

(6.4) $C_{n,0}=\alpha^{n}C_{0,0}\qquad
C^{\prime}_{n,0}=\alpha^{n}C^{\prime}_{0,0}\qquad
C_{n,1}=\alpha^{n}C_{0,1}\qquad
C^{\prime}_{n,1}=\alpha^{n}C^{\prime}_{0,1}~{}.$

and that $C_{n,0}$, $C_{n,0}^{\prime}$, $C_{n,1}$ and $C_{n,1}^{\prime}$ are
all in the complement of the image of $f$.

Figure 3. The circles $C_{n,0},C_{n,0}^{\prime},C_{n,1}$ and
$C_{n,1}^{\prime}$.

######  Lemma 6.4.

There exists constants $\epsilon,\delta>0$ such that for all $n\in\mathbb{Z}$
and all $j\in\\{0,1\\}$, $C_{n,j}\cup C_{n,j}^{\prime}$ is the asymptotic
boundary of a minimal catenoid.

###### Proof.

By (6.4) and the fact that $\alpha$ is the boundary value of an isometry of
$\mathbb{H}^{3}$ in the upper half-space model, it suffices to prove the
statement for $n=0$. We shall first choose $\epsilon$ in such a way that
$C_{0,0}\cup C_{0,0}^{\prime}$ bounds a minimal catenoid. For this, it
suffices to observe that $C_{0,0}$ and $C_{0,0}^{\prime}$ are circles of radii
$1/100$, and the Euclidean distance between their centers is
$2(2\epsilon+(1+\epsilon)/100)=2/100+402\epsilon/100$. As $\epsilon\to 0$, the
circles $C_{0,0}$ and $C_{0,0}^{\prime}$ tend to a pair of tangent circles,
each with radius $1/100$. Since the hyperbolic distance varies continuously,
and the hyperbolic distance between two totally geodesic planes tangent at
infinity equals zero, the distance between the totally geodesic planes bounded
by $C_{0,0}$ and $C_{0,0}^{\prime}$ tends to zero. Hence by Theorem 6.3, when
$\epsilon$ is sufficiently small, there exists a minimal catenoid with
asymptotic boundary $C_{0,0}\cup C_{0,0}^{\prime}$.

We shall now keep such $\epsilon$ fixed, and choose $\delta$ in such a way
that $C_{0,1}\cup C_{0,1}^{\prime}$ bounds a minimal catenoid. The reasoning
is completely analogous to the previous case. Indeed, the circles $C_{0,1}$
and $C_{0,1}^{\prime}$ have radius $\epsilon$ and the distance between their
centers is bounded above by $2\epsilon+2\delta(1+\epsilon)$. Hence $C_{0,1}$
and $C_{0,1}^{\prime}$ tend to be tangent when $\delta\to 0$, and we conclude
again by Theorem 6.3.  ∎

In the rest of the section, we will denote by $\mathrm{Cat}_{n,j}$ the minimal
catenoid whose asymptotic boundary is $C_{n,j}\cup C_{n,j}^{\prime}$, and by
$\mathrm{Cat}^{\mathrm{solid}}_{n,j}$ the closure in $\mathbb{H}^{3}$ of the
connected component of $\mathbb{H}^{3}\setminus\mathrm{Cat}_{n,j}$ that
contains the totally geodesic planes spanned by $C_{n,j}$ and
$C_{n,j}^{\prime}$.

###  6.2. Topological disks

We will now need to construct topological disks with asymptotic boundary the
Jordan curve $\Lambda=f(\mathbb{R}\mathbb{P}^{1})$, contained in the
complement of suitably chosen solid catenoids. The choice of these solid
catenoids will be encoded in a function $\iota:\mathbb{Z}\to\\{0,1\\}$. Each
such topological disk will be the starting point to construct pairwise
distinct _minimal_ disks.

######  Lemma 6.5.

For every $\iota:\mathbb{Z}\to\\{0,1\\}$, there exists a continuous injective
map $F_{\iota}$ from the upper half-plane $\mathbb{H}^{2}$ to
$\mathbb{H}^{3}$, whose image is contained in

$\mathbb{H}^{3}\setminus\bigcup_{n\in\mathbb{Z}}\mathrm{Cat}^{\mathrm{solid}}_{n,\iota(n)}$

and which extends continuously to
$f:\mathbb{R}\cup\\{\infty\\}=\partial_{\infty}\mathbb{H}^{2}\to\mathbb{C}\cup\\{\infty\\}=\partial_{\infty}\mathbb{H}^{3}$.

###### Proof.

We will divide the construction in several steps. We fix $n\in\mathbb{Z}$, and
we will first define $F_{\iota}$ on the strip
$\\{z\in\mathbb{H}^{2}\,:\,2^{n}\leq|z|\leq 2^{n}(1+3/4)\\}$ in the upper
half-plane. We will distinguish two cases here, according to whether
$\iota(n)=0$ or $\iota(n)=1$. Then we will define $F_{\iota}$ on the strip
$\\{z\in\mathbb{H}^{2}\,:\,2^{n}(1+3/4)\leq|z|\leq 2^{n+1}\\}$ in a way which
is independent of $\iota(n)$.

Suppose first that $\iota(n)=1$. We divide the strip $\\{2^{n}\leq|z|\leq
2^{n}(1+3/4)\\}$ in three portions, namely $\\{2^{n}\leq|z|\leq
2^{n}(1+1/4)\\}$, $\\{2^{n}(1+1/4)\leq|z|\leq 2^{n}(1+1/2)\\}$ and
$\\{2^{n}(1+1/2)\leq|z|\leq 2^{n}(1+3/4)\\}$, and define $F_{\iota}$ on each
of them in such a way that it extends to $f_{\iota}$ on the corresponding
interval of the real axis. Moreover, the image of $F_{\iota}$ will have to
avoid the solid catenoid $\mathrm{Cat}^{\mathrm{solid}}_{n,1}$. The idea is to
map $\\{2^{n}\leq|z|\leq 2^{n}(1+1/4)\\}$ to the convex hulls of the arcs of
circles $f_{\iota}([2^{n},2^{n}(1+1/4)])$ and
$f_{\iota}([-2^{n}(1+1/4),-2^{n}])$, and similarly $\\{2^{n}(1+1/2)\leq|z|\leq
2^{n}(1+3/4)\\}$ to the convex hull of
$f_{\iota}([2^{n}(1+1/2),2^{n}(1+3/4)])$ and
$f_{\iota}([-2^{n}(1+3/4),-2^{n}(1+1/2)])$ — these convex hulls are portions
of totally geodesic planes — and then connect them by a band connecting
$f_{\iota}([2^{n}(1+1/4),2^{n}(1+1/2)])$ and
$f_{\iota}([-2^{n}(1+1/2),-2^{n}(1+1/4)])$. See Figure 4.

Figure 4. The construction of $F_{\iota}$ if $\iota(n)=1$, in the Klein model
of $\mathbb{H}^{3}$.

Formally, let us use the upper half-space model
$\\{(z,h)\,:\,z\in\mathbb{C},h>0\\}$ of $\mathbb{H}^{3}$. As $t$ varies in
$[2^{n},2^{n}(1+1/4)]$, let $\gamma(t)$ parameterize the curve in
$\mathbb{H}^{3}$ (actually, a geodesic, although not parameterized by
arclength) contained in the vertical plane $\Re(z)=0$, in such a way that
$\lim_{x\to\pm\infty}\exp_{\gamma(t)}(xv_{\gamma(t)})=f_{\iota}(\pm t)$, where
$v_{p}$ is the vector based at $p$ and orthogonal to the vertical plane
$\Re(z)=0$, pointing towards $\Re(z)>0$. Then define

$F_{\iota}(ite^{is})=\exp_{\gamma(t)}(\chi(s)v_{\gamma(t)})$

for a fixed decreasing diffeomorphism $\chi:[-\pi/2,\pi/2]\to\mathbb{R}$. By
construction, $F_{\iota}$ extends to $f_{\iota}$ on $\pm[2^{n},2^{n}(1+1/4)]$.
Starting from $t\in[2^{n}(1+1/2),2^{n}(1+3/4)]$, we repeat exactly the same
procedure to define $F_{\iota}$ on $\\{2^{n}(1+1/2)\leq|z|\leq
2^{n}(1+3/4)\\}$, thus obtaining another portion of totally geodesic plane. We
repeat the same again to define $F_{\iota}$ on $\\{2^{n}(1+1/4)\leq|z|\leq
2^{n}(1+1/2)\\}$, simply by changing the interval of definition of $t$, with
the sole irrelevant difference that in this case the image of $\gamma$ is not
a geodesic. Putting together the three pieces, $F_{\iota}$, defined on
$\\{2^{n}\leq|z|\leq 2^{n}(1+3/4)\\}$, is continuous and piecewise smooth.
Since the minimal catenoids are contained in the convex hull of their two
asymptotic circles, it is immediate to see that the image of $F_{\iota}$ is
disjoint from the solid catenoid $\mathrm{Cat}^{\mathrm{solid}}_{n,1}$.

Having described this construction in detail when $\iota(n)=1$, we now explain
how to modify it for $\iota(n)=0$. Here we divide again the strip
$A:=\\{2^{n}\leq|z|\leq 2^{n}(1+3/4)\\}$ in three parts, but differently from
above. Let $A_{\pm}=A\cap\\{|z\mp 2^{n}(1+3/8)|=2^{n}(3/8)\\}$, and let
$A_{0}$ be the closure of $A\setminus(A_{-}\cup A_{+})$. We can use again an
auxiliary curve $\gamma:[-\pi/2,\pi/2]$ in $\mathbb{H}^{3}$, having the
property of being contained in the totally geodesic plane of symmetry between
the two totally geodesic planes with asymptotic boundaries
$\\{|z|=2^{n}(1\pm\epsilon)\\}$, such that $\gamma(\pm\pi/2)=f_{\iota}(\pm
2^{n}(1+3/8))$, and define $F_{\iota}(ie^{s})=\gamma(s)$. Then we “fill” the
strip $A$ in such a way that $F_{\iota}(A_{\pm})$ is sweeped-out by the
geodesics with endpoints $f_{\iota}(\pm 2^{n}(1+3/8+t))$, for $t\in
2^{n}(-3/8,3/8)$, and that $F_{\iota}(A_{0})$ is sweeped-out by finite
geodesic segments, in such a way that
$F_{\iota}(2^{n}ie^{s})=\exp_{p}(\chi(s)v)$ parameterizes, as in the previous
case, a geodesic with endpoints $f_{\iota}(\pm 2^{n})$, for $p$ a point in the
vertical plane $\Re(z)=0$, and analogously for
$F_{\iota}(2^{n}(1+3/4)ie^{s})$, on the other boundary half-circle of $A$. The
convex hull property will then ensure again that the image of $F_{\iota}$ is
in the complement of $\mathrm{Cat}^{\mathrm{solid}}_{n,0}$. See Figure 5. We
leave to the reader the details, which are only slightly more technical than
in the case $\iota(n)=1$.

Figure 5. The construction of $F_{\iota}$ if $\iota(n)=0$.

Finally, it remains to define $F_{\iota}$ on the strip
$\\{2^{n}(1+3/4)\leq|z|\leq 2^{n+1}\\}$. Here we take again a curve
$\gamma:[2^{n}(1+3/4),2^{n+1}]\to\mathbb{H}^{3}$ with the property that its
image is contained in $\Re(z)=0$ and that
$\lim_{x\to\pm\infty}\exp_{\gamma(t)}(xv_{\gamma(t)})=f_{\iota}(\pm t)$. We
can then define $F_{\iota}(ite^{is})=\exp_{\gamma(t)}(\chi(s)v_{\gamma(t)})$.
This definition will match the previous definitions of $F_{\iota}$ on the
common boundaries $|z|=2^{n}(1+3/4)$ or $|z|=2^{n+1}$. See Figure 6.

Figure 6. The construction of $F_{\iota}$ on the remaining strip, which is in
common for all choices of $\iota$.

Hence $F_{\iota}$ is globally well-defined, continuous and piecewise smooth.
As in the proof of Lemma 6.1, one can observe that the image of
$2^{n}\leq|z|\leq 2^{n+1}$ under the map $F_{\iota}$ is contained, in the
upper half-space, in $B(0,2^{n+1}(1+\epsilon))\setminus
B(0,2^{n}(1+\epsilon))$. Using this property, following the same lines as the
proof of Lemma 6.1 it is easy to verify that $F_{\iota}$ is injective and
$\lim_{z\to 0}F_{\iota}(z)=0$, $\lim_{z\to\infty}F_{\iota}(z)=\infty$. By
construction $F_{\iota}$ extends to $f_{\iota}$ on $\mathbb{R}$, and therefore
$F_{\iota}$ is the desired continuous and injective extension of $f_{\iota}$.
∎

###  6.3. Conclusion of the proof

We are now ready to prove Theorem 1.6.

###### Proof of Theorem 1.6.

Let $\Lambda$ be the image of the continuous injective map $f$, and fix
$\iota:\mathbb{Z}\to\\{0,1\\}$. We will construct a stable minimal disk
$\Sigma_{\iota}$ contained in

$\Omega_{\iota}:=\mathbb{H}^{3}\setminus\bigcup_{n\in\mathbb{Z}}\mathrm{Cat}^{\mathrm{solid}}_{n,\iota(n)}$

with $\partial_{\infty}\Sigma_{\iota}=\Lambda$.

Let us denote by $U_{\iota}$ the topological disk constructed in Lemma 6.5,
namely the image of the continuous injective map
$F_{\iota}:\mathbb{D}\to\mathbb{H}^{3}$, where we identify here the upper
half-plane model of $\mathbb{H}^{2}$ with the unit disc $\mathbb{D}$. We will
denote by $\mathcal{C}(U_{\iota})$ the convex hull of $\Lambda\cup U_{\iota}$
in $\mathbb{H}^{3}\cup\partial_{\infty}\mathbb{H}^{3}$. As a preliminary step,
observe that

(6.5) $\partial_{\infty}\mathcal{C}(U_{\iota})=\Lambda~{}.$

Indeed, given any point $p$ in
$\partial_{\infty}\mathbb{H}^{3}\setminus\Lambda$, since the map $F_{\iota}$
extends to a continuous injective map from $\overline{\mathbb{D}}$ to
$\mathbb{H}^{3}\cup\partial_{\infty}\mathbb{H}^{3}$, $U_{\iota}\cup\Lambda$ is
a closed subset of $\mathbb{H}^{3}\cup\partial_{\infty}\mathbb{H}^{3}$, and
one can find a neighbourhood $\mathcal{U}$ of $p$ in the topology of the
closed ball which is disjoint from $U_{\iota}\cup\Lambda$. Then a totally
geodesic plane $P$ contained in $\mathcal{U}$ disconnects $p$ from the convex
hull of $\Lambda\cup U_{\iota}$, thus showing $p\notin\mathcal{C}(U_{\iota})$.

Now, let $\gamma_{n}=F_{\iota}(\\{|z|=1-1/n\\})$. Then
$\\{\gamma_{n}\\}_{n\in\mathbb{N}}$ is a sequence of nullhomotopic simple
closed curves in $\mathcal{C}(U_{\iota})$ that converge to $\Lambda$ as
$n\to+\infty$. The proof of the existence of $\Sigma_{\iota}$ now roughly
follows the strategy of Lemma 3.3.

Fix the point $x_{o}=F_{\iota}(0)\in U_{\iota}$. For any $n$, let $R_{n}$ be a
sufficiently large radius so that $\gamma_{n}\subset B(x_{o},R_{n})$. Up to a
small perturbation of $R_{n}$, assume moreover, as a consequence of genericity
of transverse intersection, that the intersection of $\partial B(x_{o},R_{n})$
and $\partial\Omega_{\iota}$ is transverse. By [HS88, Theorem 6.3] applied to
$M_{n}:=\Omega_{\iota}\cap B(x_{o},R_{n})$, which is sufficiently convex in
the terminology of [HS88], we can solve the Plateau problem to find a least
area (hence stable) minimal disk $\Sigma_{\iota}^{n}$ contained in $M_{n}$
with $\partial\Sigma_{\iota}^{n}=\gamma_{n}$.

We will show that $\Sigma_{\iota}^{n}$ converges, up to extracting a
subsequence, smoothly on compact subsets of $\mathbb{H}^{3}$ to a properly
embedded minimal disk $\Sigma_{\iota}$ contained in $\Omega_{\iota}$. We first
observe that there exists $R_{0}>0$ such that $\Sigma_{\iota}^{n}\cap
B(x_{o},R_{0})\neq\emptyset$ for all $n$. To see this, recall that
$\mathcal{C}(U_{\iota})$ is homeomorphic to a closed ball, hence $\Lambda$
disconnects $\partial\mathcal{C}(U_{\iota})$, which is homeomorphic to a
sphere, into two topological disks which we denote
$\partial^{\pm}\mathcal{C}(U_{\iota})$. Pick a geodesic segment $\eta$
connecting two points $x_{\pm}\in\partial^{\pm}\mathcal{C}(U_{\iota})$ and
containing $x_{o}$. Since $\gamma_{n}$ and $\eta$ are contained in
$\mathcal{C}(U_{\iota})$ and linked, and the disk $\Sigma_{\iota}^{n}$ is
contained in $\mathcal{C}(U_{\iota})$, it must intersect $\eta$. Hence
$\Sigma_{\iota}^{n}$ intersects $B(x_{o},R_{0})$ for any choice of $R_{0}$
such that $B(x_{o},R_{0})$ contains $\eta$.

Now, fix any radius $R>0$ and consider the ball $B(x_{o},R)$. For $n=n(R)$
sufficiently large, $\gamma_{n}$ is contained in $\mathbb{H}^{3}\setminus
B(x_{o},R+1)$. The distance from any point of $\Sigma_{\iota}^{n}\cap
B(x_{o},R)$ and $\partial\Sigma_{\iota}^{n}=\gamma_{n}$ is thus bounded below
by $1$. By Theorem 2.3 there is a constant $C$, independent of $n$, such that
the Gaussian curvature of $\Sigma_{\iota}^{n}$ is bounded by $C$ in absolute
value. Applying Theorem 2.5, we can extract a subsequence $n_{k}$ such that
$\Sigma_{\iota}^{n_{k}}\cap B(x_{o},R)$ converges smoothly to a minimal disc.
By a diagonal argument, we can then find a subsequence of
$\\{\Sigma_{\iota}^{n}\\}_{n\in\mathbb{N}}$ that converges smoothly on compact
subsets to a minimal disk $\Sigma_{\iota}$. Moreover the limit
$\Sigma_{\iota}$ is stable since it is the smooth limit of the stable minimal
surfaces $\Sigma_{\iota}^{n}$.

An easy argument by contradiction as in the proof of Corollary 2.6 shows that
$\Sigma_{\iota}$ is properly embedded. Moreover, since $\Sigma_{\iota}^{n}$ is
contained in $\Omega_{\iota}$, then $\Sigma_{\iota}$ is contained in the
closure of $\Omega_{\iota}$. This would be sufficient for our purpose;
nonetheless the strong maximum principle actually implies that
$\Sigma_{\iota}$ is contained in $\Omega_{\iota}$ itself. Finally, since
$\Sigma_{\iota}^{n}$ is contained in $\mathcal{C}(U_{\iota})$ for all $n$, so
too is $\Sigma_{\iota}$. Then (6.5) implies that
$\partial_{\infty}\Sigma_{\iota}=\Lambda$.

It only remains to show that, if $\iota_{0}\neq\iota_{1}$, then
$\Sigma_{\iota_{0}}\neq\Sigma_{\iota_{1}}$. By hypothesis, there exists
$m\in\mathbb{Z}$ such that $\iota_{0}(m)\neq\iota_{1}(m)$. Up to switching the
roles, suppose $\iota_{0}(m)=0$ and $\iota_{1}(m)=1$. Then, by construction,
$\Sigma_{\iota_{0}}$ is contained in the complement of
$\mathrm{Cat}^{\mathrm{solid}}_{m,0}$, whereas $\Sigma_{\iota_{1}}$ is
contained in the complement of $\mathrm{Cat}^{\mathrm{solid}}_{m,1}$. But the
solid catenoids $\mathrm{Cat}^{\mathrm{solid}}_{m,0}$ and
$\mathrm{Cat}^{\mathrm{solid}}_{m,1}$ are linked in the sense of [HW15,
Section 3.4]. Then [HW15, Lemma 3.12] implies that $\Lambda$ is not
nullhomotopic in
$(\mathbb{H}^{3}\cup\partial_{\infty}\mathbb{H}^{3})\setminus(\mathrm{Cat}^{\mathrm{solid}}_{m,0}\cup\mathrm{Cat}^{\mathrm{solid}}_{m,1})$.
Now, if $\Sigma_{\iota_{0}}=\Sigma_{\iota_{1}}=:\Sigma$, then $\Sigma$ would
be a disk contained in the complement of
$\mathrm{Cat}^{\mathrm{solid}}_{m,0}\cup\mathrm{Cat}^{\mathrm{solid}}_{m,1}$,
hence contradicting that $\Lambda$ is not nullhomotopic. This concludes the
proof.  ∎

##  Appendix A Small curvatures

In this appendix, we explain how to obtain Theorems 4.2 and 4.3. These results
are well-known under the assumption that $\Sigma$ has small curvatures (see
for instance [Eps84]). However, we need to relax the hypothesis here to assume
that $\Sigma$ has only weakly small curvatures. We follow here the approach of
[EES22, Section 4].

The fundamental observation is that, if $\Sigma$ is a hypersurface in
$\mathbb{H}^{n+1}$ of weakly small curvatures and $\gamma:I\to\Sigma$ is a
geodesic for the first fundamental form of $\Sigma$, then the acceleration of
$\gamma$ is

(A.1)
$\nabla^{h}_{\gamma^{\prime}(t)}\gamma^{\prime}(t)=A(\gamma^{\prime}(t),\gamma^{\prime}(t))N(\gamma(t))$

and therefore, by the weakly small curvature assumption,

(A.2) $\|\nabla^{h}_{\gamma^{\prime}(t)}\gamma^{\prime}(t)\|\leq 1~{}.$

This leads us to study curves satisfying the bound (A.2) on the acceleration.
Let $S$ be a horosphere or a sphere in $\mathbb{H}^{n+1}$. (By sphere, here we
always mean the set of points at distance $r>0$ from a given points of
$\mathbb{H}^{n+1}$.) The complement of $S$ has two connected components, one
of which is geodesically convex, and the other is not. We call the former the
(open) convex side of $S$, and the latter the (open) concave side of $S$. The
closed convex (resp. concave) side of $S$ is the union of $S$ with the open
convex (resp. concave) side. The following lemma is an improvement of [EES22,
Lemma 4.9], following a similar strategy of proof.

######  Lemma A.1.

Let $\gamma:[a,b]\to\mathbb{H}^{n+1}$ be a smooth curve satisfying (A.2). Then

  * •

The image of $\gamma$ lies in the open concave side of any sphere tangent to
$\gamma$, except for the tangency point.

  * •

The image of $\gamma$ lies in the closed concave side of any horosphere
tangent to $\gamma$.

###### Proof.

It suffices to prove the first item, as the second item follows by taking the
limit of tangent spheres as the radius goes to infinity.

Assume that the tangency point is, in the upper half-space model
$\gamma(0)=(0,\ldots,0,1)$, that $\gamma$ is parameterized by arclength, that
$\gamma^{\prime}(0)=(1,0,\ldots,0)$ and that the tangent horosphere is
$\\{x_{n}=1\\}$. Hence the tangent spheres $S_{r}$ of radius $r$ are, in a
neighbourhood of $\gamma(0)$, graphs of functions $f_{r}:U_{r}\to\mathbb{R}$,
where $U_{r}\subset\mathbb{R}^{n}$ is a sufficiently small ball, and by
symmetry $f_{r}$ only depends on the distance from the origin in
$\mathbb{R}^{n}$.

First we show $\gamma$ lies on the open concave side of any tangent sphere
$S_{r}$ for small $t$. We use a subscript $(\cdot)_{n+1}$ to denote the last
coordinate in the upper half-space model. Thus
$\gamma_{n+1}(0)=\gamma^{\prime}_{n+1}(0)=0$. By a direct computation of the
Christoffel symbol $\Gamma_{11}^{n}=1$, we see that

$(\nabla^{h}_{\gamma^{\prime}}\gamma^{\prime})_{n+1}(0)=\gamma_{n+1}^{\prime\prime}(0)+1~{}.$

Since $\gamma$ has small acceleration, we obtain

$\gamma_{n+1}^{\prime\prime}(0)\leq|(\nabla^{h}_{\gamma^{\prime}}\gamma^{\prime})_{n+1}(0)|-1\leq\|(\nabla^{h}_{\gamma^{\prime}}\gamma^{\prime})(0)\|-1\leq
0~{}.$

Now, denote $\hat{\gamma}$ the projection of $\gamma$ to $\mathbb{R}^{n}$. By
an easy computation following similar lines,
$(f_{r}\circ\hat{\gamma})(0)=(f_{r}\circ\hat{\gamma})^{\prime}(0)=0$ and
$(f_{r}\circ\hat{\gamma})^{\prime\prime}(0)=-1+1/\tanh(r)>0$. Hence for
$t\in(0,\epsilon)$, $\gamma_{n+1}(t)<f_{r}(t)$ and this shows that $\gamma$
stays for small times in the open concave side of $S_{r}$.

It remains to show that $\gamma(t)$ stays in in the open concave side of
$S_{r}$ for every $t>0$. Suppose by contradiction that for some $t_{0}$,
$\gamma(t_{0})\in S_{r}$. Let $p$ be the center of $S_{r}$. Then the function
$t\mapsto d(p,\gamma(t))$ is equal to $r$ for $t=0$ and $t=t_{0}$, and is
larger than $r$ for $t\in(0,\epsilon)$. Hence it admits a maximum point
$t_{\max}$. This implies that $\gamma$ is tangent to the geodesic sphere
centered at $p$ of radius $d(p,\gamma(t_{\max}))$ and thus contradicts the
first part of the proof.  ∎

Having established Lemma A.1, the following lemma follows immediately.

######  Lemma A.2.

Let $\Sigma\subset\mathbb{H}^{n+1}$ be a properly immersed hypersurface of
weakly small curvatures. Then $\Sigma$ is contained in the closed concave side
of every tangent horosphere.

###### Proof.

Let $p\in\Sigma$ and let $S$ be a tangent horosphere. Given any other $q$, by
completeness of $\Sigma$, we can find a geodesic segment $\gamma$ joining $p$
and $q$. As observed in (A.2), $\gamma$ satisfies the hypothesis of Lemma A.1,
hence $q$ is contained in the closed concave side of $S$.  ∎

Now, the proof of Theorem 4.2 follows exactly the proof of [EES22, Proposition
4.15]. To prove Theorem 4.3, we need the following well-known result, which
follows from a direct computation (see for instance [HW13, formula $(2.2)$] or
[EES22, Lemma 4.5]).

######  Lemma A.3.

Let $\Sigma$ be an embedded hypersurface in $\mathbb{H}^{n+1}$ of weakly small
curvatures, and let $N$ be its unit normal vector. Then for every
$t\in\mathbb{R}$, the map $\iota_{t}:\Sigma\to\mathbb{H}^{n+1}$ defined by
$\iota_{t}(x)=\exp_{x}(tN(x))$ is an immersion, and its principal curvatures
satisfy the identity

(A.3)
$\lambda^{t}_{i}(\iota_{t}(x))={\frac{\lambda_{i}(x)-\tanh(t)}{1-\lambda_{i}(x)\tanh(t)}}~{},$

where $\lambda_{1},\ldots,\lambda_{n}$ are the principal curvatures of
$\Sigma$.

###### Proof of Theorem 4.3.

It follows from Lemma A.3 that the map $F(x,t)=\iota_{t}(x)$ has injective
differential, hence is a local diffeomorphism. We show that it is bijective.
To show that $F$ is surjective, let $p$ be any point in $\mathbb{H}^{n+1}$,
let $r=d(p,\Sigma)$ and let $x\in\Sigma$ be a point that realizes this
distance. Then the geodesic sphere of center $p$ and radius $r$ is tangent to
$\Sigma$ at $x$, and $p=F(x,r)$. For injectivity, suppose that
$p=F(x_{1},t_{1})=F(x_{2},t_{2})$. Let $S_{t_{i}}$ be the geodesic spheres
centered at $p$ of radius $t_{i}$. As before, $S_{t_{i}}$ is tangent to
$\Sigma$ at $x_{i}$. By Lemma A.1, $\Sigma$ is contained, except for the
tangency points, in the open concave side of both $S_{t_{1}}$ and $S_{t_{2}}$.
This implies that $t_{1}=t_{2}$, because otherwise the open concave side of
$S_{t_{1}}$ would intersect the convex side of $S_{t_{2}}$ (or vice versa),
leading to a contradiction. By Lemma A.1 again, $\Sigma\setminus\\{x_{1}\\}$
is contained in the open concave side of $S_{t_{1}}=S_{t_{2}}$, and thus
$x_{2}=x_{1}$.

We have thus shown that each $\iota_{t}$ is a proper embedding, and it has
weakly small curvatures by Lemma A.3. When $\Sigma$ is minimal, Lemma A.3
again implies the last statement on the sign of the mean curvature of
$\iota_{t}$, because $\lambda_{i}^{t}$ is a decreasing function of $t$.  ∎

We have already observed that, for $n=2$, if the Jordan curve $\Lambda$ spans
a surface of strongly small curvatures, then $\Lambda$ is a quasicircle, and
this in turns implies that $\Lambda$ has finite width. We conclude this
appendix by a direct proof, that works in any dimensions.

######  Lemma A.4.

Let $\Lambda$ be a topologically embedded $n$-sphere $\Lambda$ in
$\mathbb{S}_{\infty}^{n}$ and $\Sigma$ be a properly embedded hypersurface of
strongly small curvatures such that $\partial_{\infty}\Sigma=\Lambda$. Then
$\mathcal{C}(\Lambda)$ has finite width.

###### Proof.

Assume that the principal curvatures $\lambda_{1},\ldots,\lambda_{n}$ of
$\Sigma$ are smaller than $1-\epsilon$ in absolute value. By Lemma A.3, for
$t_{0}>\tanh^{-1}(1-\epsilon)$ the hypersurface $\Sigma_{t_{0}}$ (resp.
$\Sigma_{-t_{0}}$) has negative (resp. positive) principal curvatures, that
is, it is locally concave (resp. convex) with respect to the direction of the
normal vector $N$. Since $\Sigma_{\pm t_{0}}$ are properly embedded, they
bound a geodesically convex region $U$, which thus contains the
$\mathcal{C}(\Lambda)$. By Theorem 4.3, every point in $\mathcal{C}(\Lambda)$
lies on a geodesic segment of the form $t\mapsto F(x,t)$, for
$t\in[-t_{0},t_{0}]$. Hence for every $x\in\mathcal{C}(\Lambda)$,
$d(x,\partial^{+}\mathcal{C}(\Lambda))+d(x,\partial^{-}\mathcal{C}(\Lambda)\leq
2t_{0}$. This shows that $w(\Lambda)\leq 2t_{0}$.  ∎

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