# The Weyl problem for unbounded convex domains in ${\mathbb{H}}^{3}$

Jean-Marc Schlenker  Jean-Marc Schlenker: University of Luxembourg, FSTM,
Department of Mathematics, Maison du nombre, 6 avenue de la Fonte, L-4364
Esch-sur-Alzette, Luxembourg<EMAIL_ADDRESS>
(Date: v0, )

###### Abstract.

Let $K\subset{\mathbb{H}}^{3}$ be a convex subset in ${\mathbb{H}}^{3}$ with
smooth, strictly convex boundary. The induced metric on $\partial K$ then has
curvature $K>-1$. It was proved by Alexandrov that if $K$ is bounded, then it
is uniquely determined by the induced metric on the boundary, and any smooth
metric with curvature $K>-1$ can be obtained.

We propose here an extension of the existence part of this result to unbounded
convex domains in ${\mathbb{H}}^{3}$. The induced metric on $\partial K$ is
then clearly not sufficient to determine $K$. However one can consider a
richer data on the boundary including the ideal boundary of $K$. Specifically,
we consider the data composed of the conformal structure on the full boundary
of $K$ (in the Poincaré model of ${\mathbb{H}}^{3}$), together with the
induced metric on $\partial K$. We show that a wide range of “reasonable” data
of this type, satisfying mild curvature conditions, can be realized on the
boundary of a convex subset in ${\mathbb{H}}^{3}$.

Keywords: Weyl problem, hyperbolic geometry, convex domain, isometric
embedding.

Partially supported by FNR projects INTER/ANR/15/11211745, OPEN/16/11405402
and O20/14766753. The author also acknowledges support from U.S. National
Science Foundation grants DMS-1107452, 1107263, 1107367 “RNMS: GEometric
structures And Representation varieties” (the GEAR Network).

###### Contents

  1. 1 Introduction and main results
    1. 1.1 The classical Weyl problem in Euclidean and hyperbolic space
    2. 1.2 The boundary data on unbounded convex subsets
    3. 1.3 A dual question
    4. 1.4 Related results and motivations
    5. 1.5 Questions not considered here
    6. 1.6 Outline of the proofs and content of the paper
  2. 2 Background
    1. 2.1 Conformal metrics on surfaces
    2. 2.2 Surfaces in ${\mathbb{H}}^{3}$
  3. 3 Approximation by metrics on the sphere
    1. 3.1 Bounds on conformal factors
    2. 3.2 Construction of approximating metrics
    3. 3.3 A lower bound on the injectivity radius
    4. 3.4 A sequence of convex subsets
  4. 4 Bounds on principal curvatures
    1. 4.1 Principal curvature bounds for convex surfaces
    2. 4.2 Curvature bounds in the almost-flat regions
  5. 5 Proofs of the main results
    1. 5.1 Convergence of the Gauss maps
    2. 5.2 Convergence of boundary data

##  1\. Introduction and main results

###  1.1. The classical Weyl problem in Euclidean and hyperbolic space

The study of smooth or polyhedral convex surfaces in the 3-dimensional
Euclidean space ${\mathbb{R}}^{3}$ has a long history. One of the first
“modern” results on this theme was the proof by Legendre [LegII] and Cauchy
[Cau13] of the rigidity of convex polyhedra: if two polyhedra have the same
combinatorics and corresponding faces are isometric, then they are congruent.

Alexandrov [Ale05] later realized that Cauchy’s rigidity result is related to
a much deeper statement. The induced metric on the boundary of a convex
polyhedron in ${\mathbb{R}}^{3}$ is an Euclidean metric with cone
singularities with cone angle less than $2\pi$ at the vertices, and Alexandrov
proved that each metric of this type on the sphere can be realized as the
induced metric on the boundary of a unique polyhedron. This result was later
extended by Alexandrov [Ale05] to bounded convex subset of ${\mathbb{R}}^{3}$,
thus solving the “Weyl problem”, proposed by H. Weyl in 1915 and for which
substantial progress had already been made by H. Lewy [Lew35] and others.

######  Theorem 1.1 (Lewy, Alexandrov).

Any smooth metric of positive curvature on the sphere is induced on the
boundary of a unique smooth strictly convex subset of ${\mathbb{R}}^{3}$.

Alexandrov and Pogorelov also extended Theorem 1.1 to hyperbolic space.

######  Theorem 1.2 (Alexandrov, Pogorelov).

Any smooth metric of curvature $K>-1$ on the sphere is induced on the boundary
of a unique convex subset of ${\mathbb{H}}^{3}$ with smooth boundary.

Versions of Theorem 1.2 also hold for ideal polyhedra [Riv92] and for
hyperideal polyhedra [Sch98a] (see also [Sch98b] for a result on convex
domains bounded by disks, as well as a broad generalization by Luo and Wu
[LW19]). The existence of those results hints at the possibility of a wider
extension to more general unbounded convex domains in ${\mathbb{H}}^{3}$ as
considered here.

###  1.2. The boundary data on unbounded convex subsets

Before stating our main result, we define a notion of “boundary data” that is
adapted to complete convex subsets of ${\mathbb{H}}^{3}$. In the special case
of bounded convex subsets, it coincides with the induced metric in the usual
sense.

We consider a convex subsets $\Omega\subset{\mathbb{H}}^{3}$, which a priori
is unbounded. Let $\partial_{0}\Omega=\partial\rho(\Omega)$, where
$\rho:{\mathbb{H}}^{3}\to{\mathbb{R}}^{3}$ is the Poincaré model of
${\mathbb{H}}^{3}$, equipped with the conformal class $c_{0}$ of the induced
metric. So $\partial_{0}\Omega$ is the “full” boundary of $\Omega$, including
its metric boundary in ${\mathbb{H}}^{3}$ but also its ideal boundary.

We now define a notion of boundary data for a convex subset of
${\mathbb{H}}^{3}$. When this subset is bounded, it’s just the induced metric
on the boundary. For an unbounded convex domain $\Omega$, however, a boundary
data records both the induced metric on the metric boundary of $\Omega$ and
the conformal structure on the “full” boundary $\partial_{0}\Omega$, coming
from the induced metric on the metric boundary but also from the conformal
structure at infinity on the ideal boundary.

######  Definition 1.3.

Let $\Omega\subset{\mathbb{H}}^{3}$ be a convex subset. A boundary data for
$\Omega$ is a pair $(U,h)$, where $U\subset{\mathbb{C}P}^{1}$ is an open
subset and $h$ is a complete, conformal metric on $U$, such that there exists
a homeomorphism $\gamma:(\partial_{0}\Omega,c_{0})\to{\mathbb{C}P}^{1}$ such
that:

  * •

$\gamma$ is conformal on the interior of $\partial_{\infty}\Omega$,

  * •

$\gamma\circ\rho(\partial\Omega)=U$, and $\gamma\circ\rho$ is an isometry from
$\partial\Omega$ equipped with its induced metric to $(U,h)$.

By construction, $h$ has curvature $K\geq-1$ by the Gauss formula. Note that
$\partial_{0}\Omega=\partial\Omega\cup\partial_{\infty}\Omega$, and another
way to define $c_{0}$ is as the conformal structure of ${\mathbb{C}P}^{1}$ on
$\partial_{\infty}\Omega$, and the conformal structure of the induced metric
on $\partial\Omega$.

We do not consider here whether a boundary data is always unique. We note
however that one always exists under conditions that cover the hypothesis of
Theorem 1.6 below.

######  Lemma 1.4.

Let $\Omega\subset{\mathbb{H}}^{3}$ be a convex subset such that the induced
metric on $\partial\Omega$ has

  1. (1)

curvature $K\in[-1+\epsilon,1/\epsilon]$, for some $\epsilon>0$, and

  2. (2)

injectivity radius bounded from below by $\epsilon$.

Then there exists a boundary data for $\Omega$.

###### Proof.

Let $\delta:\partial_{\infty}{\mathbb{H}}^{3}\to\partial_{0}\Omega$ be the
closest-point projection – in other terms, $\delta$ is the identity on
$\partial_{\infty}\Omega$, while it projects each point of
$\partial_{\infty}{\mathbb{H}}^{3}\setminus\partial_{\infty}\Omega$ to a point
of $\partial\Omega$.

It is proved in Lemma 4.1 that, under the hypothesis considered here,
$\partial\Omega$ has principal curvatures in $[1/k_{0},k_{0}]$, for some
$k_{0}>1$.

As a consequence, $\delta$ is $K$-quasi-conformal, for a constant $K$
depending only on $k_{0}$. Applying the measurable Riemann mapping theorem
[AB60] to the Beltrami differential of $\delta$ on
$\partial_{\infty}{\mathbb{H}}^{3}\setminus\partial_{\infty}\Omega$, we obtain
a quasi-conformal homeomorphism
$q:\partial_{\infty}{\mathbb{H}}^{3}\to{\mathbb{C}P}^{1}$ which is conformal
on $\partial_{\infty}\Omega$, and has the same Beltrami differential as
$\delta$ on
$\partial_{\infty}{\mathbb{H}}^{3}\setminus\partial_{\infty}\Omega$.

The homeomorphism $\delta\circ q^{-1}:{\mathbb{C}P}^{1}\to\partial_{0}\Omega$
is then conformal by construction. The pair $((\delta\circ
q^{-1})^{-1}(\partial\Omega),(\delta\circ q^{-1})^{*}I)$ is therefore a
boundary data for $\Omega$, where $I$ denotes the induced metric on
$\partial\Omega$. ∎

Our main motivation is the following question, appearing as Question
$W^{gen}_{{\mathbb{H}}^{3}}$ in [Sch20], reformulated here in line with
Definition 1.3.

######  Question 1.5.

Let $U\subset{\mathbb{C}P}^{1}$ be an open subset, and let $h$ be a complete
metric of curvature $K\geq-1$ on $U$. Is there a unique convex subset
$\Omega\subset{\mathbb{H}}^{3}$ such that $(U,h)$ is a boundary data for
$\Omega$?

As pointed out in [Sch20], the answer to this question in such a general
setting might be negative, and additional assumptions might be necessary. We
provide here an existence result under stronger curvature conditions, and
under a mild regularity asumption on the boundary of $U$ together with a
topological hypothesis on $U$.

######  Theorem 1.6.

Let $k_{0}>0$, $U\subset{\mathbb{C}P}^{1}$ be an open subset which is a
disjoint union of $k_{0}$-quasidisks, let $\epsilon>0$, and let $h$ be a
smooth, complete metric or curvature $K\in[-1+\epsilon,-\epsilon]$ on $U$.
There exists a convex subset $\Omega\subset{\mathbb{H}}^{3}$ such that the
boundary data of $\Omega$ is $(U,h)$.

A quasidisk in ${\mathbb{C}P}^{1}$ is the image of a disk by a quasiconformal
map from ${\mathbb{C}P}^{1}$ to ${\mathbb{C}P}^{1}$.

###  1.3. A dual question

Specifically in ${\mathbb{H}}^{3}$, there is also a dual Weyl problem,
concerning the third fundamental form on the boundary of convex bodies, rather
than their induced metrics. This third fundamental form is in fact the induced
metric on the boundary of a dual convex body in the de Sitter space.

For ideal polyhedra, the third fundamental form is encoded in the (exterior)
dihedral angles, so that the dual Weyl problem corresponds to determining
which families of dihedral angles can be obtained, see [And71, Riv96]. For
bounded polyhedron, it corresponds to the “dual metric” considered by Hodgson
and Rivin [HR93], while a result is also known for smooth metrics [Sch96].

Question 1.5 has an analog concerning the extension of this “dual” Weyl
problem to unbounded convex domains with a prescribed data at infinity that
includes the third fundamental form, rather than the induced metric, on
$\partial\Omega$, see [Sch20]. We do not consider this dual problem here. It
appears plausible that the methods developed here to prove Theorem 1.6 could
also apply, in some form, to prove a dual statement, but a different
regularization procedure appears to be needed to obtain the necessary
estimates.

###  1.4. Related results and motivations

A list of related results as well as some background information can be found
[Sch20], we only list here a few results and conjectures in direct relation
with Theorem 1.6.

#### Convex subsets with a quasicircle at infinity

One of the simplest cases that can be considered is when $U$ is the complement
of a quasi-circle in ${\mathbb{C}P}^{1}$, and $h$ is a complete metric of
constant curvature $K\in(-1,0)$ on $U$. The corresponding special cases of
Theorem 1.6 (together with a dual result with the induced metric replaced by
the third fundamental form) is the main result of [BDMS19]. (Note that the
result of [BDMS19] on the induced metrics also covers the case $K=-1$, where
the convex subset $\Omega$ is the convex hull of its ideal boundary.)

The statements of Theorem A and Theorem B in [BDMS19] superficially appear
quite different from the statement of Theorem 1.6, since it concerns the
possibility to prescribe the “gluing map” at infinity between the induced
metrics on the two connected components of the boundary of a domain, with
induced metric of constant curvature, which has as boundary at infinity a
quasicircle. A moment of reflection shows however that Theorem B of [BDMS19]
is the special case of Theorem 1.6 when $U$ is the complement of a quasicircle
in ${\mathbb{C}P}^{1}$ and $h$ has constant curvature $K\in(-1,0)$.

The main motivation for this result from [BDMS19] is a conjecture made by
Thurston concerning the induced metric and the measured bending lamination on
the convex cores of geometrically finite hyperbolic manifolds. We recall those
conjectures in the quasifuchsian case for simplicity. A quasifuchsian manifold
is a complete hyperbolic manifold, homeomorphic to $S\times{\mathbb{R}}$,
where $S$ is a closed surface of genus at least $2$, containing a compact,
non-empty, geodesically convex subset. A quasifuchsian manifold $M$ contains a
smallest non-empty convex subset, its convex core $C(M)$, which is either a
totally geodesic surface (in the “Fuchsian” case) or a closed domain with
boundary the disjoint union of two surfaces, each homeomorphic to $S$, with an
induced metric of constant curvature $-1$, pleated along a measured
laminations. Thurston thus stated two conjectures:

  1. (1)

any pair of hyperbolic metrics on $S$, considered up to isotopy, can be
uniquely realized as the induced metrics on the boundary of a quasifuchsian
manifold,

  2. (2)

any pair of measured laminations on $S$ that fill together, such that no leaf
has weight larger than $\pi$, can be realized uniquely as the measured bending
lamination on the convex core of a quasifuchsian manifold.

The existence part of both statements is known, for (1) it is a consequence of
results of Labourie [Lab92] or of Epstein–Marden [EM86], while for (2) it is
proved (for geometrically finite hyperbolic manifolds) by Bonahon–Otal [BO04].

Existence and uniqueness results are known, however, for larger geodesically
convex subsets (containing the convex core) with smooth boundary. The induced
metric then has curvature $K>-1$, while the measured bending lamination is
replaced by the third fundamental form of the boundary, which has curvature
$K<1$ and closed, contractible geodesics of length $L>2\pi$. It is proved in
[Lab92] (for the existence) and [Sch06] (for the uniqueness) that, if
$h_{-},h_{+}$ are two smooth metrics of curvature $K>-1$ on a closed surface
$S$ of genus at least $2$, there exists a unique hyperbolic metric with convex
boundary on $S\times[0,1]$ such that the induced metrics on the two connected
components of the boundary are $h_{-}$ and $h_{+}$.

#### Convex domains in convex co-compact hyperbolic manifolds

The two Thurston conjectures above are in fact considered for convex co-
compact hyperbolic manifolds, and in this more general setting the existence
has been proved for both conjecture, and are due to the authors mentioned
above in the quasifuchsian setting.

Moreover, if $M$ is a closed manifold with non-empty boundary which admits a
convex co-compact hyperbolic manifold on its interior, then, again by [Lab92,
Sch06], any metric of curvature $K>-1$ on the boundary of $M$ can be uniquely
realized as the induced metric of a convex hyperbolic metric $g$ on $M$.

Theorem 1.6 can be seen as a (partial) generalization of the existence part of
this result. Indeed, given such a hyperbolic metric $g$ on $M$, the universal
cover $\tilde{M}$ of $M$, equipped with the lifted metric $\tilde{g}$, is
isometric to the interior of a convex domain $\Omega\subset{\mathbb{H}}^{3}$,
with $\partial_{\infty}\Omega$ equal to the limit set $\Lambda$ of $M$.
$\Omega$ is then invariant under the action of a representation
$\rho:\pi_{1}M\to PSL(2,{\mathbb{C}})$. The existence result from [Lab92] then
implies the existence of a convex domain $\Omega\subset{\mathbb{H}}^{3}$ with
a prescribed metric of curvature $K>-1$ on each boundary component (equal to
the lift of a metric on the corresponding boundary component of $\partial M$).
Moreover, in $\partial_{0}\Omega$ equipped with the underlying conformal
structure, each connected component of $\partial\Omega$ corresponds to a
quasidisk. So this result from [Lab92] implies a special case of Theorem 1.6,
which moreover is invariant under the representation $\rho$ (without the
asumption that the curvature is negative). Theorem 1.6 can be considered as
extending the existence result of [Lab92] without the need for a group action.
However Theorem 1.6 is more general since it allows for situations where $U$
is not dense in ${\mathbb{C}P}^{1}$.

#### Convex domains complete on one side in quasifuchsian manifolds

One such situation can be found by considering a closed, geodesically convex,
subset $\Omega$ in a quasifuchsian manifold $M$, assuming that $\Omega$ is
complete “on one side”, that is, $\Omega$ is homeomorphic to
$S\times[0,\infty)$, where $S$ is again a closed surface of genus at least $2$
and $\Omega$ is complete on the side corresponding to $\infty$, while its
metric boundary corresponds to $S\times\\{0\\}$, which is a smooth, locally
strictly convex surface $\partial\Omega$ in $M$. The induced metric on
$\partial\Omega$ then has curvature $K>-1$.

One can then ask whether given any conformal structure $c$ on $S$, and any
metric $h$ of curvature $K>-1$ on $S$ (both considered up to isotopy) there is
a unique quasifuchsian metric on $S\times{\mathbb{R}}$, containing a unique
geodesically convex subset $\Omega$ complete on one side, such that the
conformal structure on the complete boundary is $c$, while the induced metric
on the other boundary is $h$.

Lifting to the universal cover of $\Omega$, this translates as another
instance of Theorem 1.6, where $U$ is now a disk in ${\mathbb{C}P}^{1}$
(corresponding to the universal cover of $\partial\Omega$) while the interior
of the complement of $U$ corresponds to the universal cover of
$\partial_{\infty}\Omega$.

###  1.5. Questions not considered here

The Weyl problem described in Section 1.1 can also be stated in terms of
isometric immersions of locally convex surfaces in ${\mathbb{R}}^{3}$ or
${\mathbb{H}}^{3}$. This point of view leads to different extensions to
unbounded locally convex surfaces in ${\mathbb{H}}^{3}$, see [Sch20, Section
6]. We not not consider questions arising from this other point of view here.

It should be noted that the curvature hypothesis in Theorem 1.6 are quite
restrictive. On one hand, one could consider metrics of curvature $K\geq-1$,
including the special case of $K=-1$, corresponding to the case where
$\partial\Omega$ is a locally convex pleated surface, and $\Omega$ is the
convex hull of its asymptotic boundary $\partial_{\infty}\Omega$. On the other
hand, one could consider metrics with positive curvature at some points – this
is excluded here for “technical” reasons.

We also do not consider here the uniqueness in Theorem 1.6, since different
methods and tools are necessary.

###  1.6. Outline of the proofs and content of the paper

Section 2 contains a number of well-known constructions and results that are
needed at various places in the paper. Section 3 shows how a given boundary
data $(U,h)$ can be “approximated”, in a suitable sense, by a sequence
$(h_{n})_{n\in{\mathbb{N}}}$ of Riemannian metrics on $S^{2}$ which are equal
to $h$ on increasing domains in $U$, and have constant positive curvature
(going to $0$ as $n\to\infty$) on open subsets containing the complement of
$U$. The Alexandrov theorem can be applied to those metrics, yielding a
sequence $(\Omega_{n})_{n\in{\mathbb{N}}}$ of convex subsets of
${\mathbb{H}}^{3}$, and the remaining part of the paper is focused on proving
that $\Omega_{n}\to\Omega$, where $\Omega$ satisfies the conditions of Theorem
1.6.

Section 4 provides a key estimate on the principal curvatures of surfaces
under bounds on the curvature of the induced metrics, following well-
understoods arguments that need to be adapted to our context.

Section 5 then contains the proof of the main results. One key idea of the
proof of Theorem 1.6 is that, thanks to the bounds on the principal curvatures
in Section 4, the Gauss maps of the surfaces $\partial\Omega_{n}$ can be
considered as uniformly quasi-conformal diffeomorphisms from
${\mathbb{C}P}^{1}$ to ${\mathbb{C}P}^{1}$. This sequence therefore has a
subsequence converging to a quasiconformal homeomorphism, while, for each
connected component of $U$, the corresponding sequence of surfaces in
${\mathbb{H}}^{3}$ (suitably normalized by isometries) has a subsequence
converging on compact subsets of ${\mathbb{H}}^{3}$ to a limit surface. The
relative positions of those limit surfaces is controlled by the Gauss maps of
the $\partial\Omega_{n}$, leading to the proof of Theorem 1.6.

### Acknowledgements

The author is very grateful to Francesco Bonsante, Qiyu Chen, Jeff Danciger,
Sara Maloni and Andrea Seppi for many discussions related more or less closely
to the result presented here.

##  2\. Background

This section contains two elementary results, used below, recalled here for
the reader’s convenience and to avoid ambiguities in notations.

###  2.1. Conformal metrics on surfaces

Consider a surface $S$ equipped with a smooth Riemannian metric $h$. The
following lemma indicates how the curvature changes under a conformal change
of metrics.

######  Lemma 2.1.

Let $(S,h)$ be a surface equipped with a Riemannian metric, let
$u:S\to{\mathbb{R}}$ be a smooth function, and let $\bar{h}=e^{2u}h$. The
curvature of $\bar{h}$ is equal to

$\bar{K}=e^{-2u}(\Delta_{h}u+K)~{},$

where $K$ is the curvature of $h$.

Note that in this formula and elsewhere, the Laplacian of $h$ is defined as

$\Delta_{h}u=-\mbox{tr}(Ddu)~{},$

where $D$ is the Levi-Civita connection of $h$. In other terms, $\Delta_{h}$
is a positive operator on closed surfaces.

We refer to [Bes87, Theorem 1.159] for a proof.

###  2.2. Surfaces in ${\mathbb{H}}^{3}$

Let $S\subset{\mathbb{H}}^{3}$ be a smooth, oriented surface. We define its
shape operator $B$ as

$Bx=\nabla_{x}n~{},$

where $\nabla$ denotes the Levi-Civita connection of ${\mathbb{H}}^{3}$, $n$
is the unit oriented normal to $S$, and $x$ is any tangent vector to $S$. We
then define the second fundamental form of $S$ as

$II(x,y)=I(Bx,y)~{},$

where $I$ denotes the induced metric on $S$ (also classically known as the
first fundamental form) and $x,y$ are two tangent vectors at the same point,
and the third fundamental form of $S$ as

$III(x,y)=I(Bx,By)~{}.$

The second fundamental form $II$ is a symmetric bilinear form on $TS$, and it
is positive semi-definite when $S$ is locally convex, with unit normal
oriented towards the concave side. We will say that $S$ is strictly convex if
$II$ is positive definite.

The shape operator $B$ of a smooth surface $S$ in ${\mathbb{H}}^{3}$ satisfies
the Gauss-Codazzi equation:

$\det(B)=K-1~{},$

where $K$ is the curvature of the induced metric $I$, and, for any two vector
fields $x,y$ tangent to $S$,

$(\nabla_{x}B)(y)-(\nabla_{y}B)(x)=0~{}.$

##  3\. Approximation by metrics on the sphere

In this section we define a sequence of conformal metrics on
${\mathbb{C}P}^{1}$, that “approximate” in a suitable sense a metric $h$
defined on an open subset $U$, as seen in the statement of Theorem 1.6. We
then show that this sequence of approximating metrics has suitable bounds on
curvatures. This will allow us in Section 5 to prove that one can apply the
Alexandrov Theorem to those metrics and obtain a sequence of convex domains
converging (after extraction of a subsequence) to a non-compact domain having
the required boundary data.

###  3.1. Bounds on conformal factors

The main goal of this section is to obtain a bound on the gradient of the
conformal factor on each connected component of $U$. We consider a fixed
spherical metric $c_{1}$ on ${\mathbb{C}P}^{1}$, and a connected component
$U_{0}$ of $U\subset{\mathbb{C}P}^{1}$, equipped with a complete Riemannian
metric $h$ which can be written as

$h=e^{2u}c_{1}~{},$

where $u:U_{0}\to{\mathbb{R}}$. Since $c_{1}$ is a spherical metric on
$\partial_{\infty}{\mathbb{H}}^{3}$, it is the visual metric associated to a
point $m_{1}\in{\mathbb{H}}^{3}$. Recall that given a point
$m\in{\mathbb{H}}^{3}$, the visual metric $c_{m}$ on
$\partial_{\infty}{\mathbb{H}}^{3}$ associated to $m$ is the spherical metric
such that the distance between to ideal points
$\xi,\eta\in\partial_{\infty}{\mathbb{H}}^{3}$ is the angle between the
geodesic rays from $m$ to $\xi$ and $\eta$. To simplify a bit the argument, we
will assume that $m_{1}\subset CH({\mathbb{C}P}^{1}\setminus U_{0})$, where
$CH$ denotes the hyperbolic convex hull.

######  Lemma 3.1.

Assume that $h$ has curvature $K\in[-1+\epsilon,-\epsilon]$, where
$\epsilon>0$. Assume moreover that $U_{0}$ is a $k$-quasi-disk. Then there
exists a constant $C>0$, depending only on $k$, such that for all $m\in
U_{0}$, $\|du(m)\|_{h}\leq C$.

The proof is based on comparisons between four conformal metrics on $U_{0}$.
In addition to $h$ and $c_{1}$, we will consider the hyperbolic metric
$h_{-1}$ (conformal to $h$) on $U_{0}$, and the Thurston metric $h_{Th}$ on
$U_{0}$.

Recall that the Thurston metric is defined as follows: for all $m\in U_{0}$,
we let $D$ be a maximal disk containing $m$ and contained in $U_{0}$, and let
$h_{Th}$ at $m$ be equal to the complete hyperbolic metric on $D$ at $m$. This
metric $h_{Th}$ then has curvature in $[-1,0]$, see e.g. [KP94].

We will denote by $v,w,x:U_{0}\to{\mathbb{R}}$ the functions such that

$h=e^{2v}h_{-1}~{},~{}~{}h_{-1}=e^{2w}h_{Th}~{},~{}~{}h_{Th}=e^{2x}c_{1}~{}.$

The proof of Lemma 3.1 will directly follows from the following three lemmas.

######  Lemma 3.2.

Let $h_{-1}$ be the complete hyperbolic metric in the conformal class of $h$
on $U_{0}$. Suppose that $h=e^{2v}h_{-1}$, and that $h$ is complete and has
curvature in $[-1+\epsilon,-\epsilon]$. There exists a constant $C_{v}>0$,
depending only on $\epsilon$, such that, on $U_{0}$, $v\in[0,C_{v}]$ and
$\|dv\|_{h_{-1}}\leq C_{v}$.

######  Lemma 3.3.

Suppose that $U_{0}$ is a $k$-quasidisk. Then there exists a constant
$C_{w}>0$, depending only on $k$, such that, on $U_{0}$, $w\in[-C_{w},0]$ and
$\|dw\|_{h_{-1}}\leq C_{w}$.

######  Lemma 3.4.

There exists a constant $C_{x}>0$ such that for all $m\in U_{0}$,
$\|dx\|_{h_{Th}}\leq C_{x}$.

###### Proof of Lemma 3.1.

The lemma clearly follows from Lemma 3.2, Lemma 3.3 and Lemma 3.4, since by
construction $u=v+w+x$ and all three functions have bounded derivative
(measured with respect to $h$) and $v$ and $w$ are bounded, so that $h$,
$h_{-1}$ and $h_{Th}$ are within bounded multiplicative constant of each
other. ∎

###### Proof of Lemma 3.2.

The upper bound on $v$ follows from the main result in [Ahl38]. The lower
bound on $v$ is a consequence of the 1-dimensional case of the main result of
[Yau78] (see also [Roy80, Theorem 1] or [Mat03, Theorem 6]).

To obtain the bound on $dv$, we recall the equation satisfied by $v$ coming
from the fact that $h_{-1}=e^{-2v}h$, see e.g. [Bes87, §1.159]:

$\Delta_{h}v=K+e^{-2v}~{},$

where $K$ is the curvature of $h$. Since both $K$ and $v$ are bounded,
$\Delta_{h}v$ is bounded, and it then follows from standard arguments and from
the bounds on $v$ and $\Delta_{h}v$ that $\|dv\|_{h}$ is bounded. Since $v$ is
bounded, $\|dv\|_{h_{-1}}$ is also bounded.

(We outline this estimate of $\|dv\|_{h}$ for completeness. Let $x\in U_{0}$,
let $r>0$, and let $B(x,r)$ be the ball of center $x$ and radius $r$ in
$(U_{0},h)$. We can decompose the restriction of $v$ on $B(x,r)$ as

$v=v_{0}+v_{1}~{},$

where $v_{0}$ is the harmonic function on $B(x,r)$ with the same boundary
function as $v$ on $\partial B(x,r)$, while $v_{1}$ is the solution of

$\Delta_{h}v_{1}=\Delta_{h}v$

on $B(x,r)$ with Dirichlet boundary conditions. Then $\|dv_{0}(x)\|_{h}$ is
bounded by a fixed constant by the gradient estimate for harmonic functions
(because the boundary values of $v_{0}$ are bounded), while
$\|dv_{1}(x)\|_{h}$ is bounded by a fixed constant, because $|\Delta_{h}v|$ is
bounded on $B(x,r)$ and $v_{1}$ can be expressed from $\Delta_{h}v$ in terms
of Green functions of $\Delta_{h}$ on $B(x,r)$.) ∎

###### Proof of Lemma 3.3.

We denote by ${\mathbb{D}}$ the unit disk in ${\mathbb{C}}$. We first show
that for all $k>1$, there exists a constant $c_{w}$ such that if
${\mathcal{D}}$ is a $k$-quasidisk containing ${\mathbb{D}}$ as a maximal
disk, if $h_{-1}$ and $h_{Th}$ are the hyperbolic metric and Thurston metrics
on ${\mathcal{D}}$, respectively, with $h_{-1}=e^{2\bar{w}}h_{Th}$, then
$|\bar{w}|\leq c_{w}$ and $\|d\bar{w}\|_{h}\leq c_{w}$ at $0$. This follows
from a simple compactness argument. Assume on the contrary that there exists a
sequence ${\mathcal{D}}_{n}$ of $k$-quasicircles such that $|\bar{w}_{n}|\geq
n$ or $\|d\bar{w}_{n}\|_{h_{n}}\geq n$ (where the notations with index $n$ are
attached to the quasidisk ${\mathcal{D}}_{n}$). Then by the compactness
property of $k$-quasidisks (see e.g. [Ahl66]) there exists a subsequence of
$({\mathcal{D}}_{n})_{n\in{\mathbb{N}}}$ which converges in the Hausdorff
topology to a $k$-quasidisk ${\mathcal{D}}_{\infty}$ containing ${\mathbb{D}}$
as a maximal disk. Then $|\bar{w}_{\infty}|$ or
$\|d\bar{w}_{\infty}\|_{h_{\infty}}$ would need to be infinite, a
contradiction.

Now let $m\in U_{0}$, and let $D_{m}$ be a maximal disk in $U_{0}$ containing
$m$. There exists an element of $PSL(2,{\mathbb{C}})$ sending $D_{m}$ to
${\mathbb{D}}$ and $m$ to $0$. Both $h_{-1}$ and $h_{Th}$ are preserved under
Möbius transformation, and the lemma therefore follows from the estimate
obtained at $0$ for quasidisks containing ${\mathbb{D}}$ as a maximal disk. ∎

The proof of Lemma 3.4 requires a simple proposition concerning the comparison
between visual metrics at two different points in ${\mathbb{H}}^{3}$.

######  Proposition 3.5.

Let $m_{0},m_{1}\in{\mathbb{H}}^{3}$ be at distance $\delta>0$, and let
$y:\partial_{\infty}{\mathbb{H}}^{3}\to{\mathbb{R}}$ be such that
$c_{m_{1}}=e^{2y}c_{m_{0}}$. Let $\xi\in\partial_{\infty}{\mathbb{H}}^{3}$ be
such that the angle at $m_{0}$ between the segment ending at $m_{1}$ and the
geodesic ray ending at $\xi$ is $\theta$. Then

(1) $y(\xi)=-\log(\cosh(\delta)-\cos(\theta)\sinh(\delta))~{}.$

As a consequence, if $\partial_{\theta}$ is the vector at $\xi$ corresponding
to varying $\theta$ while keeping $m_{0},m_{1}$ and the plane containing
$m_{0},m_{1}$ and $\xi$ fixed, then

(2)
$dy(\partial_{\theta})=-\frac{\sin(\theta)}{\rm{cotanh}(\delta)-\cos(\theta)}~{}.$

###### Proof.

The first point is a consequence of a standard hyperbolic geometry formula
relating the values at $m_{0}$ and $m_{1}$ of the Busemann function associated
to $\xi$. Namely, if $m(t)$ is the point at distance $t$ from $m_{0}$ along
the ray $[m_{0},\xi)$, and if $l(t)=d(m_{1},m(t))$ then the triangle formula
can be written as:

$\cosh(l(t))=\cosh(t)\cosh(\delta)-\cos(\theta)\sinh(t)\sinh(\delta)~{}.$

As $t\to\infty$, it follows that

$e^{l(t)-t}\sim\cosh(\delta)-\cos(\theta)\sinh(\delta)~{},$

and the first point follows.

The second point follows from the first by a direct computation. ∎

The Thurston metric has a simple relation to the visual metric, stated for the
reader’s convenience in the next proposition. We denote by $\Sigma_{0}$ the
pleated locally convex surface facing $U_{0}$, defined as:

$\Sigma_{0}=\partial CH({\mathbb{C}P}^{1}\setminus U_{0})~{},$

where $CH$ denotes the hyperbolic convex hull.

######  Proposition 3.6.

Let $\xi\in U_{0}$, and let $m$ be the closest-point projection of $\xi$ on
$\Sigma_{0}$. Then

$h_{Th}(\xi)=c_{m}(\xi)~{}.$

###### Proof.

Let $P$ be the support plane of $CH({\mathbb{C}P}^{1}\setminus U_{0})$ at $m$
orthogonal to the direction of the geodesic ray from $m$ to $\xi$. This plane
$P$ faces a disk in ${\mathbb{C}P}^{1}$ (with the same boundary) which is a
maximal disk contained in $U_{0}$ and containing $\xi$. By construction,
$h_{Th}$ is equal at $\xi$ to the push-forward of the induced metric on $P$ by
the normal exponential map, considered as a map from $P$ to $U_{0}$.

However a simple computation (for instance done in the Poincaré model of
${\mathbb{H}}^{3}$ with $m$ at the center) shows that this push-forward metric
coincides at $\xi$ with the visual metric $c_{m}$, and the proposition
follows. ∎

###### Proof of Lemma 3.4.

To obtain the required comparison, we choose a point $\xi\in U_{0}$ and let
$m$ be the closest-point projection of $\xi$ on $\Sigma_{0}$, as above. We
first compare $h_{Th}$ to $c_{m}$ at first-order at $\xi$, and then compare
$c_{m}$ to $c_{1}$. Both comparisons will be based on (1).

For the first comparison, we again denote by $u:U_{0}\to{\mathbb{R}}$ the
function such that $h_{Th}=e^{2u}c_{m}$, and will show that $du=0$ at $\xi$.
Consider first a first-order variation of $\xi$ obtained by varying $m$ along
a flat direction of $\Sigma_{0}$ while keeping parallel the direction at $m$
of the closest-point projection line. Let $\partial_{\delta}$ be such a unit
(for $c_{m}$) tangent vector to ${\mathbb{C}P}^{1}$ at $\xi$. Then taking the
derivative of (1) with respect to $\delta$ at $\delta=0$ and $\theta=\pi/2$
shows that $dy(\partial_{\delta})=0$. Now if $m$ is located on a pleating line
of $\Sigma_{0}$ and $\partial_{\theta}$ is a vector corresponding to varying
$\theta$ while the projection to $m$ remains constant, then taking the
derivative of (1) with respect to $\theta$ with $\delta=0$ and $\theta=\pi/2$
also shows that $dy(\partial_{\theta})=0$. So we can conclude that $dy=0$ at
$\xi$, so that $h_{Th}$ coincides with $c_{m}$ at $\xi$ at first order.

The second step is to compare at $\xi$ the metric $c_{m}$ – which as we have
seen is equal to $h_{Th}$ at first order – to the spherical metric $c_{1}$,
which as any spherical metric on $\partial_{\infty}{\mathbb{H}}^{3}$ is the
visual metric of some fixed point $m_{1}\in{\mathbb{H}}^{3}$. Specifically, if
we denote (again) by $y:\partial_{\infty}{\mathbb{H}}^{3}\to{\mathbb{R}}$ the
function such that $c_{m}=e^{2y}c_{m_{1}}$, Proposition 3.5 shows that the
derivative of $y$ at $\xi$ is bounded by the right-hand side of (2), with
$m_{0}=m$. However a look at this second term shows that it is bounded from
above in absolute value as soon as $\theta$ is bounded from below, because
$\rm{cotanh}(\delta)-\cos(\theta)$ is then bounded from below since
$\rm{cotanh}(\delta)>1$.

But since we have assumed that $m_{1}\in CH({\mathbb{C}P}^{1}\setminus
U_{0})$, for all $\xi\in U_{0}$, if $m$ again denotes the closest-point
projection of $\xi$ on $\Sigma_{0}$, $m_{1}$ is on the negative side of the
support plane to $CH({\mathbb{C}P}^{1}\setminus U_{0})$ at $m$ orthogonal to
the geodesic ray from $m$ to $\xi$. (That is, $m_{1}$ is on the side of this
plane not containing $\xi$.) It follows that the angle $\theta$ occuring in
(2) is larger than $\pi/2$. This concludes the proof. ∎

###  3.2. Construction of approximating metrics

We can now define rather explicitly the sequence of approximating metrics that
will be used in the proof. The construction uses the choice of a “cut-off”
function.

######  Lemma 3.7.

There exists a smooth function $\phi:{\mathbb{R}}\to{\mathbb{R}}$ such that:

  1. (1)

$\phi(x)=x$ for $x\leq 0$,

  2. (2)

$\phi(x)=1$ for $x\geq 2$,

  3. (3)

$\phi^{\prime}(x)\in[0,1]$ and $\phi^{\prime\prime}(x)\leq 0$ for all
$x\in[0,2]$,

  4. (4)

$\phi^{\prime}(x)\leq e^{2(\phi(x)-x)}$ for all $x\in[0,2]$.

###### Proof.

Note first that $f(x)=x$ is a solution of the differential equation

$f^{\prime}(x)\leq e^{2(f(x)-x)}$

over $[0,1]$. As a consequence, one can find a function $\phi$ satisfying the
conditions above over $(-\infty,1-\epsilon)$ and arbitrarily close to
$f(x)=x$, for any $\epsilon>0$. Taking $\epsilon$ small enough, one can then
“bridge” this function to $f(x)=1$ for $x\geq 1+\epsilon$ while still
satisfying the conditions of the lemma. ∎

######  Definition 3.8.

We choose a function $\phi$ satisfying the conditions of Lemma 3.7. For all
$n\in{\mathbb{N}}$, we define $\phi_{n}:{\mathbb{R}}\to{\mathbb{R}}$ by

$\phi_{n}(x)=n+\phi(x-n)~{}.$

For the next definition, we need to remark that ${\mathbb{C}P}^{1}\setminus U$
has at least 2 points, because $U$ admits a complete, conformal metric of
negative curvature. Therefore the convex hull of ${\mathbb{C}P}^{1}\setminus
U$ contains a point $m_{1}$.

######  Definition 3.9.

Let $(U,h)$ be a boundary data on ${\mathbb{C}P}^{1}$, let $c_{1}=c_{m_{1}}$
be the visual metric of a point $m_{1}\in CH({\mathbb{C}P}^{1}\setminus U)$,
and let $u:U\to{\mathbb{R}}$ be such that $h=e^{2u}c_{1}$. We define a
sequence of conformal metrics $(h_{n})_{n\in{\mathbb{N}}}$ on
${\mathbb{C}P}^{1}$ by

$\displaystyle h_{n}=e^{2\phi_{n}(u)}c_{1}$ on $U$ $\displaystyle
h_{n}=e^{2(n+1)}c_{1}$ $\displaystyle\mbox{on ${\mathbb{C}P}^{1}\setminus
U$}~{}.$

Note that $h_{n}$ is a smooth metric on ${\mathbb{C}P}^{1}$ since, by
definition of $\phi$, $\phi_{n}(u)$ is equal to $n+1$ in a neighborhood of the
boundary of $U$. (It follows from the fact that $h$ is complete on $U$
together with Lemma 3.1 (for from Lemma 3.2) that $u\to\infty$ on $\partial
U$.)

######  Lemma 3.10.

Assume that $K\in[-1+\epsilon,-\epsilon]$, that $u$ is bounded from below, and
that $\|du\|_{h}$ is bounded from above, on $U$. There exists a constant
$K_{max}>0$ (depending on those bounds) such that the curvature $K_{n}$ of
$h_{n}$ is:

  * •

equal to $K$ on the set of points $x\in U$ where $u(x)\leq n$,

  * •

in the interval $[-1+\epsilon,K_{max}]$ on the set of points $x\in U$ where
$n\leq u(x)\leq n+2$,

  * •

equal to $e^{-2(n+1)}$ on the set of points $x\in U$ where $u(x)\geq n+2$ and
on the complement of $U$.

###### Proof.

In the region of $U$ where $u(x)\leq n$, $h_{n}=h$, and therefore $K_{n}=K$,
the curvature of $h$. On the region of $U$ where $u(x)\geq n+2$, and on the
complement of $U$, $h_{n}=e^{2(n+1)}c_{1}$, and it follows from Lemma 2.1 that
$K_{n}=e^{-2(n+1)}$.

We now focus on the region where $n\leq u(x)\leq n+2$. The conformal
transformation of curvature (Lemma 2.1) also shows that

$K_{n}=e^{-2\phi_{n}(u)}(\Delta_{c_{1}}\phi_{n}(u))+1)~{}.$

However

$d(\phi_{n}(u))=\phi^{\prime}_{n}(u)du~{},$

so that

$Dd(\phi_{n}\circ u)=\phi^{\prime}_{n}(u)Ddu+(\phi^{\prime\prime}_{n}\circ
u)du\otimes du~{},$

and therefore

$\Delta_{c_{1}}(\phi_{n}\circ u)=(\phi^{\prime}_{n}\circ
u)\Delta_{c_{1}}u-(\phi^{\prime\prime}_{n}\circ u)\|du\|^{2}_{c_{1}}~{}.$

As a consequence, and since $\|du\|^{2}_{c_{1}}=e^{2u}\|du\|^{2}_{h}$:

$\displaystyle K_{n}$ $\displaystyle=$ $\displaystyle e^{-2\phi_{n}\circ
u}((\phi^{\prime}_{n}\circ u)\Delta_{c_{1}}u-(\phi^{\prime\prime}_{n}\circ
u)\|du\|^{2}_{c_{1}}+1)$ $\displaystyle=$
$\displaystyle(\phi^{\prime}_{n}\circ u)e^{2(u-\phi_{n}\circ
u)}e^{-2u}(\Delta_{c_{1}}u+1)+e^{-2\phi_{n}\circ u}(1-(\phi^{\prime}_{n}\circ
u))-e^{2(u-\phi_{n}\circ u)}(\phi^{\prime\prime}_{n}\circ u)\|du\|^{2}_{h}$
$\displaystyle=$ $\displaystyle(\phi^{\prime}_{n}\circ u)e^{2(u-\phi_{n}\circ
u)}K+e^{-2\phi_{n}\circ u}(1-(\phi^{\prime}_{n}\circ u))-e^{2(u-\phi_{n}\circ
u)}(\phi^{\prime\prime}_{n}\circ u)\|du\|^{2}_{h}~{}.$

By the hypothesis, $K\in[-1+\epsilon,-\epsilon]$, and by point (4) in the
choice of $\phi$, $0\leq(\phi^{\prime}_{n}\circ u)e^{2(u-\phi_{n}\circ u)}\leq
1$, so the first term is in $[-1+\epsilon,0]$. By point (3) in the choice of
$\phi$, the second term is non-negative, and it is uniformly bounded because
by construction, $e^{-2\phi_{n}\circ u}$ is bounded from above. Finally, the
third term is also non-negative by point (3) in the choice of $\phi$ and
uniformly bounded by the fact that $u-\phi_{n}\circ u$ is bounded from above
and the uniform bound on $\|du\|_{h}$. The result follows. ∎

###  3.3. A lower bound on the injectivity radius

In addition to the upper bound on the curvature, the next section will make an
essential use of a lower bound on the injectivity radius of the metrics
$h_{n}$, as stated in the next lemma.

######  Lemma 3.11.

There exists $l_{0}>0$ such that the injectivity radius of $h_{n}$ is bounded
below by $l_{0}$.

We introduce additional notations for the proof, namely a decomposition of
${\mathbb{C}P}^{1}$ in three subsets:

  * •

$U_{u\leq n}$ is the set of points $x\in U$ where $u(x)\leq n$,

  * •

$U_{n<u\leq n+1}$ is the set of points $x\in U$ where $n<u(x)\leq n+1$,

  * •

$U_{u\leq n+1}=U_{u\leq n}\cup U_{n<u\leq n+1}$,

  * •

$U_{n+1<u}$ is the union of ${\mathbb{C}P}^{1}\setminus U$ with the set of
points $x\in U$ where $n+1<u(x)$.

###### Proof of Lemma 3.11.

We consider a short geodesic $\gamma$ of length $l$ for the metric $h_{n}$,
and will show that $l$ cannot be smaller than $l_{0}$, for a value of $l_{0}$
that will be determined.

The proof will consider two cases, with a similar argument used for a
different background metric in each case. In each case we will use that any
disk $\Omega$ with $\partial\Omega=\gamma$ must have total curvature $\pi$ (by
the Gauss-Bonnet Theorem) and thus, since $h_{n}$ has curvature $K\leq
K_{max}$, area bounded from below: $A_{h_{n}}(\Omega)\geq\pi/K_{max}$.

First case: $\gamma\cap U_{u\leq n}=\emptyset$. In this case $u\geq n$ on
$\gamma$, and we compare $h_{n}$ to the metric $e^{2n}c_{1}$. We have
$h_{n}\geq e^{2n}c_{1}$ on $\gamma$, so that $L_{c_{1}}(\gamma)\leq e^{-n}l$.
However we also have $h_{n}\leq e^{2(n+1)}c_{1}$ on ${\mathbb{C}P}^{1}$ and
therefore, for each of the disks $\Omega$ bounded by $\gamma$,
$A_{c_{1}}(\Omega)\geq e^{-2(n+1)}\pi/K_{max}$. For $l$ small enough, this
contradicts the spherical isoperimetric inegality.

Second case: $\gamma\cap U_{u\leq n}\neq\emptyset$. We first notice that there
exists a constant $l_{1}>0$ (depending on the constant $C$ in Lemma 3.1) such
that, if $L_{h_{n}}(\gamma)\leq l_{1}$, then $L_{h}(\gamma)\leq 2l_{1}$ and
$u\leq n+2$ on $\gamma$. Indeed, $\|du\|_{h}\leq C$, and since

$h=e^{2u}c_{1}=e^{2u-2\phi_{n}(u)}h_{n}\leq e^{2u-2n}h_{n}~{},$

we have

$\|du\|_{h_{n}}\leq e^{u-n}\|du\|_{h}\leq e^{u-n}C~{}.$

It follows from this equation, and from the fact that $u\leq n$ at a point of
$\gamma$, that if $\gamma$ is short enough, then $e^{u-n}\leq 2$ on $\gamma$,
and therefore $L_{h}(\gamma)\leq 2L_{h_{n}}(\gamma)\leq 2l$.

It then follows that $\gamma\subset U$, and therefore $\gamma$ is contained in
a connected component $U_{0}$ of $U$. Since $U$ is a disjoint union of
quasidisks, this connected component is topologically a disk, and therefore
$\gamma$ bounds a topological disk $\Omega\subset U_{0}$. Since $h\geq h_{n}$
on $U_{0}$, $A_{h}(\Omega)\geq A_{h_{n}}(\Omega)\geq\pi/K_{max}$. So for $l$
small enough, we again have a contradiction with the isoperimetric inequality,
this time for $(U_{0},h)$. ∎

###  3.4. A sequence of convex subsets

We can now apply the Alexandrov Theorem to the sequence of metrics constructed
in Section 3.2, together with the injectivity radius bounds in Section 3.3, to
obtain the following lemma.

######  Lemma 3.12.

Under the hypothesis of Theorem 1.6, there exists a sequence of bounded convex
domains $\Omega_{n}\subset{\mathbb{H}}^{3}$, together with a sequence of
diffeomorphisms $\phi_{n}:{\mathbb{C}P}^{1}\to\partial\Omega_{n}$, such that
for all $n\in{\mathbb{N}}$:

  * •

$\phi_{n}$ is a conformal map from ${\mathbb{C}P}^{1}$ to
$\partial\Omega_{n}$,

  * •

$\phi_{n}^{*}I=h$ on $U_{n}$, where $U_{n}\subset U$, $U_{n}\subset U_{m}$ for
$n\leq m$, and $\cup_{n\in{\mathbb{N}}}U_{n}=U$,

  * •

$\phi_{n}^{*}I$ has constant curvature $e^{-2n}$ on
${\mathbb{C}P}^{1}\setminus U$,

  * •

the induced metric on $\partial\Omega_{n}$ has curvature in
$[-1+\epsilon,K_{max}]$,

  * •

the injectivity radius of $(\partial\Omega_{n},I)$ is bounded from below by a
constant $l_{0}$.

##  4\. Bounds on principal curvatures

###  4.1. Principal curvature bounds for convex surfaces

The first point of this section is to find a uniform bound on the principal
curvatures on the boundaries of the domains $\Omega_{n}$ constructed in Lemma
3.12.

######  Lemma 4.1.

Let $\epsilon>0$. There exists a constant $k_{0}>0$ such that if
$\Omega\subset{\mathbb{H}}^{3}$ is a convex subset such that the induced
metric on $\partial\Omega$ has curvature $K\in[-1+\epsilon,1/\epsilon]$ and
injectivity radius bounded from below by $\epsilon$, then the principal
curvatures of $\partial\Omega$ are in $[k_{0},1/k_{0}]$.

The proof uses a result of Labourie [Lab89, Théorème D] which we recall here
for the reader’s convenience (restricting to the special case where the target
space is ${\mathbb{H}}^{3}$).

######  Theorem 4.2 (Labourie).

Let $f_{n}:S\to{\mathbb{H}}^{3}$ be a sequence of immersions of a surface $S$
such that the pullback $f_{n}^{*}(h)$ of the hyperbolic metric $h$ converges
smoothly to a metric $g_{0}$. If the integral of the mean curvature is
uniformly bounded, then a subsequence of $f_{n}$ converges smoothly to an
isometric immersion $f_{\infty}$ such that $f_{\infty}^{*}(h)=g_{0}$.

Note that Theorem 4.2 is local: no assumption of compactness of $S$ or
completeness of $f_{n}^{*}(h)$ is needed.

###### Proof of Lemma 4.1.

We proceed by contradiction and assume that there exists $\epsilon>0$ such
that for all $n$, there exists a convex subset $\Omega_{n}$ such that the
induced metric on $\partial\Omega_{n}$ has curvature
$K\in[-1+\epsilon,1/\epsilon]$ and injectivity radius bounded from below by
$\epsilon$, while the principal curvatures of $\partial\Omega_{n}$ at a point
$x_{n}$ are not in $[1/n,n]$.

We apply Theorem 4.2 to a sequence of conformal parameterizations
$\phi_{n}:{\mathbb{D}}\to\partial\Omega_{n}$ of the disk of center $x_{n}$ and
radius $\epsilon/2$, which are indeed topological disks by the asumption on
the injectivity radius. Since the induced metrics on the
$\phi_{n}({\mathbb{D}})$ have curvature $K\in[-1+\epsilon,1/\epsilon]$, there
exists a subsequence such that the induced metrics converge to a limit
$g_{0}$. Therefore, by Theorem 4.2, after extracting another subsequence,
there exists a sequence of isometries $(\psi_{n})_{n\in{\mathbb{N}}}$ of
${\mathbb{H}}^{3}$ such that

  1. (1)

either the integral mean curvature on $\phi_{n}({\mathbb{D}})$ tends to
infinity,

  2. (2)

or the $\psi_{n}\circ\phi_{n}:{\mathbb{D}}\to{\mathbb{H}}^{3}$ converge
smoothly to a smooth limit $\phi_{\infty}:{\mathbb{D}}\to{\mathbb{H}}^{3}$.

Clearly the second hypothesis is excluded by the hypothesis on the principal
curvatures of $\partial\Omega_{n}$ at $x_{n}$, so it is sufficient to exclude
also the first case.

Let $\Omega_{n}^{1}$ be the $1$-neighborhood of $\Omega_{n}$:

$\Omega_{n}^{1}=\\{x\in{\mathbb{H}}^{3}~{}|~{}d(x,\Omega_{n})\leq 1\\}~{},$

and let $\rho:\partial\Omega_{n}^{1}\to\partial\Omega_{n}$ be the closest-
point projection. Since $\Omega_{n}$ is convex and smooth, $\Omega_{n}^{1}$ is
also convex, and $\rho$ is a diffeomorphism. A well-known “tube” formula
indicates that $\rho$ contracts the area of $\partial\Omega_{n}^{1}$ at a
point $x\in\partial\Omega_{n}^{1}$ by a factor
$\cosh(1)^{2}+\cosh(1)\sinh(1)\mbox{tr}(B)+\sinh(1)^{2}\det(B)$, where $B$
denotes the shape operator of $\Omega_{n}$ at $\rho(x)$. As a consequence, the
area of $\rho^{-1}(\phi_{n}({\mathbb{D}}))$ is equal to

$A(\rho^{-1}(\phi_{n}({\mathbb{D}})))=\int_{\phi_{n}({\mathbb{D}})}\cosh(t)^{2}+\cosh(t)\sinh(t)\mbox{tr}(B)+\sinh(t)^{2}\det(B)da_{I}~{}.$

Let $B(x_{n},2)$ be the ball of center $x_{n}$ and radius $2$. The projection
from

$(\partial B(x_{n},2))\cap({\mathbb{H}}^{3}\setminus\Omega^{1}_{n})$

to $\partial\Omega^{1}_{n}$ is contracting, and its image contains
$\phi_{n}({\mathbb{D}})$. As a consequence, the area of
$\rho^{-1}\circ\phi_{n}({\mathbb{D}})$ is at most equal to the area of
$\partial B(x_{n},2)$, so that

$\int_{\phi_{n}({\mathbb{D}})}\cosh(1)^{2}+\cosh(1)\sinh(1)\mbox{tr}(B)+\sinh(1)^{2}\det(B)da_{I}\leq
4\pi\sinh(2)^{2}~{}.$

Since all three terms are non-negative, this simplies in particular that the
integral of the mean curvature is bounded on $\phi_{n}({\mathbb{D}})$, and the
first alternative above is impossible. This concludes the proof of the lemma.
∎

###  4.2. Curvature bounds in the almost-flat regions

In the proof of Theorem 1.6 it will be necessary to show that the conformal
structure on the boundary at infinity of the convex set $\Omega_{\infty}$
which is the limit of the $\Omega_{n}$ is identified – under the limit map
from ${\mathbb{C}P}^{1}$ to $\overline{{\mathbb{H}}^{3}}$ – with the conformal
structure at infinity of ${\mathbb{C}P}^{1}$. This will be based on the fact
that the regions in the $\partial\Omega_{n}$ which are the image of the
complement of $U$ are (asymptotically) umbilic, with principal curvatures
going to $1$ as $n\to\infty$. This section is focused on proving this result.

Specifically, we will need the following lemma.

######  Lemma 4.3.

For all $k_{0}>0$, there exists $R>0$ and $K_{0}>0$ such that if
$\Omega\subset{\mathbb{H}}^{3}$ is a convex domain and $x\in\partial\Omega$
such that the injectivity radius of $(\partial\Omega,I)$ at $x$ is at least
$R$, with curvature in $[-K_{0},K_{0}]$ on the ball of center $x$ and radius
$R$, then the principal curvatures of $\partial\Omega$ at $x$ are in
$[1-k_{0},1+k_{0}]$.

The proof uses the following proposition.

######  Proposition 4.4.

Let $\Omega\subset{\mathbb{H}}^{3}$. Assume that $(\partial\Omega,I)$ is flat,
and that its injectivity radius at a point $x\in\partial\Omega$ is infinite.
Then $\Omega$ is a horosphere.

###### Proof.

This follows directly from a result of Volkov and Vladimirova [VV71] who
proved that an isometric immersion of the Euclidean plane in
${\mathbb{H}}^{3}$ is either a parameterization of a horosphere, or a multiple
cover of a cylinder. ∎

###### Proof of Lemma 4.3.

Assume by contradiction that there exists $k_{0}>0$ such that no such $R$ and
$K_{0}$ exist. Then for all $n\in{\mathbb{N}}_{>0}$ there exists a convex
subset $\Omega_{n}$ and a point $x_{n}\in\partial\Omega_{n}$ such that the
injectivity radius of $\partial\Omega_{n}$ at $x_{n}$ is at least $n$, and the
curvature of $\partial\Omega_{n}$ on the ball of center $x_{n}$ and radius $n$
is in $[-1/n,1/n]$. However one of the principal curvatures of
$\partial\Omega_{n}$ at $x_{n}$ is not in $[1-k_{0},1+k_{0}]$.

For each $n$, let $\phi_{n}\in\rm{Isom}({\mathbb{H}}^{3})$ be an isometry
sending $x_{n}$ to $x_{0}$ (a fixed point). We can extract from
$(\phi_{n}(\Omega_{n}))_{n\in{\mathbb{N}}}$ a sequence of convex subset
$\Omega^{\prime}_{n}\subset{\mathbb{H}}^{3}$ which converges in the Hausdorff
topology on compact subsets of ${\mathbb{H}}^{3}$ to a limit $\Omega_{\infty}$
(see [NQS79] and references therein for topological properties of the space of
convex subsets of a linear space).

Then $x_{0}\in\partial\Omega_{\infty}$ by construction. Moreover
$\partial\Omega^{\prime}_{n}\to\partial\Omega_{\infty}$, and the induced
metric on $\partial\Omega^{\prime}_{n}$ converges to the induced metric on
$\partial\Omega_{\infty}$ (see [BBI01, Section 10.2]). As a consequence,
$\partial\Omega_{\infty}$ is intrisically flat, with infinite injectivity
radius since the injectivity radius of $\partial\Omega_{n}$ at $x_{n}$ is at
least $n$. It follows from Proposition 4.4 that it is a horosphere. This
contradicts the hypothesis that the principal curvatures of
$\partial\Omega^{\prime}_{n}$ are not in $[1-k_{0},1+k_{0}]$ at $x_{0}$. ∎

##  5\. Proofs of the main results

We are now ready to prove Theorem 1.6.

###  5.1. Convergence of the Gauss maps

We consider the sequence of convex domains $\Omega_{n}$ and conformal
diffeomorphisms $\phi_{n}:{\mathbb{C}P}^{1}\to\partial\Omega_{n}$ described in
Lemma 3.12.

For each $n$, we denote by
$G_{n}:\partial\Omega_{n}\to\partial_{\infty}{\mathbb{H}}^{3}$ the hyperbolic
Gauss map of $\partial\Omega_{n}$, that is, the map sending
$x\in\partial\Omega_{n}$ to the endpoint at infinity of the geodesic ray
starting from $x$ directed by the exterior normal of $\Omega$ at $x$.

######  Lemma 5.1.

There exists a constant $K_{G}>1$ such that for all $n\in{\mathbb{N}}$, the
map $G_{n}\circ\phi_{n}:{\mathbb{C}P}^{1}\to\partial_{\infty}{\mathbb{H}}^{3}$
is $K_{G}$-quasi-conformal.

###### Proof.

Let $x\in\partial\Omega_{n}$, for some $n\in{\mathbb{N}}$. We can identify
$T_{x}\partial\Omega_{n}$ to $T_{G_{n}(x)}\partial_{\infty}{\mathbb{H}}^{3}$
by the exponential map at $x$,
$\exp_{x}:T_{x}{\mathbb{H}}^{3}\to\partial_{\infty}{\mathbb{H}}^{3}$, using
the natural identification of $T_{x}\partial\Omega_{n}$ with the normal in
$T_{x}{\mathbb{H}}^{3}$ of the unit normal to $\partial\Omega_{n}$ at $x$.
Under this identification, the differential of $G_{n}$ at $x$ is equal to
$Id+B$, where $Id$ is the identity, and $B$ the shape operator of
$\partial\Omega_{n}$.

Since the principal curvatures of $\partial\Omega_{n}$ are uniformly bounded,
the differentials of the Gauss map $G_{n}$ are uniformly bounded. Since the
$\phi_{n}$ are conformal, it follows that the maps $G_{n}\circ\phi_{n}$ are
uniformly quasi-conformal. ∎

######  Corollary 5.2.

After extracting a subsequence and composing with a sequence of Möbius
transformations $u_{n}$, the sequence $(G_{n}\circ\phi_{n})_{n}$ converges
$C^{0}$ to a $K_{G}$-quasi-conformal homeomorphism
$\psi_{\infty}:{\mathbb{C}P}^{1}\to\partial_{\infty}{\mathbb{H}}^{3}$.

###### Proof.

This follows from the compactness of the space of $K$-quasi-conformal
homeomorphisms from ${\mathbb{C}P}^{1}$ to ${\mathbb{C}P}^{1}$, see e.g.
[Ahl66]. ∎

Note that the Möbius transformations appearing in this corollary are boundary
values of hyperbolic isometries. If $u\in PSL(2,{\mathbb{C}})$ is a hyperbolic
isometry, and $\partial u$ is its boundary action, considered as a Möbius
transformation, then composing $G_{n}\circ\phi_{n}$ on the left by a Möbius
transformation $\partial u$ is equivalent to composing $\phi_{n}$ on the left
by $u$ and then composing with its Gauss map.

We will assume in the last section below that the $\phi_{n}$ have already been
composed on the left by a sequence of Möbius transformations, so that
$G_{n}\circ\phi_{n}\to\psi_{\infty}$.

###  5.2. Convergence of boundary data

Recall that we denote by $\rho:{\mathbb{H}}^{3}\to B(0,1)$ the Poincaré model
of hyperbolic space. We consider the sequence of maps
$(\phi_{n})_{n\in{\mathbb{N}}}$ considered above.

######  Lemma 5.3.

After extracting a subsequence,

  1. (1)

the sequence of restrictions of the $\phi_{n}$ to $U$ converge $C^{\infty}$ to
an isometric embedding $\phi_{\infty}$ of $(U,h)$ in ${\mathbb{H}}^{3}$,

  2. (2)

the sequence $(\rho\circ\phi_{n})_{n\in{\mathbb{N}}}$ converges to a map
$\Phi_{\infty}:{\mathbb{C}P}^{1}\to B(0,1)$ which is conformal on $U$ and on
the interior of ${\mathbb{C}P}^{1}\setminus U$,

  3. (3)

$\phi_{\infty}(U)$ is the boundary in ${\mathbb{H}}^{3}$ of a convex subset
$\Omega$.

###### Proof.

Let $U_{0}$ be a connected component of $U$. It follows from the arguments
already used in Section 4.1 that there exists a sequence
$(u^{0}_{n})_{n\in{\mathbb{N}}}$ of hyperbolic isometries such that the
sequence of restrictions of the $u^{0}_{n}\circ\phi_{n}$ to $U_{0}$ converges
$C^{\infty}$ (after extracting a subsequence) to a limit $\phi_{0,\infty}$. It
follows that the Gauss maps of the $u^{0}_{n}\circ\phi_{n}$ converge to the
Gauss map of $\phi_{0,\infty}$ on $U_{0}$, that is,

$\partial u_{n}^{0}\circ(G_{n}\circ\phi_{n})\to
G_{0,\infty}\circ\phi_{0,\infty}~{},$

where $G_{0,\infty}$ denotes the Gauss map of $\phi_{0,\infty}$.

However we already know by Corollary 5.2 that the $G_{n}\circ\phi_{n}$
converge to a limit $\psi_{\infty}$, so that

$\lim_{n\to\infty}\partial
u_{n}^{0}\circ\psi_{\infty}=G_{0,\infty}\circ\phi_{0,\infty}~{}.$

As a consequence, $(u_{n}^{0})_{n\in{\mathbb{N}}}$ converges to a hyperbolic
isometry $u^{0}$. So, to obtain the convergence of
$u_{n}^{0}\circ\phi_{n}\circ\phi_{0,\infty}$ above, it is sufficient to take
$u_{n}^{0}$ constant and equal to $u^{0}$. This means that after replacing the
$\phi_{n}$ by $u^{0}\circ\phi_{n}$, the sequence
$(\phi_{n})_{n\in{\mathbb{N}}}$ converges $C^{\infty}$ (without normalization
by the sequence $(u^{0}_{n})_{n\in{\mathbb{N}}}$). This proves the first
point.

The second point is less obvious than it might appear. We already know that on
$U$, $\phi_{n}\to\phi_{\infty}$, where $\phi_{\infty}$ is an isometry from
$(U,h)$ to ${\mathbb{H}}^{3}$ and is therefore conformal. However the
conformality of the limit map is less clear on ${\mathbb{C}P}^{1}\setminus U$.
This is because the Poincaré model of ${\mathbb{H}}^{3}$ does not preserve
convexity, so that the $\rho(\Omega)$ are not necessarily convex, and a
sequence of conformal embeddings might well converge to a limit which is not
conformal. This could conceivably happen at parts of $\partial\Omega_{n}$
converging to infinity in ${\mathbb{H}}^{3}$.

To avoid this issue for regions converging to
$\partial_{\infty}{\mathbb{H}}^{3}$, we note that for all
$x\in{\mathbb{C}P}^{1}\setminus U$, $(\phi_{n}(x))_{n\in{\mathbb{N}}}$ has the
same limit in $\partial_{\infty}{\mathbb{H}}^{3}$ as
$(G_{n}\circ\phi_{n}(x))_{n\in{\mathbb{N}}}$. Let $x\in{\mathbb{C}P}^{1}$ be
contained in the interior of the complement of $U$, then Lemma 4.3 shows that
the principal curvatures of $\phi_{n}$ at $x$ converge to $1$ as $n\to\infty$.
It follows that the pull-back by $G_{n}\circ\phi_{n}$ of the conformal metric
at infinity is quasi-conformal to $h_{n}$ in the neighborhood of $x$, with
quasiconformal constant going to $1$ as $n\to\infty$. So $\phi_{\infty}$ is
conformal on $U$ and on the interior of ${\mathbb{C}P}^{1}\setminus U$.

The third point follows from the construction, since
$\phi_{\infty}(U)=\partial\Omega_{\infty}$, where $\Omega_{\infty}$ is the
Hausdorff limit (on compact subsets of ${\mathbb{H}}^{3}$) of the
$\Omega_{n}$, which are convex subsets of ${\mathbb{H}}^{3}$. ∎

###### Proof of Theorem 1.6.

Let $(U,h)$ be a boundary data, with $U\subset{\mathbb{C}P}^{1}$ a disjoint
union of $k_{0}$-quasidisks, and let $h$ be a complete metric of curvature
$K\in[-1+\epsilon,-\epsilon]$ on $U$. We define a sequence
$(h_{n})_{n\in{\mathbb{N}}}$ of approximating metrics following Definition
3.9, and those metrics have curvature $K_{n}\in[-1+\epsilon,K_{max}]$ by Lemma
3.10. (Note that the constant $K_{max}$ does not depend on the connected
component of $U$ since each is a $k_{0}$-quasidisk, and the curvature
estimates on $h_{n}$ are valid for all.)

The Alexandrov Theorem (Theorem 1.2) therefore realizes each of the $h_{n}$ as
the induced metric on the boundary of a convex subset
$\Omega_{n}\subset{\mathbb{H}}^{3}$, as see in Section 3.4.

Lemma 5.3 then shows that, after normalizing by a sequence of isometries and
extracting a subsequence, $(\Omega_{n})_{n\in{\mathbb{N}}}$ converges to a
convex subset $\Omega_{\infty}$ realizing the boundary data $(U,h)$. ∎

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