# The geometric data on the boundary of convex subsets of hyperbolic manifolds

Qiyu Chen  Qiyu Chen: School of Mathematics, South China University of
Technology, 510641, Guangzhou, P. R. China<EMAIL_ADDRESS>and  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: v1, )

###### Abstract.

Let $N$ be a geodesically convex subset in a convex co-compact hyperbolic
manifold $M$ with incompressible boundary. We assume that each boundary
component of $N$ is either a boundary component of $\partial_{\infty}M$, or a
smooth, locally convex surface in $M$. We show that $N$ is uniquely determined
by the boundary data defined by the conformal structure on the boundary
components at infinity, and by either the induced metric or the third
fundamental form on the boundary components which are locally convex surfaces.
We also describe the possible boundary data. This provides an extension of
both the hyperbolic Weyl problem and the Ahlfors-Bers Theorem.

Using this statement for quasifuchsian manifolds, we obtain existence results
for similar questions for convex domains $\Omega\subset{\mathbb{H}}^{3}$ which
meets the boundary at infinity $\partial_{\infty}{\mathbb{H}}^{3}$ either
along a quasicircle or along a quasidisk. The boundary data then includes
either the induced metric or the third fundamental form in ${\mathbb{H}}^{3}$,
but also an additional “gluing” data between different components of the
boundary, either in ${\mathbb{H}}^{3}$ or in
$\partial_{\infty}{\mathbb{H}}^{3}$.

Keywords: hyperbolic geometry, Weyl problem, convex surfaces, isometric
embeddings.

Q. Chen was partially supported by NSFC No: 12101244, Guangdong Basic and
Applied Basic Research Foundation No: 2022A1515012225, and Science and
Technology Program of Guangzhou, China No: 202201010464.

J.-M. S. was partially supported by FNR projects INTER/ANR/15/11211745 and
O20/14766753.

###### Contents

  1. 1 Introduction and main results
    1. 1.1 The classical Weyl problem in Euclidean and hyperbolic spaces
    2. 1.2 The Weyl problem for unbounded convex subsets in hyperbolic space
    3. 1.3 Convex domains in quasifuchsian manifolds
    4. 1.4 Convex domains in convex co-compact manifolds
  2. 2 Background material
    1. 2.1 The hyperbolic and de Sitter spaces
    2. 2.2 The $sl_{2}({\mathbb{C}})$-bundle over $\overline{{\mathbb{H}}^{3}}$
    3. 2.3 The convex co-compact structures and their deformations
      1. 2.3.1 The deformation vector field
      2. 2.3.2 The deformation section
      3. 2.3.3 The deformation one-form
    4. 2.4 The bundle of Euclidean Killing fields over ${\mathbb{R}}^{3}$
    5. 2.5 The infinitesimal Pogorelov map
    6. 2.6 The duality between ${\mathbb{H}}^{3}$ and ${{\rm d}\mathbb{S}}^{3}$
  3. 3 Local rigidity
    1. 3.1 General overview
      1. 3.1.1 From $M$ to ${\mathbb{H}}^{3}$
      2. 3.1.2 From ${\mathbb{H}}^{3}$ to ${\mathbb{R}}^{3}$
      3. 3.1.3 The boundary behavior of the convex domains $\bar{\Omega}$ and $\bar{\Omega}^{*}$.
      4. 3.1.4 The Vekua lemma
    2. 3.2 Estimates on $\omega$
      1. 3.2.1 The deformation one-form on $\partial_{i}M$
      2. 3.2.2 The equidistant foliations of hyperbolic ends
      3. 3.2.3 The metric at infinity associated to an equidistant foliation of hyperbolic ends
      4. 3.2.4 Proof of Lemma 3.13.
    3. 3.3 Estimates on $\bar{\omega}$
    4. 3.4 A deformation vector field on $\partial\bar{\Omega}\setminus\bar{\Lambda}$
    5. 3.5 Hölder estimate on $\bar{v}$
      1. 3.5.1 Distance estimates on $\partial\bar{\Omega}\setminus\bar{\Lambda}$ in ${\mathbb{R}}^{3}$
      2. 3.5.2 Estimates on $(\bar{\tau},\bar{\sigma})$
      3. 3.5.3 Estimates on projection angles to $T\partial\bar{\Omega}$ and $T\partial\bar{\Omega}^{*}$ of radial and lateral vectors
    6. 3.6 Analytical properties of $\bar{v}$
  4. 4 Convex co-compact manifolds
    1. 4.1 Metrics of constant curvatures on the boundary
    2. 4.2 Proof of Theorem 1.6
  5. 5 Convex domains in ${\mathbb{H}}^{3}$
    1. 5.1 The classical conformal welding and generalized gluing maps
      1. 5.1.1 The classical conformal welding of a Jordan curve in ${\mathbb{C}P}^{1}$
      2. 5.1.2 The gluing maps for unbounded convex domains in ${\mathbb{H}}^{3}$ of the first type
      3. 5.1.3 The gluing maps for unbounded convex domains in ${\mathbb{H}}^{3}$ of the second type
      4. 5.1.4 Approximation of quasisymmetric maps
    2. 5.2 Fuchsian approximations of non-positively curved metrics
    3. 5.3 Fuchsian approximations of metrics with $K>-1$
    4. 5.4 Proof of Theorem 1.2
      1. 5.4.1 Equivalent statements of Theorem 1.2
      2. 5.4.2 Convex domains in quasifuchsian case
      3. 5.4.3 Approximation of convex domains
  6. 6 Applications
    1. 6.1 New parametrizations of Bers slice
    2. 6.2 Proof of Theorem 1.5

##  1\. Introduction and main results

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

The classical Weyl problem, going back to Hermann Weyl [48], asks whether any
smooth metric of positive curvature on the sphere can be obtained uniquely as
the induced metric on the boundary of a bounded convex subset in
${\mathbb{R}}^{3}$. A proof was obtained by Lewy [27] for analytic metrics,
and later by Alexandrov [4] for polyhedral metrics. The result was then
further extended by Pogorelov [31] to non-smooth metrics: the induced metric
on the boundary of any bounded convex subset of ${\mathbb{R}}^{3}$ (even if
non-smooth) is a geodesic distance of curvature $K\geq 0$ in the sense of
Alexandrov, and each such metric on the sphere is realized on the boundary of
a unique bounded convex body.

The same question was considered by Alexandrov [4] and Pogorelov [31] for
surfaces in the 3-dimensional hyperbolic space ${\mathbb{H}}^{3}$. Given a
bounded convex subset $\Omega\subset{\mathbb{H}}^{3}$, the induced metric on
its boundary has curvature $K\geq-1$. Conversely, Alexandrov and Pogorelov
proved that any smooth metric of curvature $K>-1$ is induced on the boundary
of a unique bounded convex subset of ${\mathbb{H}}^{3}$ with smooth boundary.
A similar result holds for bounded polyhedra, for which the induced metric on
the boundary is a hyperbolic metric with cone singularities of angles less
than $2\pi$ [4].

In the hyperbolic space, there is also a dual Weyl problem, describing the
possible third fundamental forms of the boundaries of bounded convex subsets.
For a smooth surfaces $S\subset{\mathbb{H}}^{3}$, the third fundamental form
is classically defined for two tangent vectors $u,v$ to $S$ at a point $x$ by:

$III(u,v)=I(\nabla_{u}N,\nabla_{v}N),$

where $I$ is the induced metric on $S$, $\nabla$ is the Levi-Civita connection
of ${\mathbb{H}}^{3}$, and $N$ is a unit normal vector field to $S$. This
third fundamental form can also be described as the induced metric on a dual
surface in the de Sitter space (see Section 2.6), and under this definition
extends to the boundary of a convex domain with non-smooth boundary, for
instance polyhedra in ${\mathbb{H}}^{3}$. For polyhedra in ${\mathbb{H}}^{3}$,
the third fundamental form is also called the dual metric, e.g. in [23].

If $\Omega\subset{\mathbb{H}}^{3}$ is a convex bounded domain, the third
fundamental form on its boundary is $CAT(1)$: it has curvature $K\leq 1$, and
its closed geodesics have length $L>2\pi$ [15]. Conversely, any smooth metric
on $S^{2}$ with curvature $K<1$ and closed geodesics of length $L>2\pi$ is
realized as the third fundamental form on the boundary of a unique bounded
convex subset in ${\mathbb{H}}^{3}$ with smooth boundary, see [37]. A similar
result was proved by Rivin and Hodgson [32, 23]: any spherical metric on
$S^{2}$ with cone singularities of angles greater than $2\pi$, with closed
geodesics of length $L>2\pi$, is the dual metric of a unique bounded
polyhedron in ${\mathbb{H}}^{3}$. This description of the third fundamental
forms on the boundary of bounded or ideal polyhedra is also related to the
description of the dihedral angles of those polyhedra, see e.g. [6, 7, 38, 33,
35].

###  1.2. The Weyl problem for unbounded convex subsets in hyperbolic space

We are interested here in the Weyl problem for convex subsets that are not
bounded. More specifically, we will consider here what are perhaps the
simplest cases of such a convex subset, of two possible types. The first case
is a convex unbounded subset $\Omega\subset{\mathbb{H}}^{3}$ such that

  * •

the boundary of $\Omega$ in ${\mathbb{H}}^{3}$ is an open disk, which we
denote by $\partial\Omega$,

  * •

the ideal boundary of $\Omega$ in the ideal boundary
$\partial_{\infty}{\mathbb{H}}^{3}$ (identified with ${\mathbb{C}P}^{1}$) of
${\mathbb{H}}^{3}$ is a closed disk, which we denote by
$\partial_{\infty}\Omega$.

Clearly, in this setting, the induced metric (or third fundamental form) on
$\partial\Omega$ does not uniquely determine $\Omega$ (up to isometries). For
instance, if $\Omega$ is the hyperbolic convex hull of
$\partial_{\infty}\Omega$ (assuming that $\partial_{\infty}\Omega$ is not a
round disk), then $\partial\Omega$ is developable, and $(\partial\Omega,I)$ is
always isometric to the hyperbolic plane. In the same setting, for any
$K\in(-1,0)$, the boundary (in $\partial_{\infty}{\mathbb{H}}^{3}$) of
$\partial_{\infty}\Omega$ bounds a unique $K$-surface $\Sigma_{K}$ bounding a
convex domain with asymptotic boundary $\partial_{\infty}\Omega$, see [34],
and $\Sigma_{K}$ equipped with its induced metric is always isometric to the
hyperbolic plane with the metric scaled by $1/|K|$. The same construction can
also be used for the third fundamental form, since $\Sigma_{K}$ also has a
third fundamental form of constant curvature $K^{*}=K/(K+1)$, so it is always
isometric to the hyperbolic plane with the metric scaled by $1/|K^{*}|$.

The second type of unbounded convex subset of ${\mathbb{H}}^{3}$ that we
consider here is a convex domain $\Omega\subset{\mathbb{H}}^{3}$ such that
$\partial\Omega$ is the disjoint union of two complete disks, denoted here by
$\partial_{-}\Omega$ and $\partial_{+}\Omega$, which “meet” at infinity along
a Jordan curve. In the spirit of the Weyl problem and its dual, we would like
to prescribe on $\partial_{-}\Omega$ and $\partial_{+}\Omega$ either the
induced metric or the third fundamental form. However this data is again not
sufficient to uniquely determine $\Omega$, as shown in [10] for metrics of
constant curvature: one needs also to prescribe a “gluing map” at infinity
between $\partial_{-}\Omega$ and $\partial_{+}\Omega$.

Our results are obtained only when the limit curve at infinity – the boundary
of $\partial_{\infty}\Omega$ in the first case, the common boundary of
$\partial_{-}\Omega$ and $\partial_{+}\Omega$ in the second case – is a quasi-
circle, which corresponds to the gluing maps being quasi-symmetric. We can
state those results as follows, with points (i) and (ii) corresponding to the
first case above, and points (iii)-(v) corresponding to the second case.
Before stating the result, however, we need a simple definition.

######  Definition 1.1.

Let $h$ be a conformal metric on the open unit disk ${\mathbb{D}}^{2}$, and
let $h_{0}$ be the conformal hyperbolic metric on ${\mathbb{D}}^{2}$. We say
that $h$ is bounded if there exists a constant $C>1$ such that at all points
of ${\mathbb{D}}^{2}$, $h\leq Ch_{0}$.

Note that if $h$ is complete and has curvature $K\geq-c$ for some $c>0$, then
$h\geq(1/c)h_{0}$ (see [29, Theorem 2.8], where the argument is based on
[16]).

######  Theorem 1.2.

Let $D_{-},D_{+}\subset{\mathbb{C}P}^{1}$ be disjoint quasi-disks with
$\partial D_{-}=\partial D_{+}$. Let $\epsilon\in(0,1)$ and let $h_{-}$ (resp.
$h_{+}$) be a smooth, complete, bounded conformal metric of curvature varying
in $[-1+\epsilon,1/\epsilon]$ on $D_{-}$ (resp. $D_{+}$), and $h^{*}_{-}$
(resp. $h^{*}_{+}$) be a smooth, complete, bounded conformal metric of
curvature varying in $[-1/\epsilon,0]$ on $D_{-}$ (resp. $D_{+}$). Then

  1. (i)

There exists a convex subset $\Omega\subset{\mathbb{H}}^{3}$ with
$\partial_{\infty}\Omega$ a closed disk and a continuous map
$\phi:{\mathbb{C}P}^{1}\to\partial\Omega\cup\partial_{\infty}\Omega$ sending
$D_{+}$ to $\partial_{\infty}\Omega$, which is conformal on $D_{+}$ and such
that $\phi^{*}I=h_{-}$ on $D_{-}$.

  2. (ii)

There exists a convex subset $\Omega\subset{\mathbb{H}}^{3}$ with
$\partial_{\infty}\Omega$ a closed disk and a continuous map
$\phi:{\mathbb{C}P}^{1}\to\partial\Omega\cup\partial_{\infty}\Omega$ sending
$D_{+}$ to $\partial_{\infty}\Omega$, which is conformal on $D_{+}$ and such
that $\phi^{*}III=h^{*}_{-}$ on $D_{-}$.

  3. (iii)

There exists a convex subset $\Omega\subset{\mathbb{H}}^{3}$ with
$\partial_{\infty}\Omega$ a Jordan curve, and a continuous map
$\phi:{\mathbb{C}P}^{1}\to\partial\Omega\cup\partial_{\infty}\Omega$, such
that $\phi^{*}I=h_{-}$ on $D_{-}$ and $\phi^{*}I=h_{+}$ on $D_{+}$.

  4. (iv)

There exists a convex subset $\Omega\subset{\mathbb{H}}^{3}$ with
$\partial_{\infty}\Omega$ a Jordan curve, and a continuous map
$\phi:{\mathbb{C}P}^{1}\to\partial\Omega\cup\partial_{\infty}\Omega$, such
that $\phi^{*}III=h^{*}_{-}$ on $D_{-}$ and $\phi^{*}III=h^{*}_{+}$ on
$D_{+}$.

  5. (v)

There exists a convex subset $\Omega\subset{\mathbb{H}}^{3}$ with
$\partial_{\infty}\Omega$ a Jordan curve, and a continuous map
$\phi:{\mathbb{C}P}^{1}\to\partial\Omega\cup\partial_{\infty}\Omega$, such
that $\phi^{*}I=h_{-}$ on $D_{-}$ and $\phi^{*}III=h_{+}^{*}$ on $D_{+}$.

The hypothesis on the metrics $h^{*}_{\pm}$ are probably not optimal – one
could expect similar results only assuming that those metrics have curvature
in $[-1/\epsilon,1-\epsilon]$, and that their closed geodesics have length
$L>2\pi$. However the arguments used in Section 5 only function (at this
point) with the stronger assumption considered here. The hypothesis that those
metrics are “bounded” in the sense considered here is also necessary for
“technical” reasons in the proofs.

Theorem 1.2 extends the main result of [10], which dealt only with constant
curvature metrics, and has a non-empty intersection with the main result of
[43], which only deals with the induced metric. However it should be pointed
out that the argument used here is quite different, since the approximation
was done in [43] by metrics on the sphere, while here we use a quasifuchsian
approximation argument. One could also find additional motivations in [42].

###  1.3. Convex domains in quasifuchsian manifolds

The proof of Theorem 1.2 is based on an approximation argument, where complete
metrics on a disk together with quasi-symmetric homeomorphisms from
${\mathbb{RP}}^{1}$ to itself are approximated, in a suitable sense, by
metrics and quasi-symmetric homeomorphisms which are invariant (resp.
equivariant) with respect to the action of surface groups. Those invariant
metrics (and equivariant gluing maps) correspond to convex surfaces embedded
in quasifuchsian hyperbolic manifolds.

In this quasifuchsian setting, and more generally in the convex co-compact
setting discussed below, we will obtain an existence but also a uniqueness
statement. We first discuss the quasifuchsian setting here for simplicity, and
for reference below.

Let $S$ be a closed, oriented connected surface of genus at least $2$ and let
${\mathcal{T}}_{S}$ denote the Teichmüller space of $S$, which is the space of
hyperbolic metrics (or conformal structures) on $S$ (up to isotopies). The
following statement is a consequence of Theorem 1.6 below.

######  Corollary 1.3.

Let $h$ be a Riemannian metric of curvature greater than $-1$ on $S$, and let
$c\in{\mathcal{T}}_{S}$ be a conformal structure on $S$. Then there is a
unique (up to isotopy) hyperbolic manifold homeomorphic to
$S\times[0,+\infty)$ which has a convex boundary at $S\times\\{0\\}$ with
induced metric isotopic to $h$, and has a conformal structure at infinity on
$S\times\\{+\infty\\}$ isotopic to $c$.

We now consider a “dual” statement to Corollary 1.3, where we prescribe the
third fundamental form rather than the induced metric.

######  Corollary 1.4.

Let $h^{*}$ be a Riemannian metric of curvature less than $1$ on $S$ with
closed, contractible geodesics of length greater than $2\pi$, and let
$c\in{\mathcal{T}}_{S}$ be a conformal structure on $S$. Then there is a
unique (up to isotopy) hyperbolic manifold homeomorphic to
$S\times[0,+\infty)$ which has a convex boundary at $S\times\\{0\\}$ with the
third fundamental form isotopic to $h^{*}$, and has a conformal structure at
infinity on $S\times\\{+\infty\\}$ isotopic to $c$.

Note that the above hyperbolic manifold can be isometrically embedded in a
unique (up to isometries) quasifuchsian manifold.

A consequence of Corollary 1.3 concerns quasifuchsian manifolds with induced
metric prescribed on one side of the convex core, while the conformal
structure at infinity is prescribed on the other side. We only claim existence
in that statement, since proving uniqueness would request obtaining a local
rigidity statement that appears difficult. The statement is presented here as
a theorem since it is not a special case of Theorem 1.6.

######  Theorem 1.5.

Let $h\in{\mathcal{T}}_{S}$ be a hyperbolic metric on $S$, and let
$c\in{\mathcal{T}}_{S}$ be a conformal structure on $S$. Then there is a
quasifuchsian hyperbolic manifold with the conformal structure on its upper
boundary component at infinity isotopic to $c$, and with the induced metric on
the lower boundary component of its convex core isotopic to $h$.

###  1.4. Convex domains in convex co-compact manifolds

We finally state a general result on the existence and uniqueness of convex
co-compact hyperbolic manifold containing a geodesically convex subset such
that the induced metric, third fundamental form, or conformal metric at
infinity is prescribed for each end.

######  Theorem 1.6.

Let $M$ be a compact 3-manifold with non-empty, incompressible boundary which
admits a convex co-compact hyperbolic structure on its interior and let
$\partial_{l}M$ ( $l=1,\ldots,n$, $n\geq 2$ ) denote its boundary components.
Let $\mathcal{I}\sqcup\mathcal{J}\sqcup\mathcal{K}=\\{1,\ldots,n\\}$ be a
partition, where $\mathcal{I}$, $\mathcal{J}$, or $\mathcal{K}$ might be an
empty set. Let $c_{i}$ ($i\in\mathcal{I}$) be a conformal structure on
$\partial_{i}M$, $h_{j}$ ($j\in\mathcal{J}$) be a Riemannian metric of
curvature greater than $-1$ on $\partial_{j}M$ and $h_{k}^{*}$
($k\in\mathcal{K}$) be a Riemannian metric of curvature less than $1$ on
$\partial_{k}M$, with closed, contractible geodesics of length greater than
$2\pi$. Then there exists a unique (up to isotopy) convex co-compact
hyperbolic structure on $M$ which is complete on $\partial_{i}M$ (if
$\mathcal{I}\not=\emptyset$) with conformal structure isotopic to $c_{i}$, and
with induced metric isotopic to $h_{j}$ on a surface isotopic to
$\partial_{j}M$ (if $\mathcal{J}\not=\emptyset$), and with third fundamental
form isotopic to $h_{k}^{*}$ on a surface isotopic to $\partial_{k}M$ (if
$\mathcal{K}\not=\emptyset$).

Corollaries 1.3 and 1.4 clearly follow from this theorem.

##  2\. Background material

###  2.1. The hyperbolic and de Sitter spaces

Recall that the hyperboloid models of the 3-dimensional hyperbolic space and
de Sitter (dS) space are respectively

$H^{3}:=\\{x\in{\mathbb{R}}^{3,1}|\langle x,x\rangle_{3,1}=-1,x_{0}>0\\},$

and

$dS^{3}:=\\{x\in{\mathbb{R}}^{3,1}|\langle x,x\rangle_{3,1}=1\\},$

where $\langle x,y\rangle_{3,1}=-x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}$.
We denote by ${\mathbb{H}}^{3}$ (resp. ${{\rm d}\mathbb{S}}^{3}$) the
projection of $H^{3}$ (resp. $dS^{3}$) to ${\mathbb{RP}}^{3}$ (with the metric
induced from the projection), which is the _projective model_ of the
hyperbolic (resp. dS) 3-space (we use this model throughout the paper, unless
otherwise stated).

Let $\mathbb{HS}^{3}$ denote the space ${\mathbb{RP}}^{3}$ equipped with the
geometric structure induced by the bilinear form
$\langle\cdot,\cdot\rangle_{3,1}$, called the 3-dimensional _HS space of
signature $(3,0)$_. By construction its geometry is invariant under
$PO_{0}(3,1)$. It has a component - corresponding to the time-like vectors
with $x_{0}>0$ in ${\mathbb{R}}^{3,1}$ isometric to ${\mathbb{H}}^{3}$, while
the complement of the closure of this component is isometric to ${{\rm
d}\mathbb{S}}^{3}$. The signature of the metric induced on $\mathbb{HS}^{3}$
is in fact $(3,0)$ on ${\mathbb{H}}^{3}$ and $(2,1)$ on ${{\rm
d}\mathbb{S}}^{3}$.

Let $\phi:{\mathbb{H}}^{3}\cup{{\rm
d}\mathbb{S}}^{3}\rightarrow{\mathbb{R}}^{3,1}$ be a local section of the
canonical projection $\pi:{\mathbb{R}}^{3,1}\rightarrow{\mathbb{RP}}^{3}$ with
$\phi({\mathbb{H}}^{3})=H^{3}$ and $\phi({{\rm d}\mathbb{S}}^{3})\subset
dS^{3}$. For each point $x\in{\mathbb{H}}^{3}\cup{{\rm d}\mathbb{S}}^{3}$, we
define the _HS scalar product_ on $T_{x}\mathbb{HS}^{3}$ as

$\langle v,w\rangle_{HS}:=\langle\phi_{*}(v),\phi_{*}(w)\rangle_{3,1}$

for all $v,w\in T_{x}\mathbb{HS}^{3}$. The restriction of
$\langle\cdot,\cdot\rangle_{HS}$ to $T{\mathbb{H}}^{3}$ (resp. $T{{\rm
d}\mathbb{S}}^{3}$) exactly induces the hyperbolic (resp. dS) metric.

The boundary in ${\mathbb{RP}}^{3}$ of ${\mathbb{H}}^{3}$ or ${{\rm
d}\mathbb{S}}^{3}$ coincides with the projectivization in ${\mathbb{RP}}^{3}$
of the set of light-like vectors in ${\mathbb{R}}^{3,1}$, which is conformally
equivalent to ${\mathbb{C}P}^{1}$ in a canonical way. Let
$\overline{{\mathbb{H}}^{3}}={\mathbb{H}}^{3}\cup{\mathbb{C}P}^{1}$ denote the
natural (conformal) compactification of ${\mathbb{H}}^{3}$ by
${\mathbb{C}P}^{1}$. Note that the action of isometries of ${\mathbb{H}}^{3}$
can extend to ${\mathbb{C}P}^{1}$ as projective automorphisms (or Möbius
transformations) and conversely, Möbius transformations on ${\mathbb{C}P}^{1}$
can extend to isometries inside ${\mathbb{H}}^{3}$ (using the Poincaré upper
half-space model) [28]. For simplicity, we denote by ${\rm
PSL}_{2}({\mathbb{C}})$ the group of orientation-preserving isometries of
${\mathbb{H}}^{3}$ as well as the group of Möbius transformations on
${\mathbb{C}P}^{1}$. Let $sl_{2}({\mathbb{C}})$ denote the Lie algebra of
${\rm PSL_{2}({\mathbb{C}})}$, which can be interpreted as the space of
infinitesimal automorphisms of $\overline{{\mathbb{H}}^{3}}$. These are vector
fields on $\overline{{\mathbb{H}}^{3}}$ whose flows are isometries (resp.
Möbius transformations) on ${\mathbb{H}}^{3}$ (resp.
$\partial\overline{{\mathbb{H}}^{3}}\cong{\mathbb{C}P}^{1}$), called _Killing_
(resp. projective) vector fields (see e.g. [22, 12]).

###  2.2. The $sl_{2}({\mathbb{C}})$-bundle over $\overline{{\mathbb{H}}^{3}}$

Let $E$ denote the trivial $sl_{2}({\mathbb{C}})$-bundle over
$\overline{{\mathbb{H}}^{3}}$. Let $\kappa\in\Gamma(E)$ be a section of $E$.
If $x\in{\mathbb{H}}^{3}$, the value of $\kappa$ at $x$, denoted by
$\kappa_{x}$, is a Killing vector field on ${\mathbb{H}}^{3}$, which has the
following decomposition (see e.g. [22, 40]):

$\kappa_{x}:=u_{x}+v_{x},$

where $u_{x}$ is the _infinitesimal pure translation at $x$_ (namely,
$u_{x}(x)=\kappa_{x}(x)$ and $\nabla_{X}u_{x}=0$ for all $X\in
T_{x}{\mathbb{H}}^{3}$), while $v_{x}$ is the _infinitesimal pure rotation at
$x$_ (namely, $v_{x}(x)=0$ and there exists a vector say $\sigma(x)\in
T_{x}{\mathbb{H}}^{3}$ such that $\nabla_{X}v_{x}=\sigma(x)\times X$ for all
$X\in T_{x}{\mathbb{H}}^{3}$), where $\nabla$ is the Levi-Civita connection of
${\mathbb{H}}^{3}$. As a consequence, we can identify $\kappa_{x}$ with a pair
of tangent vectors at $x$, say $(\tau(x),\sigma(x))\in
T_{x}{\mathbb{H}}^{3}\times T_{x}{\mathbb{H}}^{3}$, which satisfy that

$\tau(x)=\kappa_{x}(x),\quad\quad\nabla_{X}\kappa_{x}=\sigma(x)\times X,$

for all $X\in T_{x}{\mathbb{H}}^{3}$. The _curl_ of a vector field $X$ on
${\mathbb{H}}^{3}$ is defined as

(1) ${\rm
curl}X:=\sum_{i\in{\mathbb{Z}}/3{\mathbb{Z}}}(\langle\nabla_{e_{i}}X,e_{i+1}\rangle-\langle\nabla_{e_{i+1}}X,e_{i}\rangle)e_{i+2},$

where $(e_{1},e_{2},e_{3})$ is an orthonormal frame of $T{\mathbb{H}}^{3}$
(see e.g. [45, Section 11.1]). A direct computation shows that the above
$\sigma(x)$ is one half the curl of $\kappa_{x}$ at $x$, similarly for the
euclidean case in Section 2.4.

In this way, the restriction to ${\mathbb{H}}^{3}$ of a section
$\kappa\in\Gamma(E)$ is identified with a pair of vector fields
$(\tau,\sigma)\in\Gamma(T{\mathbb{H}}^{3})\times\Gamma(T{\mathbb{H}}^{3})$.
For convenience, we call $\tau$ (resp. $\sigma$) the _translation_ (resp.
_rotation_) _component_ of $\kappa$. Under this identification and by a direct
computation of the flat connection $D$ for $sl_{2}({\mathbb{C}})$-bundle over
${\mathbb{H}}^{3}$, we conclude that: for any $x\in{\mathbb{H}}^{3}$ and any
$X\in T_{x}{\mathbb{H}}^{3}$, the one-form $\omega=d^{D}\kappa$ can be written
as (see [22] or [40, Section 3]):

(2) $\omega(X)=d^{D}_{X}(\tau,\sigma)=\big{(}\nabla_{X}\tau-\sigma(x)\times
X,\nabla_{X}\sigma+\tau(x)\times
X\big{)}=:\big{(}\omega_{\tau}(X),\omega_{\sigma}(X)\big{)}.$

We call $\omega_{\tau}$ (resp. $\omega_{\sigma}$) the _translation_ (resp.
_rotation_) _part_ of $\omega$. Note that $\kappa$ is a constant section if
and only if $d^{D}\kappa=0$, also, if and only if $\tau$ and $\sigma$ are
hyperbolic Killing fields.

If $x\in{\mathbb{C}P}^{1}$, $\kappa_{x}$ is a projective vector field on
${\mathbb{C}P}^{1}$. Let $\big{(}\phi_{t}(z)\big{)}_{t\in[0,\varepsilon]}$ be
a one-parameter family of Möbius transformations which generate $\kappa_{x}$.
By computing the derivative at $t=0$ of $\phi_{t}(z)$, $\kappa_{x}$ has the
following form in an affine chart $z$ near $x$ (see e.g. [14, Section 3.2] and
[30, Remark 4.4]):

$\kappa_{x}(z)=P(z)\partial_{z},$

where $P(z)$ is a complex polynomial of degree at most 2 (called _quadratic
polynomials_ for simplicity). Moreover, there is a unique Killing field on
${\mathbb{H}}^{3}$, determined by the above
$(\phi_{t}(z))_{t\in[0,\varepsilon]}$, that extends continuously to
$\kappa_{x}$ (see the proof of Lemma 3.14 for instance).

###  2.3. The convex co-compact structures and their deformations

Let $M$ be a compact 3-manifold with non-empty, incompressible boundary. A
_$({\rm PSL}_{2}({\mathbb{C}}),\overline{{\mathbb{H}}^{3}})$ -structure_ on
$M$ is a maximal atlas of charts $\\{(U_{\alpha},\varphi_{\alpha})\\}$ to
$\overline{{\mathbb{H}}^{3}}$ with transition maps the restrictions of
elements of ${\rm PSL}_{2}({\mathbb{C}})$, which is a _hyperbolic structure_
when restricted to the interior of $M$, and a _complex projective structure_
when restricted to $\partial M$ (see e.g. [12]). Under this condition, the
interior of $M$ is equipped with a complete, convex co-compact hyperbolic
metric, also called a _convex co-compact structure_. (To see this, let
$(D_{i})_{i=1,\dots,n}$ be a cover of $\partial M$ by embedded open disks, for
the complex projective structure at infinity. Each of the $D_{i}$ bounds a
half-space $H_{i}$ embedded in $M$. The complement $\Omega$ in $M$ of the
union of the $H_{i}$ is then by construction a non-empty, geodesically convex
subset of $M$, in the sense that any geodesic segment in $M$ with endpoints in
$\Omega$ is contained in $\Omega$. This proves that the hyperbolic structure
on the interior of $M$ is convex co-compact.)

Let ${\mathcal{CC}}(M)$ denote the space of convex co-compact structures on
$M$ (up to isotopy). In particular, if $M$ is homeomorphic to the product of a
surface $S$ (of genus at least $2$) by an interval, the manifold $M$ equipped
with a convex co-compact structure is also called a _quasifuchsian_ hyperbolic
manifold. In this case, the two boundary components of $M$ are homeomorphic to
$S$.

####  2.3.1. The deformation vector field

Let $(U_{\alpha},\varphi_{\alpha}^{t})_{t\in[0,\epsilon]}$ be a smooth one-
parameter family of convex co-compact structures on $M$. Let $dev_{t}$ and
$hol_{t},t\in[0,\epsilon]$, be the corresponding developing maps and holonomy
representations, respectively, where
$dev_{t}:\tilde{M}\rightarrow\overline{{\mathbb{H}}^{3}}$ and
$hol_{t}:\pi_{1}(M)\rightarrow{\rm PSL}_{2}({\mathbb{C}})$ with $\tilde{M}$
the universal cover of $M$. This defines a _deformation vector field_ say $u$,
on $\tilde{M}$, which associates to each $x\in\tilde{M}$ a tangent vector at
$t=0$, say $u(x)$, of the curve $t\mapsto dev_{0}^{-1}(dev_{t}(x))$. The
equivariance property $dev_{t}(\gamma\cdot x)=hol_{t}(\gamma)\circ dev_{t}(x)$
for all $\gamma\in\pi_{1}(M)$ implies that the vector field $\gamma_{*}u-u$ is
Killing (resp. _projective_) on $\tilde{M}\setminus\partial\tilde{M}$ (resp.
$\partial\tilde{M}$):

(3) $(\gamma_{*})^{-1}u(\gamma\cdot
x)=u(x)+dev_{0}^{*}\big{(}\dot{hol}_{0}(\gamma)(dev_{0}(x))\big{)},$

where $\dot{hol}_{0}(\gamma)\in sl_{2}({\mathbb{C}})$ is the derivative of
$hol_{t}(\gamma)$ at $t=0$, which is a Killing (resp. projective) vector field
on ${\mathbb{H}}^{3}$ (resp. ${\mathbb{C}P}^{1}$). We call such $u$
automorphic. Moreover, we say $u$ is _equivariant_ (with respect to the action
of $\pi_{1}(M)$) if in particular $\dot{hol}_{0}(\gamma)=0$ (in this case,
$dev_{t}$ is obtained by pre-composing $dev_{0}$ with the lift of
diffeomorphisms $f_{t}$ of $M$ isotopic to the identity), see e.g. [22, 14,
30]

Note that if the above first-order deformation preserves the conformal
structure at infinity on $\partial_{l}M$ ($1\leq l\leq n$), then $u$ is a
conformal vector field when restricted to $\partial_{l}\tilde{M}$.

####  2.3.2. The deformation section

Let $\Lambda\subset{\mathbb{C}P}^{1}$ be the limit set of the Klein group
$hol_{0}(\pi_{1}(M))$. For simplicity, we identify the universal cover
$\tilde{M}$ with its image $\overline{{\mathbb{H}}^{3}}$ under the developing
map. Now we introduce a _deformation section_ of $E$ over
$\overline{{\mathbb{H}}^{3}}\setminus\Lambda$ (associated to a deformation
vector field $u$) as follows (see [14, 30]).

For $x\in{\mathbb{H}}^{3}$, we define a Killing vector field $s_{x}\in
sl_{2}({\mathbb{C}})$ that “best approximates” the vector field $u$ at $x$:

(4) $s_{x}(x)=u(x),\quad\quad{\rm{curl}}(s_{x})(x)={\rm curl}(u)(x).$

For $z_{0}\in{\mathbb{C}P}^{1}\setminus\Lambda$, let $u(z)=f(z)\partial_{z}$
be the local expression of $u$ in an affine chart near $z_{0}$, we define a
projective vector field $s_{z_{0}}$ whose two-jets agree with $f$ at $z_{0}$.
More precisely,

(5)
$s_{z_{0}}(z)=\big{(}f(z_{0})+f_{z}(z_{0})(z-z_{0})+f_{zz}(z_{0})\frac{(z-z_{0})^{2}}{2}\big{)}\partial_{z}.$

Note that the above $s_{x}$ (resp. $s_{z_{0}}$) is uniquely determined by the
equations (4) (resp. the equation (5)). We call such a section the canonical
lift of $u$ and denote it by $s_{u}$. By definition, the canonical lift of a
vector field in $sl_{2}({\mathbb{C}})$ is a constant section. As a
consequence, the canonical lift $s_{u}$ is automorphic, that is, for each
$\gamma\in\pi_{1}(M)$, $(s_{u})_{\gamma\cdot x}-\gamma_{*}(s_{u})_{x}$ is the
same element in $sl_{2}({\mathbb{C}})$ for all
$x\in\overline{{\mathbb{H}}^{3}}\setminus\Lambda$. In particular, if $u$ is
equivariant, then $s_{u}$ is equivariant, in the sense that
$(s_{u})_{\gamma\cdot x}=\gamma_{*}(s_{u})_{x}$ for all $\gamma\in\pi_{1}(M)$
and $x\in\overline{{\mathbb{H}}^{3}}\setminus\Lambda$.

####  2.3.3. The deformation one-form

Recall that $E$ has a natural flat connection $D$. Let $d^{D}$ be the de Rham
differential twisted by $D$. We now introduce a _deformation one-form_ :
$\tilde{\omega}=d^{D}s_{u}$ on $\overline{{\mathbb{H}}^{3}}\setminus\Lambda$,
see e.g. [22, 14, 30]. One can check that $\tilde{\omega}$ is an _equivariant_
$E$-valued one-form, which measures how far the deformation vector field $u$
differs from a Killing (resp. projective) field on ${\mathbb{H}}^{3}$ (resp.
${\mathbb{C}P}^{1}$). Indeed,

$\tilde{\omega}-\gamma_{*}\tilde{\omega}=d^{D}s_{u}-\gamma_{*}d^{D}s_{u}=d^{D}(s_{u}-\gamma_{*}s_{u})=0.$

So $\tilde{\omega}$ descends to an $E_{M}$-valued closed one-form
$\omega\in\Omega^{1}(M,E_{M})$ on $M$ (since it is locally exact), where
$E_{M}$ is the quotient of
$(\overline{{\mathbb{H}}^{3}}\setminus\Lambda)\times sl_{2}({\mathbb{C}})$ by
$hol_{0}(\pi_{1}(M))$, which acts on
$\overline{{\mathbb{H}}^{3}}\setminus\Lambda$ by deck transformations and on
$sl_{2}({\mathbb{C}})$ via adjoint representation.

A first-order deformation of a convex co-compact structure
$g\in{\mathcal{CC}}(M)$ is _trivial_ if the corresponding deformation vector
field $u$ belongs to $sl_{2}({\mathbb{C}})$ or is equivariant, or
equivalently, the deformation section $s_{u}$ is constant or equivariant, or
equivalently, the deformation one-form $\tilde{\omega}=d^{D}s_{u}$ is exact.
As a consequence, there is a one-to-one correspondence between the space of
infinitesimal deformations of $M$ modulo trivial deformations and the first-
cohomology group $H^{1}(M,E_{M})$ (see [22, 14, 30]).

######  Definition 2.1.

Let $\kappa$ be a Killing vector field on ${\mathbb{H}}^{3}$ (which can be
viewed as a constant section of $E$ over ${\mathbb{H}}^{3}$). We define the
norm of $\kappa$ at $x$ as

$\|\kappa\|_{x}:=\|\tau(x)\|_{{\mathbb{H}}^{3}}+\|\sigma(x)\|_{{\mathbb{H}}^{3}},$

where $\tau$ and $\sigma$ are respectively the translation and rotation
components of $\kappa$. Let $\omega\in H^{1}(M,E_{M})$. The norm of $\omega$
at $p\in M$ is defined as

$\|\omega\|_{p}:=\sup_{X\in T_{p}M:\,\|X\|_{T_{p}M}=1}\|\omega(X)\|_{p},$

where $\omega(X)\in E_{M}$ is a local Killing field on a neighbourhood in $M$
of $p$. We say $\omega$ is bounded on a subset $N\subset M$ if
$\|\omega\|_{p}$ is uniformly bounded for all $p\in N$.

###  2.4. The bundle of Euclidean Killing fields over ${\mathbb{R}}^{3}$

Let $\bar{E}$ denote the trivial bundle of Euclidean Killing fields over
${\mathbb{R}}^{3}$. The identification between $\bar{E}$ and
$T{\mathbb{R}}^{3}\times T{\mathbb{R}}^{3}$ can be done along similar lines as
hyperbolic case, see [40, Section 3]. Let $\bar{\kappa}\in\Gamma(\bar{E})$ and
$\bar{\kappa}_{\bar{x}}$ be its value at $\bar{x}\in{\mathbb{R}}^{3}$. We can
write

$\bar{\kappa}_{\bar{x}}:=\bar{u}_{\bar{x}}+\bar{v}_{\bar{x}},$

where $\bar{u}_{\bar{x}}$ is the _infinitesimal pure translation at $\bar{x}$_
(that is, $\bar{u}_{\bar{x}}(\bar{x})=\bar{\kappa}_{\bar{x}}(\bar{x})$ and
$\bar{\nabla}_{\bar{X}}\bar{u}_{\bar{x}}=0$ for all $\bar{X}\in
T_{\bar{x}}{\mathbb{R}}^{3}$), while $\bar{v}_{\bar{x}}$ is the _infinitesimal
pure rotation at $\bar{x}$_ (that is, $\bar{u}_{\bar{x}}(\bar{x})=0$ and there
exists a vector say $\bar{\sigma}(\bar{x})\in T_{\bar{x}}{\mathbb{R}}^{3}$
such that
$\bar{\nabla}_{\bar{X}}\bar{v}_{\bar{x}}=\bar{\sigma}(\bar{x})\times\bar{X}$
for all $\bar{X}\in T_{\bar{x}}{\mathbb{R}}^{3}$), here $\bar{\nabla}$ is the
Levi-Civita connection of ${\mathbb{R}}^{3}$. Therefore, we can identify a
Euclidean Killing field $\bar{\kappa}_{\bar{x}}$ with a pair of tangent
vectors at $\bar{x}$, say $(\bar{\tau}(\bar{x}),\bar{\sigma}(\bar{x}))\in
T_{\bar{x}}{\mathbb{R}}^{3}\times T_{\bar{x}}{\mathbb{R}}^{3}$, satisfying
that

$\bar{\tau}(\bar{x}):=\bar{\kappa}_{\bar{x}}(\bar{x}),\quad\quad\bar{\nabla}_{\bar{X}}\bar{\kappa}_{\bar{x}}=\bar{\sigma}(\bar{x})\times\bar{X},$

for all $\bar{X}\in T_{\bar{x}}{\mathbb{R}}^{3}$.

In this way, a section $\bar{\kappa}\in\Gamma(\bar{E})$ is identified with a
pair of vector fields
$(\bar{\tau},\bar{\sigma})\in\Gamma(T{\mathbb{R}}^{3})\times\Gamma(T{\mathbb{R}}^{3})$.
we call $\bar{\tau}$ (resp. $\bar{\sigma}$) the _translation_ (resp.
_rotation_) _component_ of $\bar{\kappa}$. Under this identification and by a
direct computation of the flat connection $\bar{D}$ for the bundle $\bar{E}$,
it follows that for any $\bar{x}\in{\mathbb{R}}^{3}$ and any $\bar{X}\in
T_{\bar{x}}{\mathbb{R}}^{3}$, the one-form
$\bar{\omega}=d^{\bar{D}}\bar{\kappa}$ can be written as (see [40, Section
3]).

(6)
$\bar{\omega}(\bar{X})=d^{\bar{D}}_{\bar{X}}(\bar{\tau},\bar{\sigma})=\big{(}\bar{\nabla}_{\bar{X}}\bar{\tau}-\bar{\sigma}(\bar{x})\times\bar{X},\bar{\nabla}_{\bar{X}}\bar{\sigma}\big{)}=:\big{(}\bar{\omega}_{\bar{\tau}}(\bar{X}),\bar{\omega}_{\bar{\sigma}}(\bar{X})\big{)}.$

We call $\bar{\omega}_{\bar{\tau}}$ (resp. $\bar{\omega}_{\bar{\sigma}}$) the
_translation_ (resp. _rotation_) _part_ of $\bar{\omega}$. Note that
$\bar{\kappa}$ is a constant section if and only if
$d^{\bar{D}}\bar{\kappa}=0$, also, if and only if $\bar{\tau}$ and
$\bar{\sigma}$ are Euclidean Killing fields.

###  2.5. The infinitesimal Pogorelov map

We first recall the definition of the infinitesimal Pogorelov map and its key
properties (see e.g. [31, 38] and in particular [40, Definitions 1.9 and
1.12]).

Choose an affine chart $x_{0}=1$ and denote by $O^{*}$ the projective plane in
${{\rm d}\mathbb{S}}^{3}$ dual to the center $O:=[1,0,0,0]\in{\mathbb{H}}^{3}$
in $\mathbb{HS}^{3}$. Let
$U=\mathbb{HS}^{3}\setminus(O^{*}\cup\partial\overline{{\mathbb{H}}^{3}})$.
Then $U$ is the intersection of
$\mathbb{HS}^{3}\setminus\partial\overline{{\mathbb{H}}^{3}}$ with the
aforementioned affine chart of ${\mathbb{RP}}^{3}$. Let ${\mathbb{R}}^{3}$
denote the 3-dimensional Euclidean space and let
$\iota:U\rightarrow{\mathbb{R}}^{3}$ be an inclusion that sends $O$ to the
origin $\bar{O}$ of ${\mathbb{R}}^{3}$, with $\iota({\mathbb{H}}^{3})$ the
unit open ball $\mathbb{D}^{3}$. It can be checked directly from the
hyperbolic metric on ${\mathbb{H}}^{3}$ that $\iota$ is an isometry on the
tangent space at $O$.

For any $x\in U$ and any vector $v\in T_{x}U$, we can write
$v=v^{r}+v^{\perp}$, where $v^{r}$ is tangent to the geodesic in
$\mathbb{HS}^{3}$ going through $O$ and $x$ (called the _radial_ component of
$v$) and $v^{\perp}$ is orthogonal to $v^{r}$ (called the _lateral_ component
of $v$). Let $\Upsilon:TU\rightarrow T{\mathbb{R}}^{3}$ be the bundle map over
the inclusion $\iota$, which is defined in the following way:

$\Upsilon(v)=d\iota(v)$

for all $v\in T_{O}U$, and

$\Upsilon(v)=\sqrt{\frac{\langle\hat{x},\hat{x}\rangle_{HS}}{\langle
d\iota(\hat{x}),d\iota(\hat{x})\rangle_{{\mathbb{R}}^{3}}}}d\iota(v^{r})+d\iota(v^{\perp}),$

for all $v\in T_{x}U$ with $x\in U$, where $\hat{x}$ is a non-zero radial
vector, and $\langle\cdot,\cdot\rangle_{HS}$ is the HS scalar product. We call
this bundle map an _infinitesimal Pogorelov map_.

Note that $\Upsilon$ sends radial (resp. lateral) directions to radial (resp.
lateral) directions. In particular, the radial component of $\Upsilon(v)$ has
the same norm as the radial component of $v$ whenever $x\in{\mathbb{H}}^{3}$
or ${{\rm d}\mathbb{S}}^{3}$ (with respect to corresponding metrics), while
$\Upsilon$ acts on the lateral component as the differential of the projective
model map: the norm of the lateral component of $\Upsilon(v)$ is
$1/{\cosh(\rho)}$ (resp. $1/\sinh(\rho)$) times that of the lateral component
of $v$ for $x\in{\mathbb{H}}^{3}$ (resp. $x\in{{\rm d}\mathbb{S}}^{3}$), where
$\rho$ is the hyperbolic distance from $x$ to $O$ (resp. the dS distance from
$x$ to $O^{*}$), see e.g. [40, Propositions 1.2 and 1.3].

We say that a vector field on a surface $S\subset{\mathbb{H}}^{3}$ (resp. a
spacelike surface $S^{\prime}\subset{{\rm d}\mathbb{S}}^{3}$) is _isometric_
if it preserves the induced metric on $S\subset{\mathbb{H}}^{3}$ (resp.
$S^{\prime}\subset{{\rm d}\mathbb{S}}^{3}$) at first order. Similarly for that
on a surface $\bar{S}\subset{\mathbb{R}}^{3}$. We stress the following
property of the bundle map $\Upsilon$, see [40, Lemma 1.10], which is crucial
for the local rigidity problem here.

######  Lemma 2.2.

The following statements are true:

  1. (1)

Let $Z$ be a vector field on $U$. Then $Z$ is a Killing vector field (for the
HS metric) if and only if $\Upsilon(Z)$ is a Killing vector field (for the
Euclidean metric).

  2. (2)

Let $v$ be a vector field on an embedded surface $S\subset U$. Then $v$ is an
isometric vector field on $S$ if and only if $\Upsilon(v)$ is an isometric
vector field on $\iota(S)$.

As a consequence of Lemma 2.2, the infinitesimal Pogorelov map $\Upsilon$
induces the following map between sections of $E$ (restricted to
${\mathbb{H}}^{3}$) and $\bar{E}$ (see e.g. [40, Definition 3.5]):

$\begin{split}\Psi:E&\rightarrow\bar{E}\\\
\kappa&\mapsto\bar{\kappa},\end{split}$

which takes a hyperbolic Killing field $\kappa_{x}$ (associated to $x$) to a
Euclidean Killing field $\bar{\kappa}_{\bar{x}}$ (associated to
$\bar{x}=\iota(x)$), with
$\bar{\kappa}_{\bar{x}}(\bar{y}):=\Upsilon(\kappa_{x}(y))$ for all
$y\in{\mathbb{H}}^{3}$ and $\bar{y}=\iota(y)$. For convenience, we also denote
by $\Psi:\Gamma(E)\rightarrow\Gamma(\bar{E})$ the restriction of $\Psi$ to
$\Gamma(E)$. In particular, $\Psi$ sends flat sections of $E$ for $D$ to flat
sections of $\bar{E}$ for $\bar{D}$ and therefore, it respects the connections
$D$ and $\bar{D}$ [40, Proposition 3.6]. We have

(7) $\Psi(d^{D}_{X}\kappa):=d^{\bar{D}}_{d\iota(X)}(\Psi(\kappa)),$

for any section $\kappa\in\Gamma(E)$ and $X\in T{\mathbb{H}}^{3}$. For
simplicity, we denote $\Psi_{*}(d^{D}\kappa):=d^{\bar{D}}(\Psi(\kappa))$.

###  2.6. The duality between ${\mathbb{H}}^{3}$ and ${{\rm d}\mathbb{S}}^{3}$

The (polar) duality between a point (resp. a totally geodesic plane) in
${\mathbb{H}}^{3}$ and a totally geodesic plane (resp. a point) in ${{\rm
d}\mathbb{S}}^{3}$ is defined in the following way.

Consider the hyperboloid models $H^{3}$ and $dS^{3}$ in ${\mathbb{R}}^{3,1}$.
Given a point $x\in H^{3}$ (which is the intersection with $H^{3}$ of the line
$l_{x}\subset{\mathbb{R}}^{3,1}$ in the direction of $x$), we define the
_dual_ of $x$ as the intersection $(l_{x})^{\perp}\cap dS^{3}$, where
$(l_{x})^{\perp}$ is the orthogonal complement in ${\mathbb{R}}^{3,1}$ of
$l_{x}$ with respect to the Minkowski metric
$\langle\cdot,\cdot\rangle_{3,1}$. Given a totally geodesic plane $P\subset
H^{3}$ (which is obtained from the intersection with $H^{3}$ of a hyperspace
$H_{P}\subset{\mathbb{R}}^{3,1}$), we define the _dual_ of $P$ as the
intersection $(H_{P})^{\perp}\cap dS^{3}$. Similarly, we can define the dual
of a point $x^{\prime}\in dS^{3}$ and the dual of a totally geodesic spacelike
plane $P^{\prime}\subset dS^{3}$ (note that the spacelike condition ensures
that the intersection $(H_{P^{\prime}})^{\perp}\cap H^{3}$ is non-empty and
the dual of $P^{\prime}$ is thus well-defined). Through the projective maps,
the duality is naturally defined in the projective models ${\mathbb{H}}^{3}$
and ${{\rm d}\mathbb{S}}^{3}$.

Using this duality, one can define the _dual surface_ , denoted by $S^{*}$, of
a convex embedded surface $S\subset{\mathbb{H}}^{3}$ as the set of points
which are dual to the tangent planes $P_{x}$ to $S$ at all $x\in S$.
Similarly, one can define the _dual surface_ , denoted by $(S^{\prime})^{*}$,
of a convex embedded spacelike surface $S^{\prime}\subset{{\rm
d}\mathbb{S}}^{3}$. In particular, we denote by $x^{*}$ the point in $S^{*}$
which is dual to the tangent plane to $S$ at $x\in S$.

We list the following relations and properties between dual surfaces, see e.g.
[38, Section 3].

######  Proposition 2.3.

Let $S\subset{\mathbb{H}}^{3}$ be a strictly convex embedded surface. Then

  1. (1)

The dual surface $S^{*}$ of $S$ is a strictly convex space-like surface in
${{\rm d}\mathbb{S}}^{3}$.

  2. (2)

The dual surface $(S^{*})^{*}$ of $S^{*}$ is exactly $S$.

  3. (3)

The pull-back of the induced metric on $S^{*}$ through the duality map is the
third fundamental form of $S$ and vice versa.

  4. (4)

$S$ has curvature $K\in(-1,+\infty)$ at a point $x\in S$ if and only if
$S^{*}$ has curvature $K^{*}=K/(K+1)\in(-\infty,1)$ at the dual point
$x^{*}\in S^{*}$.

##  3\. Local rigidity

###  3.1. General overview

Let $g\in{\mathcal{CC}}(M)$ be a convex co-compact hyperbolic structure on $M$
with conformal structure at infinity $c_{i}$ on $\partial_{i}M$ (for each
$i\in\mathcal{I}$), with induced metric $h_{j}$ on a smooth, strictly convex
surface $S_{j}\subset M$ isotopic to $\partial_{j}M$ (for each
$j\in\mathcal{J}$), and with third fundamental form $h^{*}_{k}$ on a smooth,
strictly convex surface $S_{k}\subset M$ isotopic to $\partial_{k}M$ (for each
$k\in\mathcal{K}$). In this section, we aim to show the infinitesimal rigidity
of $g$ with respect to $c_{i}$, $h_{j}$ and $h^{*}_{k}$ over
$i\in\mathcal{I},j\in\mathcal{J},k\in\mathcal{K}$ (as stated below).

######  Proposition 3.1.

For any first-order deformation $\dot{g}\in T_{g}{\mathcal{CC}}(M)$, if
$\dot{g}$ preserves the conformal structure at infinity $c_{i}$ on
$\partial_{i}M$ (for all $i\in\mathcal{I}$), the induced metric $h_{j}$ on
$S_{j}$ (for all $j\in\mathcal{J}$) and the third fundamental form $h^{*}_{k}$
on $S_{k}$ (for all $k\in\mathcal{K}$) at first order, then $\dot{g}$ is
trivial.

####  3.1.1. From $M$ to ${\mathbb{H}}^{3}$

For simplicity, we denote by $M$ the hyperbolic manifold with convex co-
compact structure $g\in{\mathcal{CC}}(M)$. Let $N\subset M$ be the
geodesically convex subset with the conformal boundary at infinity
$\partial_{i}M$ and the smooth, strictly convex boundary surfaces $S_{j}$ and
$S_{k}$ over $i\in\mathcal{I}$, $j\in\mathcal{J}$ and $k\in\mathcal{K}$. We
denote $\partial_{i}N:=\partial_{i}M$, $\partial_{j}N:=S_{j}$ and
$\partial_{k}N:=S_{k}$.

Let $\Omega$ denote the the lift of $N$ in the universal cover of $M$ (which
is identified with $\overline{{\mathbb{H}}^{3}}$). Up to isometries, we always
assume that the center $O$ of ${\mathbb{H}}^{3}$ is contained in the interior
of the convex hull of the limit set $\Lambda$ of the Klein group of $M$ (which
descends to the convex core $C_{M}$ of $M$), and is thus contained in the
interior of $\Omega$. Moreover, we assume that the family
$(dev_{t})_{t\in[0,\epsilon]}$ of developing maps associated to the first
order deformation $\dot{g}$ (see Section 2.3.1) is normalized such that the
deformation section vanishes at $O$ (namely, $\tau(O)=\sigma(O)=0$). This will
help to simplify some estimates we need later (see Lemma 3.18 for instance).

We denote by $\partial_{i}\Omega\subset\partial\overline{{\mathbb{H}}^{3}}$
($i\in\mathcal{I}$) the union of all the lifts in
$\partial\overline{{\mathbb{H}}^{3}}$ of the conformal boundary component
$\partial_{i}M$, by $\partial_{j}\Omega\subset{\mathbb{H}}^{3}$
($j\in\mathcal{J}$) the union of all the lifts in ${\mathbb{H}}^{3}$ of
$S_{j}$, and by $\partial_{k}\Omega\subset{\mathbb{H}}^{3}$
($k\in\mathcal{K}$) the union of all the lifts in ${\mathbb{H}}^{3}$ of
$S_{k}$. Let $(\partial_{k}\Omega)^{*}$ denote the dual surface in ${{\rm
d}\mathbb{S}}^{3}$ of the strictly convex surface
$\partial_{k}\Omega\subset{\mathbb{H}}^{3}$ and let $\Omega^{*}$ denote the
convex domain in ${\mathbb{RP}}^{3}$, sharing the same boundary with $\Omega$,
except with $\partial_{k}\Omega$ replaced by $(\partial_{k}\Omega)^{*}$.

####  3.1.2. From ${\mathbb{H}}^{3}$ to ${\mathbb{R}}^{3}$

Recall that $\iota$ is an inclusion in ${\mathbb{R}}^{3}$ that sends the
projective model ${\mathbb{H}}^{3}$ to the unit open ball $\mathbb{D}^{3}$
(see Section 2.5). We denote $\bar{\Omega}:=\iota(\Omega)$,
$\partial_{l}\bar{\Omega}=\iota(\partial_{l}\Omega)$ ($1\leq l\leq n$) and
$\bar{\Lambda}=\iota(\Lambda)$. Note that the center $\bar{O}$ of
$\mathbb{D}^{3}$ is contained in the interior of $\bar{\Omega}$ (by the above
assumption) and

$\partial\bar{\Omega}=(\cup_{i\in\mathcal{I}}\partial_{i}\bar{\Omega})\cup(\cup_{j\in\mathcal{J}}\partial_{j}\bar{\Omega})\cup(\cup_{k\in\mathcal{K}}\partial_{k}\bar{\Omega})\cup\bar{\Lambda}.$

Let $\bar{\Omega}^{*}=\iota(\Omega^{*})$ and denote
$\partial_{k}\bar{\Omega}^{*}:=\iota((\partial_{k}\Omega)^{*})$,
$k\in\mathcal{K}$. Then

$\partial\bar{\Omega}^{*}=(\cup_{i\in\mathcal{I}}\partial_{i}\bar{\Omega})\cup(\cup_{j\in\mathcal{J}}\partial_{j}\bar{\Omega})\cup(\cup_{k\in\mathcal{K}}\partial_{k}\bar{\Omega}^{*})\cup\bar{\Lambda}.$

In the case that $M$ is quasifuchsian, the limit set $\bar{\Lambda}$ is a
Jordan curve and $\partial\bar{\Omega}\setminus\bar{\Lambda}$ has two
connected components. In the case that $M$ has $n>2$ boundary components,
$\partial\bar{\Omega}\setminus\bar{\Lambda}$ is a countably infinite union of
disjoint open disks, and for each $i\in\mathcal{I}$ (resp. $j\in\mathcal{J}$,
$k\in\mathcal{K}$), $\partial_{i}\bar{\Omega}$ (resp.
$\partial_{j}\bar{\Omega}$, $\partial_{k}\bar{\Omega}$ and
$\partial_{k}\bar{\Omega}^{*}$) has countably infinitely many connected
components.

####  3.1.3. The boundary behavior of the convex domains $\bar{\Omega}$ and
$\bar{\Omega}^{*}$.

The following describes the behavior of the principal curvatures of the image
in ${\mathbb{D}}^{3}\subset{\mathbb{R}}^{3}$ (resp.
${\mathbb{R}}^{3}\setminus\overline{{\mathbb{D}}^{3}}$) under $\iota$ of a
strictly convex surface in ${\mathbb{H}}^{3}$ (resp. a strictly convex
spacelike surface in ${{\rm d}\mathbb{S}}^{3}$), based on the fact that the
second fundamental form of the surface changes conformally (see [40, Lemmas
1.4 and 2.8]).

######  Lemma 3.2.

Let $S$ be a smooth surface in ${\mathbb{H}}^{3}$ or a spacelike smooth
surface in ${{\rm d}\mathbb{S}}^{3}$. If $S$ has principal curvatures in
$[k_{min},k_{max}]$ for some $k_{max}>k_{min}>0$. Then $\iota(S)$ is a convex
surface in ${\mathbb{R}}^{3}$, with principal curvatures in
$[k^{\prime}_{min},k^{\prime}_{max}]$ for some
$k^{\prime}_{max}>k^{\prime}_{min}>0$.

######  Remark 3.3.

It is shown by Sullivan [44] that $\bar{\Lambda}$ has Hausdorff dimension in
$[1,2)$. Moreover, Bishop has proved that the Hausdorff dimension and
Minkowski dimension are the same for the limit set of analytically finite
Klein groups [9].

######  Remark 3.4.

The surface $\partial\bar{\Omega}\setminus\bar{\Lambda}$ (resp.
$\partial\bar{\Omega}^{*}\setminus\bar{\Lambda}$) is smooth, while
$\partial\bar{\Omega}$ (resp. $\partial\bar{\Omega}^{*}$) is $C^{1,1}$ (see
e.g. [40, Corollaries 2.6 and 2.9]). In particular, each connected component
of $\partial_{i}\bar{\Omega}$ (resp. $\partial_{j}\bar{\Omega}$,
$\partial_{k}\bar{\Omega}$, $\partial_{k}\bar{\Omega}^{*}$) is contained in
$\partial{\mathbb{D}}^{3}$ (resp. tangent to $\partial{\mathbb{D}}^{3}$ along
its boundary).

Note that we do not claim that $\partial\bar{\Omega}$ (resp.
$\partial\bar{\Omega}^{*}$) is $C^{2}$, however the principal curvatures are
well-defined almost everywhere and we do claim that they are bounded:

######  Lemma 3.5.

$\partial\bar{\Omega}$ (resp. $\partial\bar{\Omega}^{*}$) is in the class
$D_{2,\infty}$, namely, it satisfies the following:

  1. (1)

The principal curvatures on $\partial\bar{\Omega}$ (resp.
$\partial\bar{\Omega}^{*}$) has positive lower and upper bounds almost
everywhere.

  2. (2)

$\partial\bar{\Omega}$ (resp. $\partial\bar{\Omega}^{*}$) can be locally
written as the graph of a function with bounded measurable second derivatives.

###### Proof.

Statement (1) follows from Lemma 3.2, Remarks 3.3 and 3.4, and the fact that
$\partial_{i}\Omega$ (resp. $\partial_{j}\Omega$, $\partial_{k}\Omega$ and
$(\partial_{k}\Omega)^{*}$) descends to a closed, strictly convex surface in
$M$, with principal curvatures bounded between two positive constants.
Statement (2) follows from Statement (1), the $C^{1,1}$-smoothness (see Remark
3.4) and strictly convexity of $\partial\bar{\Omega}$ (resp.
$\partial\bar{\Omega}^{*}$). ∎

For each $\bar{x}\in\partial\bar{\Omega}\setminus\bar{\Lambda}$ (resp.
$\bar{x}\in\partial\bar{\Omega}^{*}\setminus\bar{\Lambda}$), let
$\theta_{\bar{x}}$ denote the angle between the radial direction along
$\bar{O}\bar{x}$ at $\bar{x}$ and the outward-pointing normal direction to
$\partial\bar{\Omega}\setminus\bar{\Lambda}$ (resp.
$\partial\bar{\Omega}^{*}\setminus\bar{\Lambda}$) at $\bar{x}$ and let
$\delta_{\partial\bar{\Omega}}(\bar{x},\bar{\Lambda})$ (resp.
$\delta_{\partial\bar{\Omega}^{*}}(\bar{x},\bar{\Lambda})$ ) denote the
distance from the point $\bar{x}$ to the limit set $\bar{\Lambda}$ along
$\partial\bar{\Omega}$ (resp. $\partial\bar{\Omega}^{*}$).

Since $\partial_{i}\bar{\Omega}\subset\partial{\mathbb{D}}^{3}$, it is clear
that $\theta_{\bar{x}}=0$ for all $\bar{x}\in\partial_{i}\bar{\Omega}$
($i\in\mathcal{I}$). For the other
$\bar{x}\in\partial\bar{\Omega}\cap{\mathbb{D}}^{3}$ or
$\bar{x}\in\partial\bar{\Omega}^{*}\setminus\overline{{\mathbb{D}}^{3}}$, the
estimate of the angle $\theta_{\bar{x}}$ and the distances
$\delta_{\partial\bar{\Omega}}(\bar{x},\bar{\Lambda})$,
$\delta_{\partial\bar{\Omega}^{*}}(\bar{x},\bar{\Lambda})$ in [40, Lemmas 2.4
and 2.7] still work and we conclude the following:

######  Lemma 3.6.

There exists a constant $C>1$, such that

  1. (1)

For any $\bar{x}\in\partial\bar{\Omega}\cap{\mathbb{D}}^{3}$ with $\bar{x}$
close enough to $\bar{\Lambda}$, we have $\theta_{\bar{x}}\leq
C\sqrt{1-\|\bar{x}\|}$ and moreover,

$\frac{\sqrt{1-\|\bar{x}\|}}{C}\leq\delta_{\partial\bar{\Omega}}(\bar{x},\bar{\Lambda})\leq
C\sqrt{1-\|\bar{x}\|}.$

  2. (2)

For any
$\bar{x}\in\partial\bar{\Omega}^{*}\setminus\overline{{\mathbb{D}}^{3}}$ with
$\bar{x}$ close enough to $\bar{\Lambda}$, we have $\theta_{\bar{x}}\leq
C\sqrt{\|\bar{x}\|-1}$ and moreover,

$\frac{\sqrt{\|\bar{x}\|-1}}{C}\leq\delta_{\partial\bar{\Omega}^{*}}(\bar{x},\bar{\Lambda})\leq
C\sqrt{\|\bar{x}\|-1}.$

######  Remark 3.7.

Let $\tilde{S}^{i}_{t}$ be the lift in $\Omega$ of the equidistant surface
$S^{i}_{t}$ (which is a closed strictly convex surface) at distance $t$ from
$\partial_{i}C_{M}$ (see Subsection 3.2.2) and let
$\bar{S}^{i}_{t}:=\iota(\tilde{S}^{i}_{t})$. Note that for each $t\geq 1$, the
principal curvatures of $\bar{S}^{i}_{t}$ have a uniform positive lower and
upper bound for all $i\in\mathcal{I}$ (by Lemma 3.2 and Remark 3.15).
Statement (1) in Lemma 3.6 still holds for any
$\bar{x}\in\cup_{i\in\mathcal{I}}\bar{S}^{i}_{t}$ (with $\partial\bar{\Omega}$
replaced by $\cup_{i\in\mathcal{I}}\bar{S}^{i}_{t}$) close enough to
$\bar{\Lambda}$.

####  3.1.4. The Vekua lemma

Let $\bar{I}$ (resp. $\bar{II}$, $\bar{III}$) and $\bar{B}$ denote the first
(resp. second, third) fundamental form and the shape operator of an embedded
strictly convex surface $\bar{S}$ in ${\mathbb{R}}^{3}$. We define

$\bar{B}(\bar{X}):=-\bar{\nabla}_{\bar{X}}\bar{n},\quad\quad\bar{II}(\bar{X},\bar{Y})=\bar{I}(\bar{B}(\bar{X}),\bar{Y}),\quad\quad\bar{III}(\bar{X},\bar{Y})=\bar{I}(\bar{B}(\bar{X}),\bar{B}(\bar{Y})),$

for $\bar{X},\bar{Y}\in T\bar{S}$, where $\bar{n}$ is the unit normal vector
field on $\bar{S}$ oriented such that the eigenvalues of $\bar{B}$ are
positive, and $\bar{\nabla}$ is the Levi-Civita connection of
${\mathbb{R}}^{3}$.

Let $\bar{J}$ be the complex structure associated to $\bar{II}$ and let
$\nabla^{\bar{III}}$ denote the Levi-Civita connection for $\bar{III}$. By
computation (see [41, Proposition 3.1]),

$\nabla^{\bar{III}}_{\bar{X}}\bar{Y}=\bar{B}^{-1}\nabla^{\bar{I}}_{\bar{X}}(\bar{B}\bar{Y}),$

where $\bar{X},\bar{Y}\in T\bar{S}$ and $\nabla^{\bar{I}}$ is the Levi-Civita
connection for $\bar{I}$.

Let $\bar{\partial}_{\bar{III}}$ be the operator that takes a vector field
$\bar{W}$ over $\bar{S}$ to a section $\bar{\partial}_{\bar{III}}\bar{W}$ of
the bundle of one-forms with values in $T\bar{S}$, defined by

$(\bar{\partial}_{\bar{III}}\bar{W})(\bar{X}):={\nabla}^{\bar{III}}_{\bar{X}}\bar{W}+\bar{J}{\nabla}^{\bar{III}}_{\bar{J}\bar{X}}\bar{W},$

for all $\bar{X}\in T\bar{S}$. It follows from a direct computation that

$(\bar{\partial}_{\bar{III}}\bar{W})(\bar{J}\bar{X})=-\bar{J}(\bar{\partial}_{\bar{III}}\bar{W})(\bar{X})~{}.$

Let $\bar{v}$ be a vector field on $\bar{S}$. The following equation will play
a central role, so we give it a specific name.

(E) $\bar{\partial}_{\bar{III}}(\bar{B}^{-1}\bar{v})=0~{}.$

It ensures that the Lie derivative of the induced metric $\bar{I}$ on
$\bar{S}\subset{\mathbb{R}}^{3}$ with respect to $\bar{v}$ is parallel to
$\bar{II}$ (see the proof of Lemma 3.8 below for more details).

The following states a crucial relation between an isometric deformation
vector field on an embedded surface $\bar{S}\subset{\mathbb{R}}^{3}$ and
equation (E).

######  Lemma 3.8.

Let $\bar{U}$ be an isometric deformation vector field on an embedded convex
surface $\bar{S}\subset{\mathbb{R}}^{3}$ (with the shape operator $\bar{B}$)
and let $\bar{V}$ be its component tangent to $\bar{S}$. Then $\bar{V}$
satisfies equation $(E)$. Conversely, let $\bar{V}$ be a solution of equation
(E), there exists a unique isometric deformation vector field of $\bar{S}$
whose component tangent to $\bar{S}$ is $\bar{V}$.

###### Proof.

The first statement was shown in [41, Proposition 3.8] – note that we refer
here to the arxiv version (v4) rather than to the published version.

It suffices to show the second statement. Assume that $\bar{V}$ is a solution
of equation (E), then by a direct computation (see also [41, Proposition
3.8]), we have

(8)
$\begin{split}\bar{II}(\partial_{\bar{III}}(\bar{B}^{-1}\bar{V})\bar{X},\bar{X})=\bar{I}(\nabla^{\bar{I}}_{\bar{X}}\bar{V},\bar{X})-\bar{I}(\nabla^{\bar{I}}_{\bar{J}\bar{X}}\bar{V},\bar{J}\bar{X})=0,\\\
\bar{II}(\partial_{\bar{III}}(\bar{B}^{-1}\bar{V})\bar{X},\bar{J}\bar{X})=\bar{I}(\nabla^{\bar{I}}_{\bar{X}}\bar{V},\bar{J}\bar{X})+\bar{I}(\nabla^{\bar{I}}_{\bar{J}\bar{X}}\bar{V},\bar{X})=0.\\\
\end{split}$

Since $\bar{J}$ is the complex structure associated to $\bar{II}$, then
$\bar{II}(\bar{X},\bar{J}\bar{X})=0$. Note that

$(L_{\bar{V}}\bar{I})(\bar{X},\bar{Y})=\bar{I}(\nabla^{\bar{I}}_{\bar{X}}\bar{V},\bar{Y})+\bar{I}(\bar{X},\nabla^{\bar{I}}_{\bar{Y}}\bar{V}).$

Assume that $\bar{X}\not=0$. When $\bar{Y}=\bar{J}\bar{X}$, combined with (8),
then

$(L_{\bar{V}}\bar{I})(\bar{X},\bar{J}\bar{X})=0=\bar{II}(\bar{X},\bar{J}\bar{X})=2\frac{\bar{I}(\nabla^{\bar{I}}_{\bar{X}}\bar{V},\bar{X})}{\bar{II}(\bar{X},\bar{X})}\bar{II}(\bar{X},\bar{J}\bar{X}).$

When $\bar{Y}=\bar{X}$, using (8) again, then

$(L_{\bar{V}}\bar{I})(\bar{X},\bar{X})=2\bar{I}(\nabla^{\bar{I}}_{\bar{X}}\bar{V},\bar{X})=2\frac{\bar{I}(\nabla^{\bar{I}}_{\bar{X}}\bar{V},\bar{X})}{\bar{II}(\bar{X},\bar{X})}\bar{II}(\bar{X},\bar{X}).$

Define
$f(\bar{x}):=\bar{I}(\nabla^{\bar{I}}_{\bar{Z}_{\bar{x}}}\bar{V},\bar{Z}_{\bar{x}})/\bar{I}I(\bar{Z}_{\bar{x}},\bar{Z}_{\bar{x}})$,
where $\bar{Z}_{\bar{x}}\in T_{\bar{x}}\bar{S}$ is an arbitrary non-zero
vector. By (8) and the fact that
$\bar{I}I(\bar{X},\bar{X})=\bar{I}I(\bar{J}\bar{X},\bar{J}\bar{X})$, it can be
directly checked that $f$ depends only on the base point of
$\bar{Z}_{\bar{x}}$ and is thus a well-defined function on $\bar{S}$.
Therefore,

(9) $(L_{\bar{V}}\bar{I})(\bar{X},\bar{Y})=2f\bar{II}(\bar{X},\bar{Y}).$

Let $\bar{U}:=\bar{V}+f\bar{n}$, where $\bar{n}$ is the unit normal vector
field on $\bar{S}$ defined above. Then by (9),

$(L_{\bar{U}}\bar{I})(\bar{X},\bar{Y})=(L_{\bar{V}+f\bar{n}}\bar{I})(\bar{X},\bar{Y})=(L_{\bar{V}}\bar{I})(\bar{X},\bar{Y})-2f\bar{II}(\bar{X},\bar{Y})=0.$

As a consequence, $\bar{V}$ is the tangent component of the unique isometric
vector field $\bar{U}$ on $\bar{S}$. ∎

######  Remark 3.9.

Note that in Lemma 3.8, the surface $\bar{S}$ can be locally written as the
graph of a function say $u=f(x,y)$. If $\bar{U}$ is an isometric deformation
vector field on $\bar{S}$, then the tangent component $\bar{V}$ of $\bar{U}$
satisfies the generalized Beltrami equation:

(10) $w_{\bar{z}}-\mu(z)(w_{z}+\bar{w}_{\bar{z}})=0,$

where $w(z)$ is the local expression of the tangent component $\bar{V}$ in a
local chart $z=x+iy$, and $\mu(z)=(1/2)(f_{\bar{z}\bar{z}}/f_{z\bar{z}})$ with
$\|\mu\|_{\infty}<1$ (see e.g. [47, V.2.2, Eq (2.9) p.397]). By Lemma 3.8, the
tangent component $\bar{V}$ satisfies equation (E) if and only if it satisfies
equation (10).

A solution of equation (E) is said to be _trivial_ if it is the tangent
component (restricted to $\bar{S}$) of a global Killing field of
${\mathbb{R}}^{3}$, which corresponds to a trivial infinitesimal deformation
of $\bar{S}$ (see [41, Section 5] and [47, V.1, p. 394]).

######  Definition 3.10.

Assume that a vector field $\bar{v}$ on $\bar{S}$ has the expression
$\bar{v}=f(z)\partial_{z}$ in local conformal coordinates $z=x+iy$. We say
$\bar{v}$ is in the class $D_{1,p}(T\bar{S})$ (where $p\geq 1$) if the
generalized derivative of $f$ is in $L^{p}(\bar{S})$, in terms of the
integration of the product of $f$ and the derivative (with respect to $z$ or
$\bar{z}$) of any $C^{1}$ function with compact support.

We now introduce a result of Vekua [47, V.2.2, p. 393-402], which is crucial
for the proof of Proposition 3.1.

######  Lemma 3.11 (Vekua).

Let $\bar{\Omega}\subset{\mathbb{R}}^{3}$ be a bounded convex subset such that
$\partial\bar{\Omega}$ is in $D_{2,\infty}$. Suppose that $\bar{v}$ is a
solution of (E) which is in $D_{1,p}(T\partial\bar{\Omega})$. Then $\bar{v}$
is trivial.

We will use this lemma through [40, Lemma 5.5], which we recall here in
slightly different notations.

######  Lemma 3.12.

Let $\bar{\Omega}\subset{\mathbb{R}}^{3}$ be a bounded convex subset. Let
$\bar{\Lambda}\subset\partial\bar{\Omega}$ be a closed subset of Hausdorff
dimension $D_{\bar{\Lambda}}\in(1,2)$. Let $p>2$ and let
$f:\partial\bar{\Omega}\rightarrow{\mathbb{R}}$ be a function such that

  * •

$f$ is everywhere $C^{\alpha}$ for some $\alpha\in(D_{\bar{\Lambda}}-1,1)$.

  * •

$f$ is smooth outside $\bar{\Lambda}$.

  * •

$\int_{\partial\bar{\Omega}\setminus\bar{\Lambda}}\|df\|^{p}da<\infty$, where
$da$ is the area form of the induced metric on
$\partial\bar{\Omega}\subset{\mathbb{R}}^{3}$.

Then $df$, considered as an $L^{p}$ section (defined on
$\partial\bar{\Omega}\setminus\bar{\Lambda}$ as the usual derivative of $f$,
and as $0$ on $\bar{\Lambda}$), is the generalized derivative of $f$ on
$\partial\bar{\Omega}$, and $f$ is in $D_{1,p}(T\partial\bar{\Omega})$.

###  3.2. Estimates on $\omega$

In this subsection, we aim to show the following statement:

######  Lemma 3.13.

Let $\dot{g}$ be a first-order deformation of the hyperbolic structure
$g\in{\mathcal{CC}}(M)$ which leaves invariant the conformal metric at
infinity $c_{i}$ on $\partial_{i}M=\partial_{i}N$, $i\in\mathcal{I}$, the
induced metric $h_{j}$ on $S_{j}=\partial_{j}N$, $j\in\mathcal{J}$, and the
third fundamental form $h_{k}$ on $S_{k}=\partial_{k}N$, $k\in\mathcal{K}$, at
first order. There exists a representative $\omega$ of $\dot{g}$, which is a
closed one-form in $\Omega^{1}(M,E_{M})$ such that:

  1. (i)

$\omega$ vanishes along the geodesic rays starting orthogonally from a support
plane to $\partial C_{M}$, at all points at distance at least $1$ from
$C_{M}$.

  2. (ii)

for each $x\in N$, $\|\omega\|_{x}\leq C\exp(-2d(x,C_{M}))$ for a constant
$C>0$ depending on $\omega$.

####  3.2.1. The deformation one-form on $\partial_{i}M$

Let $\sigma_{i}$ ($i\in\mathcal{I}$) denote the complex projective structure
on $\partial_{i}M$. By assumption, the first-order deformation $\dot{g}$ of
$M$ preserves the conformal structure at infinity on $\partial_{i}M$, so the
deformation vector field $u$ restricted to $\partial_{i}\Omega$, say $u^{i}$,
is holomorphic and has the form

$u^{i}(z)=f_{i}(z)\partial_{z}~{},$

where $f_{i}(z)$ is a holomorphic function in an affine chart of $\sigma_{i}$.
The (infinitesimal) Schwarzian derivative of $u^{i}$ has the following
expression (see e.g. [13, Section 2]):

$\mathcal{S}(u^{i})(z)=f^{\prime\prime\prime}_{i}(z)dz^{2}~{}.$

One can check that $\mathcal{S}(u^{i})(\gamma\cdot z)=\mathcal{S}(u^{i})(z)$
for all $\gamma\in PSL_{2}({\mathbb{C}})$, so it determines a quadratic
differential on $\partial_{i}M$. Moreover, $\mathcal{S}(u^{i})\equiv 0$ if and
only if $f_{i}(z)$ is a quadratic polynomial.

Note that the canonical lift $s_{u^{i}}$ of $u^{i}$ has local expression in
the form of (5). Define $\omega^{i}:=d^{D}s_{u^{i}}$. Then for each
$z_{0}\in\partial_{i}\Omega$ and $U=u(z)\partial_{z}\in
T_{z_{0}}{\mathbb{C}}$,

(11)
$\begin{split}\omega^{i}(U)(z_{0})=(d^{D}_{U}s_{u^{i}})(z_{0})&=\big{(}u(z)\partial_{z}\big{)}|_{z_{0}}\big{(}(s_{u^{i}})(z)\big{)}\\\
&=u(z_{0})\partial_{z_{0}}\big{(}f_{i}(z_{0})+f^{\prime}_{i}(z_{0})(z-z_{0})+\frac{f^{\prime\prime}_{i}(z_{0})}{2}(z-z_{0})^{2}\big{)}\partial_{z}\\\
&=\left(\frac{1}{2}u(z_{0})f^{\prime\prime\prime}_{i}(z_{0})(z-z_{0})^{2}\right)\partial_{z}~{}.\end{split}$

As a result, $(d^{D}{s_{u^{i}}})$ takes values in the vector space of
homogeneous polynomials of degree 2 (see also [30, Lemma 4.8]).

####  3.2.2. The equidistant foliations of hyperbolic ends

It is well-known that $M\setminus C_{M}$ has $n$ connected components, each
called a _hyperbolic end_ of $M$, denoted by ${\mathcal{E}}^{l}$ ($1\leq l\leq
n$). The connected component of the boundary of $C_{M}$ as a pleated boundary
surface of ${\mathcal{E}}^{l}$ is denoted by $\partial_{l}C_{M}$. Moreover,
each ${\mathcal{E}}^{l}$ is foliated by a family of equidistant surfaces
$(S^{l}_{t})_{t>0}$ lying at a distance $t$ from
$S^{l}_{0}:=\partial_{l}C_{M}$. Since $\partial C_{M}$ is convex, for each
point $p\in{\mathcal{E}}^{l}$, there is a unique geodesic ray (say $r_{q}$) in
${\mathcal{E}}^{l}$ starting from $q\in\partial_{l}C_{M}$ and orthogonal to a
support plane to $\partial_{l}C_{M}$ at $q$ and going through $p$. We define
$G:M\setminus C_{M}\rightarrow\partial_{\infty}M$ to be the hyperbolic Gauss
map that takes each point $p\in{\mathcal{E}}^{l}$ to the endpoint at infinity
of the geodesic ray $r_{q}$. In particular, the union
$\cup_{q\in\partial_{l}C_{M}}r_{q}$ of these “vertical” geodesic rays forms
another foliation of ${\mathcal{E}}^{l}$.

Let $\tilde{C}_{M}$ (resp. $\tilde{{\mathcal{E}}}^{l}$, $\tilde{S}^{l}_{t}$)
denote the lift in ${\mathbb{H}}^{3}$ of $C_{M}$ (resp. ${\mathcal{E}}^{l}$,
$S^{l}_{t}$). Let
$\tilde{G}:{\mathbb{H}}^{3}\setminus\tilde{C}_{M}\rightarrow{\mathbb{C}P}^{1}\setminus\Lambda$
be the lifting map of the Gauss map $G$.

####  3.2.3. The metric at infinity associated to an equidistant foliation of
hyperbolic ends

There are several ways to define a conformal metric at infinity on the
boundary $\partial_{l}M$ of a hyperbolic end ${\mathcal{E}}^{l}$, by using the
equidistant foliation of ${\mathcal{E}}^{l}$ (see e.g. [30, Section 3, 4]).
Here we consider the metric at infinity on $\partial_{i}M$
($i\in\mathcal{I}$), defined as

(12) $g_{i,\infty}:=4e^{-2t}G_{*}I^{*}_{t}~{},$

where $I^{*}_{t}=I_{t}+2II_{t}+III_{t}$ is the “horospherical metric” on an
equidistant surface $S^{i}_{t}$ ($t\geq 1$), with first, second and third
fundamental forms on $S^{i}_{t}$ denoted by $I_{t},II_{t},III_{t}$. We stress
that the metric $g_{i,\infty}$ is independent of the choice of $t$, and
moreover, its conformal structure is the same as that underlies the
${\mathbb{C}P}^{1}$ structure on $\partial_{i}M$ (see [39] and [10, Section
3.3] for more details). (This metric $g_{i,\infty}$ is then the Thurston
metric at infinity.)

Let $x_{t}\in\tilde{S}^{i}_{t}$ ($t\geq 1$). Let $z_{0}=\tilde{G}(x_{t})$ and
$z_{1}$ be the other endpoint at infinity of the complete geodesic passing
through $x_{t}$ and going towards $z_{0}$ on one side. We choose an affine
chart $z$ near $z_{0}$ such that $z_{0}$ is identified with $0$, while $z_{1}$
is identified with $\infty$ in the Poincaré upper half-space model
${\mathbb{U}}^{3}:=\\{(z,s):z\in{\mathbb{C}},s>0\\}$, and moreover, the metric
$\tilde{g}_{i,\infty}$, as the lift of $g_{i,\infty}$ to $\partial_{i}\Omega$,
has the following expression on $T_{z_{0}}(\partial_{i}\Omega)\subset
T_{z_{0}}{\mathbb{C}P}^{1}$:

(13) $\tilde{g}_{i,\infty}:=4e^{-2t}\tilde{G}_{*}\tilde{I}^{*}_{t}=|dz|^{2},$

where $\tilde{I}^{*}_{t}$ is the lifting metric on $\tilde{S}^{i}_{t}$ of
$I^{*}_{t}$.

Note that the deformation one form on $\partial_{i}\Omega$ takes values in the
projective fields given by homogeneous polynomials of degree 2, see (11). We
now consider the extension of such projective fields to Killing fields in
${\mathbb{H}}^{3}$, which will be used later in the proof of Lemma 3.13.

######  Lemma 3.14.

Let $x_{t}\in\tilde{S}^{i}_{t}$ ($t\geq 1$) and let $z$ be an affine chart
near $z_{0}$ as chosen above. Let $P(z)=\lambda(z-z_{0})^{2}\partial_{z}$
($\lambda\in{\mathbb{C}}$ and $\lambda\not=0$) be a projective vector field on
${\mathbb{C}P}^{1}\setminus\\{z_{1}\\}$. Then the norm of the Killing field
extending $P(z)$, at $x_{t}$, is $4|\lambda|e^{-t}$. In particular, the norm
decreases to zero as $x_{t}$ tends to $z_{0}$ along the geodesic ray
$r_{x_{0}}$ orthogonal to $\tilde{S}^{i}_{0}$ at $x_{0}$.

###### Proof.

We consider the model ${\mathbb{U}}^{3}:=\\{(z,s):z\in{\mathbb{C}},~{}s>0\\}$,
with the metric $(|dz|^{2}+ds^{2})/s^{2}$. Let
$\big{(}\phi_{t}(z)\big{)}_{t\in[0,\epsilon]}$ be a family of Möbius
transformations generating $P(z)$, with $\phi_{0}(z)=\rm{Id}$ and

$\phi_{t}(z):=\frac{a(t)z+b(t)}{c(t)z+d(t)},$

where $a(t),b(t),c(t),d(t)\in{\mathbb{C}}$ and $a(t)d(t)-b(t)c(t)=1$. A
computation shows that

$\frac{\partial}{\partial
t}|_{t=0}\phi_{t}(z)=-c^{\prime}(0)z^{2}+(a^{\prime}(0)-d^{\prime}(0))z+b^{\prime}(0).$

Therefore,

(14) $-c^{\prime}(0)=\lambda,\quad
a^{\prime}(0)-d^{\prime}(0)=b^{\prime}(0)=0.$

Moreover, $a(t)d(t)-b(t)c(t)\equiv 1$. Differentiating on both sides of this
identity, we obtain $a^{\prime}(0)+d^{\prime}(0)=0$. Combined with (14), we
have that $a^{\prime}(0)=d^{\prime}(0)=0$.

Now we construct a Killing field on ${\mathbb{U}}^{3}$ whose extension to
${\mathbb{C}P}^{1}\setminus\\{z_{1}\\}$ is $P(z)$. Note that if $c(t)\not=0$,
there is a canonical way to extend $\phi_{t}(z)$ to an isometry on
$\mathbb{U}^{3}$ (see [28, Chapter 1.1]) by

(15)
$\tilde{\phi}_{t}(z,s):=\Big{(}-\frac{\overline{z+\big{(}d(t)/c(t)\big{)}}}{c^{2}(t)\big{(}|z+(d(t)/c(t))|^{2}+s^{2}\big{)}}+\frac{a(t)}{c(t)},\,\,\frac{s}{|c(t)|^{2}\big{(}|z+\big{(}d(t)/c(t)\big{)}|^{2}+s^{2}\big{)}}\Big{)}.$

Taking the derivative of (15) at $t=0$, the desired Killing field, say
$\kappa$, on $\mathbb{U}^{3}$ is

$\frac{\partial}{\partial t}|_{t=0}\tilde{\phi}_{t}(z,s)=\big{(}\lambda
z^{2}-\overline{\lambda}s^{2},\,\,2s\cdot{\rm Re}(\lambda z)\big{)}.$

This is a parabolic Killing field that vanishes at $0$ and preserves
(globally) the horospheres centered at $0$. Using the orthonormal frame
$(e_{1},e_{2},e_{3}):=(s\partial_{x},s\partial_{y},s\partial_{s})$ of
$T\mathbb{U}^{3}$ and by a direct computation, we have

$\begin{split}\nabla_{e_{1}}e_{1}&=e_{3},\quad\nabla_{e_{2}}e_{1}=0,\quad\nabla_{e_{3}}e_{1}=0,\\\
\nabla_{e_{1}}e_{2}&=0,\quad\nabla_{e_{2}}e_{2}=e_{3},\quad\nabla_{e_{3}}e_{2}=0,\\\
\nabla_{e_{1}}e_{3}&=-e_{1},\quad\nabla_{e_{2}}e_{3}=-e_{2},\quad\nabla_{e_{3}}e_{3}=0.\\\
\end{split}$

Combined with the formula (1), we obtain

$\frac{1}{2}{\rm curl}(\kappa)=\big{(}-i\lambda
z^{2}-i\overline{\lambda}s^{2},\,\,2s\cdot{\rm Im}(\lambda z)\big{)}.$

Thus the norm of $\kappa$ at the point $x_{t}=(0,s(x_{t}))$ is

$\|\kappa\|_{x_{t}}=\|\tau(x_{t})\|_{\mathbb{U}^{3}}+\|\sigma(x_{t})\|_{\mathbb{U}^{3}}=\|(-\overline{\lambda}s^{2}(x_{t}),0)\|_{\mathbb{U}^{3}}+\|((-i\overline{\lambda}s^{2}(x_{t}),0)\|_{\mathbb{U}^{3}}=2|\lambda|s(x_{t}),$

which decreases to zero as $x_{t}$ tends to $z_{0}$ along the geodesic ray
orthogonal to $\tilde{S}^{i}_{0}$ at $x_{0}$ (note that $s(x_{t})$ tends to
zero as $t\rightarrow\infty$).

By (13), under the chosen affine chart $z$ near $z_{0}$,

$\tilde{G}_{*}(\tilde{I}^{*}_{t}|_{T_{x_{t}}\tilde{S}^{i}_{t}})=\frac{1}{4}(e^{2t}|dz|^{2})|_{T_{z_{0}}(\partial_{i}\Omega)}.$

Note that the coordinate $s(x_{t})$ is uniquely determined by the above choice
of affine chart $z$. Since the horospherical metric $\tilde{I}_{t}^{*}$ on
$T_{x_{t}}\tilde{S}^{i}_{t}$ is equal to the metric on the tangent space of
the horosphere $\\{s=s(x_{t})\\}$ at $x_{t}$, we obtain

$\frac{1}{4}e^{2t}|dz|^{2}=\frac{1}{s(x_{t})^{2}}|dz|^{2}~{}.$

It follows that $s(x_{t})=2e^{-t}$.

As a consequence, the norm of the Killing field extending $P(z)$, at $x_{t}$,
is equal to $4|\lambda|e^{-t}$. The lemma follows. ∎

######  Remark 3.15.

Let $G_{t}:S^{i}_{t}\rightarrow\partial_{i}M$ ($t\geq 1$, $i\in\mathcal{I}$)
denote the hyperbolic Gauss map. Since $S^{i}_{1}$ is strictly convex, by a
direct computation, the principal curvatures $\lambda_{t}(p)$, $\mu_{t}(p)$ of
the equidistant surface $S^{i}_{t}$ ($t\geq 1$) are respectively

$\lambda_{t}(p)=\frac{\lambda_{1}(p)+\tanh t}{1+\lambda_{1}(p)\tanh
t},\quad\quad\mu_{t}(p)=\frac{\mu_{1}(p)+\tanh t}{1+\mu_{1}(p)\tanh t}~{}.$

Therefore, the principal curvatures of the closed surface $S^{i}_{t}$ have a
uniform positive upper and lower bound (and thus the second and third
fundamental forms $II_{t}$ and $III_{t}$ are uniformly bounded) for all $t\geq
1$, say $k^{i}_{\rm max}$ and $k^{i}_{\rm min}$. Recall that the horospherical
metric on $S^{i}_{t}$ is $I^{*}_{t}=I_{t}+2II_{t}+III_{t}$. As a result, on
$T_{p}S^{i}_{t}$,

$I_{t}\leq I_{t}^{*}\leq(1+k)^{2}I_{t},$

where $k=\max_{i\in\mathcal{I}}\\{k^{i}_{\rm max}\\}$.

####  3.2.4. Proof of Lemma 3.13.

We are now ready to give the proof.

###### Proof of Lemma 3.13.

Let $(g_{t})_{t\in[0,\varepsilon]}$ be a one-parameter deformation of
$g\in{\mathcal{CC}}(M)$ with derivative at $t=0$ corresponding to $\dot{g}$
(up to a trivial deformation). We now construct the desired representative
one-form $\omega$ by the following steps:

  1. Step 1

Let $u_{0}$ be a deformation (automorphic) vector field on $\Omega$ associated
to $\dot{g}$, whose restriction to $\partial_{i}\Omega$ is a holomorphic
(automorphic) vector field, denoted by $u^{i}_{0}$. Let $s_{0}$ denote the
(automorphic) deformation section associated to $u_{0}$ and in particular, let
$s^{i}_{0}$ denote the deformation section associated to $u_{0}^{i}$ (see
Section 2.3). For each point $y\in\tilde{{\mathcal{E}}}^{i}$
($i\in\mathcal{I}$), there is a unique point $x\in\tilde{S}^{i}_{0}$ such that
$y$ lies in the geodesic ray $r_{x}$ orthogonal to a support plane to
$\tilde{S}^{i}_{0}$ at $x$. Let $z\in\partial_{i}\Omega$ be the endpoint at
infinity of $r_{x}$ and let $(s^{i}_{0})_{z}$ be the projective vector field
on $\partial_{i}\Omega$, which is the value in $sl_{2}({\mathbb{C}})$ at $z$
of $s^{i}_{0}$. There is a unique Killing field on $\Omega$, denoted by
$\kappa^{i}_{x}$, whose extension to $\partial_{i}\Omega$ is
$(s^{i}_{0})_{z}$. For each $y\in r_{x}$, we associate to $y$ the same Killing
field $\kappa^{i}_{x}$. Then we obtain a section $\kappa$ over
$\cup_{i\in\mathcal{I}}\tilde{{\mathcal{E}}}^{i}$, which is constant along
$r_{x}$ for each $x\in\tilde{S}^{i}_{0}$. Besides, $\kappa$ is automorphic,
since each section $s^{i}_{0}$ is automorphic.

  2. Step 2

Let $\eta(t)$ be a smooth cut-off function which vanishes for
$t\leq\delta_{0}$ (where $\delta_{0}<1$ is taken small enough) and is equal to
1 for $t\geq 1$. Let $\chi_{E}$ be the characteristic function of
$E\subset\Omega$, which is equal to $1$ on $E$ and to $0$ on the complement of
$E$. Now we construct a new section, say $s$, over $\Omega$:

$s:=\big{(}1-\eta\circ
d(\cdot,C_{M})\cdot\chi_{\cup_{i\in\mathcal{I}}\tilde{{\mathcal{E}}}^{i}}\big{)}s_{0}+\big{(}\eta\circ
d(\cdot,C_{M})\cdot\chi_{\cup_{i\in\mathcal{I}}\tilde{{\mathcal{E}}}^{i}}\big{)}\kappa.$

Then $s$ coincides with $\kappa$ near $\partial_{i}\Omega$ for all
$i\in\mathcal{I}$ and coincides with $s_{0}$ near the convex subset
$\tilde{C}_{M}$. It is clear that $s$ is automorphic.

  3. Step 3

Let $\tilde{\omega}=d^{D}s$. Then $\tilde{\omega}$ is a one-form on $\Omega$,
which descends to a closed one-form say $\omega$, on $N$, which is
cohomologous to the one-form, say $\omega_{0}$, that descends from
$\tilde{\omega}_{0}:=d^{D}s_{0}$. Indeed,

$\tilde{\omega}-\tilde{\omega}_{0}=d^{D}(s-s_{0})=\big{(}-\eta\circ
d(\cdot,C_{M})\cdot\chi_{\cup_{i\in\mathcal{I}}\tilde{{\mathcal{E}}}^{i}}\big{)}d^{D}(s_{0}-\kappa),$

where $d^{D}(s_{0}-\kappa)$ is exact on
$\cup_{i\in\mathcal{I}}\tilde{{\mathcal{E}}}^{i}$.

By construction, $\omega$ vanishes along each geodesic ray starting from
$S^{i}_{1}$ orthogonally for all $i\in\mathcal{I}$, which implies Condition
(i). It remains to show $\omega$ also satisfies Condition (ii).

We first estimate the norm of $\omega$ on the subset $\cup_{t\geq 1}S^{i}_{t}$
of ${\mathcal{E}}^{i}$ ($i\in\mathcal{I}$). Let $p_{t}\in S^{i}_{t}$ and $X\in
T_{p_{t}}M$. By Condition (i), it suffices to consider $\omega(X)$ for $X\in
T_{p_{t}}S^{i}_{t}$. Denote $z_{0}:=G(p_{t})\in\partial_{i}M$ and
$U:=G_{*}X\in T_{z_{0}}(\partial_{i}M)$. By (12), the horospherical metric
$I^{*}_{t}$ satisfies

$G_{*}(I^{*}_{t}|_{T_{p_{t}}S^{i}_{t}}):=\frac{1}{4}e^{2t}g_{i,\infty}|_{T_{z_{0}}(\partial_{i}M)}~{}.$

We now choose an affine chart $z$ near $z_{0}$ such that
$g_{i,\infty}|_{T_{z_{0}}(\partial_{i}M)}=|dz|^{2}$, as in (13). In
particular,

(16)
$\|U\|_{g_{i,\infty}}=\|G_{*}X\|_{g_{i,\infty}}=\|X\|_{G^{*}g_{i,\infty}}=2e^{-t}\|X\|_{I^{*}_{t}}.$

Assume that $U=u(z)\partial_{z}$ in the above chosen affine chart $z$ and the
holomorphic deformation vector field $u_{0}^{i}$ on $\partial_{i}M$ is
expressed as $u_{0}^{i}(z)=f_{i}(z)\partial_{z}$, where $f_{i}(z)$ is a
holomorphic function. It is clear that $\omega=\omega_{0}$ on
$\cup_{i\in\mathcal{I}}\partial_{i}M$. Then by (11),

$\omega(U)=\omega_{0}(U)=d^{D}_{U}s_{0}=\frac{1}{2}u(z_{0})f^{\prime\prime\prime}_{i}(z_{0})(z-z_{0})^{2}\partial_{z}.$

Note that $\omega(X)$ is the Killing field extending the projective field
$\omega(U)$. By Lemma 3.14, we have

$\begin{split}\|\omega(X)\|_{p_{t}}&=4e^{-t}\cdot|u(z_{0})|\cdot\frac{1}{2}|f^{\prime\prime\prime}_{i}(z_{0})|\\\
&=2e^{-t}\cdot\|U\|_{g_{i,\infty}}\cdot|f^{\prime\prime\prime}_{i}(z_{0})|\\\
&=4e^{-2t}\cdot\|X\|_{I^{*}_{t}}\cdot|f^{\prime\prime\prime}_{i}(z_{0})|\\\
&\leq
4C_{0}e^{-2t}\cdot\|X\|_{I_{t}}\cdot|f^{\prime\prime\prime}_{i}(z_{0})|\\\
&\leq C_{1}e^{-2t}\cdot\|X\|_{I_{t}}.\end{split}$

The second equality follows from the fact that
$g_{i,\infty}|_{T_{z_{0}}(\partial_{i}M)}=|dz|^{2}$, the third one follows
from (16), while the first inequality follows from Remark 3.15, where
$C_{0}=1+k$. Since $f^{\prime\prime\prime}_{i}(z)dz^{2}$ is a quadratic
differential on $\partial_{i}\Omega$ and under the above affine chart (see
Section 3.2.1), $|f^{\prime\prime\prime}_{i}(z)|$ descends to a well-defined
smooth function on the closed surface $\partial_{i}M$ and is thus bounded. The
last inequality follows by taking a uniform bound for all
$|f^{\prime\prime\prime}_{i}(z)|$ over all $i\in\mathcal{I}$ (note that
$\mathcal{I}$ is a finite set). By Definition 2.1, we have
$\|\omega\|_{p_{t}}\leq C_{1}e^{-2t}$, where $t=d(\cdot,C_{M})$.

On the other hand, $\omega$ is uniformly bounded in the convex compact subset
$N_{0}:=N\setminus(\cup_{i\in\mathcal{I}}\cup_{t>1}S^{i}_{t})$ of $N$. Note
that the function $d(\cdot,C_{M})$ has a uniform upper bound on $N_{0}$. As a
consequence, there is a constant $C>0$ depending on $\omega$, such that

$\|\omega\|_{x}\leq C\exp(-2d(x,C_{M}))~{},$

for all $x\in N$ (compare with the first paragraph in the proof of Theorem 3.5
in [14] or the proof of Lemma 3.2, p. 106 of [30]). So $\omega$ also satisfies
Condition (ii). This concludes the proof of the lemma. ∎

For convenience, henceforth, we denote by $\omega$ the representative one-
forms (constructed in Lemma 3.13) on the submanifold $N\subset M$ and its
universal cover $\Omega$.

###  3.3. Estimates on $\bar{\omega}$

The following describes the images of the infinitesimal pure translation and
rotation under the map $\Psi$ (see [40, Lemma 3.7]).

######  Lemma 3.16.

Let $x\in(\Omega\cap{\mathbb{H}}^{3})\setminus\\{O\\}$ and
$\bar{x}:=\iota(x)\in(\bar{\Omega}\cap{\mathbb{D}}^{3})\setminus\\{\bar{O}\\}$.
Let $R$ (resp. $\bar{R}:=\Upsilon(R)$) denote the unit radial vector in
$T_{x}{\mathbb{H}}^{3}$ (resp. $T_{\bar{x}}{\mathbb{R}}^{3}$) pointing away
from the center. Let $v$ (resp. $\bar{v}:=(\cosh\rho)\Upsilon(v)$) denote a
unit lateral vector in $T_{x}{\mathbb{H}}^{3}$ (resp.
$T_{\bar{x}}{\mathbb{R}}^{3}$), where $\rho:=d_{{\mathbb{H}}^{3}}(x,O)$.

Set $r:=d_{{\mathbb{R}}^{3}}(\bar{x},\bar{O})$. Then

$\begin{split}\Psi(R,0)&=(\bar{R},0),\\\ \Psi(0,R)&=(0,\bar{R}),\\\
\Psi(v,0)&=(\sqrt{1-r^{2}}\,\bar{v},-\frac{r}{\sqrt{1-r^{2}}}\,\bar{R}\times\bar{v}),\\\
\Psi(0,v)&=(0,\frac{1}{\sqrt{1-r^{2}}}\,\bar{v}),\end{split}$

where $(R,0)$ represents the infinitesimal pure translation whose value at $x$
is $R$, while $(0,R)$ represents the infinitesimal pure rotation whose curl at
$x$ is $2R$. Similarly for the other pairs of tangent vectors.

Let $\omega=d^{D}s$ be the one-form on $\Omega$ constructed in the proof of
Lemma 3.13 and let $\bar{\omega}:=\Psi_{*}(\omega)$. We denote by
$\omega^{r}_{\tau}$ and $\omega^{\perp}_{\tau}$ (resp. $\omega^{r}_{\sigma}$
and $\omega^{\perp}_{\sigma}$) the radial and lateral components of the
translation (resp. rotation) part $\omega_{\tau}$ (resp. $\omega_{\sigma}$) of
$\omega$, and by $\bar{\omega}^{r}_{\bar{\tau}}$ and
$\bar{\omega}^{\perp}_{\bar{\tau}}$ (resp. $\bar{\omega}^{r}_{\bar{\sigma}}$
and $\bar{\omega}^{\perp}_{\bar{\sigma}}$) the radial and lateral components
of the translation (resp. rotation) part $\bar{\omega}_{\bar{\tau}}$ (resp.
$\bar{\omega}_{\bar{\sigma}}$) of $\bar{\omega}$. The following is the
estimates of their norms in the unit radial and lateral directions. The proof
is similar to Lemma 3.8 in [40]. We include it here for completeness. For
simplicity, we denote by $\|\cdot\|$ the Euclidean norm henceforth.

######  Lemma 3.17.

Let $x\in(\Omega\cap{\mathbb{H}}^{3})\setminus\\{O\\}$ and
$\bar{x}:=\iota(x)\in(\bar{\Omega}\cap{\mathbb{D}}^{3})\setminus\\{\bar{O}\\}$.
Let $R$, $\bar{R}$, $v$, $\bar{v}$, $\rho$ and $r$ be the same notations as in
Lemma 3.16. Then there is a constant $C>0$ depending only on $\omega$, such
that

$\begin{split}&\|\bar{\omega}^{r}_{\bar{\tau}}(\bar{R})\|\leq\frac{C}{1-r},\quad\quad\|\bar{\omega}^{r}_{\bar{\tau}}(\bar{v})\|\leq\frac{C}{\sqrt{1-r}},\\\
&\|\bar{\omega}^{r}_{\bar{\sigma}}(\bar{R})\|\leq\frac{C}{1-r},\quad\quad\|\bar{\omega}^{r}_{\bar{\sigma}}(\bar{v})\|\leq\frac{C}{\sqrt{1-r}},\\\
&\|\bar{\omega}^{\perp}_{\bar{\tau}}(\bar{R})\|\leq\frac{C}{\sqrt{1-r}},\quad\quad\|\bar{\omega}^{\perp}_{\bar{\tau}}(\bar{v})\|\leq
C,\\\
&\|\bar{\omega}^{\perp}_{\bar{\sigma}}(\bar{R})\|\leq\frac{2C}{(\sqrt{1-r})^{3}},\quad\quad\|\bar{\omega}^{\perp}_{\bar{\sigma}}(\bar{v})\|\leq\frac{2C}{1-r}.\end{split}$

###### Proof.

By the assumption and (7),
$\bar{\omega}(d\iota(X))=(\Psi_{*}\omega)(d\iota(X))=\Psi(\omega(X))$ for all
$X\in T\Omega$. In particular, $\bar{\omega}^{r}_{\bar{\tau}}(d\iota(X))$ is
equal to the radial component of the translation part, denoted by
$\big{(}\Psi(\omega(X))\big{)}^{r}_{\bar{\tau}}$, of $\Psi(\omega(X))$.
Similarly for $\bar{\omega}^{\perp}_{\bar{\tau}}$,
$\bar{\omega}^{r}_{\bar{\sigma}}$ and $\bar{\omega}^{\perp}_{\bar{\sigma}}$
acting on $d\iota(X)$. Note that

$\bar{R}=\Upsilon(R)=\cosh^{2}\rho\cdot
d\iota(R),\quad\quad\bar{v}=\cosh\rho\cdot\Upsilon(v)=\cosh\rho\cdot
d\iota(v),\quad\quad r=\tanh{\rho}.$

We divide the discussion into the following four cases:

Case 1: We consider the case of $\bar{\omega}^{r}_{\bar{\tau}}$. By the above
facts, we obtain

$\begin{split}\|\bar{\omega}^{r}_{\bar{\tau}}(\bar{R})\|&=\|\bar{\omega}^{r}_{\bar{\tau}}\big{(}\cosh^{2}\rho\cdot
d\iota(R)\,\big{)}\|=\cosh^{2}{\rho}\,\|\big{(}\Psi({\omega}(R)\big{)}^{r}_{\bar{\tau}}\|\\\
&=\cosh^{2}{\rho}\,\|\big{(}\Psi(\omega^{r}_{\tau}(R))\big{)}^{r}_{\bar{\tau}}\|=\cosh^{2}{\rho}\,\|\omega^{r}_{\tau}(R)\|\\\
&\leq\cosh^{2}{\rho}\,\|\omega_{\tau}\|_{x}\cdot\|R\|_{{\mathbb{H}}^{3}}\leq\frac{\|\omega\|_{x}}{1-r^{2}}\\\
&\leq\frac{C\exp(-2d(x,C_{M}))}{1-r^{2}}\leq\frac{C}{1-r}.\end{split}$

Here the first equality on the second line follows from Lemma 3.16, which
shows that the radial component of the translation part of $\Psi(\omega(R))$
only comes from the radial component of the translation part of $\omega(R)$.
The second equality on the second line follows from the first equality in
Lemma 3.16. The last line is a consequence of Lemma 3.13 and the fact that
$1-r^{2}\geq 1-r$. The estimate for
$\|\bar{\omega}^{r}_{\bar{\tau}}(\bar{v})\|$ follows in the same way by
replacing $\bar{R}$ (resp. $R$ and $\cosh^{2}\rho$) by $\bar{v}$ (resp. $v$
and $\cosh\rho$).

Case 2: For the estimates for $\bar{\omega}^{r}_{\bar{\sigma}}$, we have

$\begin{split}\|\bar{\omega}^{r}_{\bar{\sigma}}(\bar{R})\|&=\|\bar{\omega}^{r}_{\bar{\sigma}}\big{(}\cosh^{2}\rho\cdot
d\iota(R)\,\big{)}\|=\cosh^{2}{\rho}\,\|\big{(}\Psi({\omega}(R)\big{)}^{r}_{\bar{\sigma}}\|\\\
&=\cosh^{2}{\rho}\,\|\big{(}\Psi(\omega^{r}_{\sigma}(R))\big{)}^{r}_{\bar{\sigma}}\|=\cosh^{2}{\rho}\,\|\omega^{r}_{\sigma}(R)\|\\\
&\leq\cosh^{2}{\rho}\,\|\omega_{\sigma}\|_{x}\cdot\|R\|_{{\mathbb{H}}^{3}}\leq\frac{\|\omega\|_{x}}{1-r^{2}}\\\
&\leq\frac{C\exp(-2d(x,C_{M}))}{1-r^{2}}\leq\frac{C}{1-r}.\end{split}$

Here the first equality on the second line follows from Lemma 3.16, which
shows that the radial component of the rotation part of $\Psi(\omega(R))$ only
comes from the radial component of the rotation part of $\omega(R)$. The
second equality on the second line follows from the second equality in Lemma
3.16. The estimate for $\|\bar{\omega}^{r}_{\bar{\sigma}}(\bar{v})\|$ follows
in the same way by replacing $\bar{R}$ (resp. $R$ and $\cosh^{2}\rho$) by
$\bar{v}$ (resp. $v$ and $\cosh\rho$).

Case 3: Concerning the estimates for $\bar{\omega}^{\perp}_{\bar{\tau}}$, we
obtain

$\begin{split}\|\bar{\omega}^{\perp}_{\bar{\tau}}(\bar{R})\|&=\|\bar{\omega}^{\perp}_{\bar{\tau}}\big{(}\cosh^{2}\rho\cdot
d\iota(R)\,\big{)}\|=\cosh^{2}{\rho}\,\|\big{(}\Psi({\omega}(R)\big{)}^{\perp}_{\bar{\tau}}\|\\\
&=\cosh^{2}{\rho}\,\|\big{(}\Psi(\omega^{\perp}_{\tau}(R))\big{)}^{\perp}_{\bar{\tau}}\|=\cosh^{2}{\rho}\cdot\sqrt{1-r^{2}}\,\|\omega^{\perp}_{\tau}(R)\|\\\
&\leq\frac{1}{\sqrt{1-r^{2}}}\,\|\omega_{\tau}\|_{x}\cdot\|R\|_{{\mathbb{H}}^{3}}\leq\frac{\|\omega\|_{x}}{\sqrt{1-r^{2}}}\\\
&\leq\frac{C\exp(-2d(x,C_{M}))}{\sqrt{1-r^{2}}}\leq\frac{C}{\sqrt{1-r}}.\end{split}$

Here the first equality on the second line follows from Lemma 3.16, which
shows that the lateral component of the translation part of $\Psi(\omega(R))$
only comes from the lateral component of the translation part of $\omega(R)$.
The second equality on the second line follows from the third equality in
Lemma 3.16. The estimate for $\|\bar{\omega}^{\perp}_{\bar{\tau}}(\bar{v})\|$
follows in the same way by replacing $\bar{R}$ (resp. $R$ and $\cosh^{2}\rho$)
by $\bar{v}$ (resp. $v$ and $\cosh\rho$).

Case 4: As for the estimates for $\bar{\omega}^{\perp}_{\bar{\sigma}}$, we
have

$\begin{split}\|\bar{\omega}^{\perp}_{\bar{\sigma}}(\bar{R})\|&=\|\bar{\omega}^{\perp}_{\bar{\sigma}}\big{(}\cosh^{2}\rho\cdot
d\iota(R)\,\big{)}\|=\cosh^{2}{\rho}\,\|\big{(}\Psi({\omega}(R)\big{)}^{\perp}_{\bar{\sigma}}\|\\\
&=\cosh^{2}{\rho}\,\|\big{(}\Psi(\omega^{\perp}_{\tau}(R)+\omega^{\perp}_{\sigma}(R))\big{)}^{\perp}_{\bar{\sigma}}\|\\\
&\leq\cosh^{2}{\rho}\cdot\big{(}\frac{r}{\sqrt{1-r^{2}}}\|\omega^{\perp}_{\tau}(R)\|+\frac{1}{\sqrt{1-r^{2}}}\|\omega^{\perp}_{\sigma}(R)\|\big{)}\\\
&\leq\frac{1}{(\sqrt{1-r^{2}})^{3}}\big{(}\|\omega_{\tau}\|_{x}\cdot\|R\|_{{\mathbb{H}}^{3}}+\|\omega_{\sigma}\|_{x}\cdot\|R\|_{{\mathbb{H}}^{3}})\leq\frac{2\|\omega\|_{x}}{(\sqrt{1-r^{2}})^{3}}\\\
&\leq\frac{2C\exp(-2d(x,C_{M}))}{(\sqrt{1-r^{2}})^{3}}\leq\frac{2C}{(\sqrt{1-r})^{3}}.\end{split}$

Here the equality on the second line follows from Lemma 3.16, which shows that
the lateral component of the rotation part of $\Psi(\omega(R))$ comes from the
lateral components of both the translation and rotation parts of $\omega(R)$.
The first inequality follows from the triangle inequality and the last two
equalities in Lemma 3.16. The estimate for
$\|\bar{\omega}^{\perp}_{\bar{\sigma}}(\bar{v})\|$ follows in the same way by
replacing $\bar{R}$ (resp. $R$ and $\cosh^{2}\rho$) by $\bar{v}$ (resp. $v$
and $\cosh\rho$).

∎

Let $s$ be the deformation section defined in the proof of Lemma 3.13. The
following is an estimate on the radial and lateral components of the
translation and rotation components of the deformation section
$\bar{s}:=\Psi(s)$, which is identified with
$(\bar{\tau},\bar{\sigma})\in\Gamma(T{\mathbb{R}}^{3}\times
T{\mathbb{R}}^{3})$ (see also [40, Corollary 3.9]).

######  Lemma 3.18.

Let $x\in(\Omega\cap{\mathbb{H}}^{3})\setminus\\{O\\}$ and
$\bar{x}:=\iota(x)\in(\bar{\Omega}\cap{\mathbb{D}}^{3})\setminus\\{\bar{O}\\}$.
Let $\bar{\tau}^{r}$ (resp. $\bar{\tau}^{\perp}$) denote the radial (resp.
lateral) component of the vector field $\bar{\tau}$ on
$\bar{\Omega}\cap{\mathbb{D}}^{3}$. Let $\bar{\sigma}^{r}$ (resp.
$\bar{\sigma}^{\perp}$) denote the radial (resp. lateral) component of the
vector field $\bar{\sigma}$ on $\bar{\Omega}\cap{\mathbb{D}}^{3}$. Set
$r:=d_{{\mathbb{R}}^{3}}(\bar{x},\bar{O})$. Then at $\bar{x}$,

$\begin{split}&\|\bar{\tau}^{r}(\bar{x})\|\leq
C|\log(1-r)|,\quad\quad\|\bar{\tau}^{\perp}(\bar{x})\|\leq C,\\\
&\|\bar{\sigma}^{r}(\bar{x})\|\leq
C|\log(1-r)|,\quad\quad\|\bar{\sigma}^{\perp}(\bar{x})\|\leq\frac{C}{\sqrt{1-r}}.\\\
\end{split}$

###### Proof.

By assumption, the infinitesimal deformation $\dot{g}$ of $g$ is normalized
such that $\tau(O)=\sigma(O)=0$ (see Section 3.1.1). Let
$\gamma:[0,L]\rightarrow\Omega$ be the geodesic parameterized at speed 1, with
$\gamma(0)=O$ and $\gamma(L)=x$. Now we consider the image $\bar{s}$ under the
Pogerelov map $\Psi$ of the deformation section $s$. Denote
$\bar{\gamma}=\iota(\gamma)$, $\bar{O}=\iota(O)=\bar{\gamma}(0)$ and
$\bar{x}=\iota(x)=\bar{\gamma}(r)$, where $r=\tanh L$ and
$\bar{\gamma}:[0,r]\rightarrow\bar{\Omega}$ is the geodesic parameterized at
speed 1. By assumption, $\bar{\tau}(\bar{O})=\tau(O)=0$,
$\bar{\sigma}(\bar{O})=\sigma(O)=0$, then combined with the formula (6),

$\bar{\tau}(\bar{x})=\int^{r}_{0}\bar{\nabla}_{\bar{\gamma}^{\prime}(s)}\bar{\tau}~{}ds=\int^{r}_{0}\big{(}\bar{\omega}_{\bar{\tau}}(\bar{\gamma}^{\prime}(s))+\bar{\sigma}(\bar{\gamma}(s))\times\bar{\gamma}^{\prime}(s)\big{)}ds,\quad\quad\bar{\sigma}(\bar{x})=\int^{r}_{0}\bar{\nabla}_{\bar{\gamma}^{\prime}(s)}\bar{\sigma}~{}ds=\int^{r}_{0}\bar{\omega}_{\bar{\sigma}}(\bar{\gamma}^{\prime}(s))ds.$

In particular, the radial and lateral components of $\bar{\tau}$ and
$\bar{\sigma}$ satisfy that

$\begin{split}&\bar{\tau}^{r}(\bar{x})=\int^{r}_{0}\bar{\omega}^{r}_{\bar{\tau}}(\bar{\gamma}^{\prime}(s))ds,\quad\quad\bar{\tau}^{\perp}(\bar{x})=\int^{r}_{0}\big{(}\bar{\omega}^{\perp}_{\bar{\tau}}(\bar{\gamma}^{\prime}(s))+\bar{\sigma}^{\perp}(\bar{\gamma}(s))\times\bar{\gamma}^{\prime}(s)\big{)}ds,\\\
&\bar{\sigma}^{r}(\bar{x})=\int^{r}_{0}\bar{\omega}^{r}_{\bar{\sigma}}(\bar{\gamma}^{\prime}(s))ds,\quad\quad\bar{\sigma}^{\perp}(\bar{x})=\int^{r}_{0}\bar{\omega}^{\perp}_{\bar{\sigma}}(\bar{\gamma}^{\prime}(s))ds.\\\
\end{split}$

Combined with Lemma 3.17, we obtain

$\begin{split}&\|\bar{\sigma}^{r}(\bar{x})\|\leq\int^{r}_{0}\|\bar{\omega}^{r}_{\bar{\sigma}}(\bar{\gamma}^{\prime}(s))\|ds\leq\int^{r}_{0}\frac{C}{1-s}ds\leq
C|\log(1-r)|,\\\
&\|\bar{\sigma}^{\perp}(\bar{x})\|\leq\int^{r}_{0}\|\bar{\omega}^{\perp}_{\bar{\sigma}}(\bar{\gamma}^{\prime}(s))\|ds\leq\int^{r}_{0}\frac{C}{(\sqrt{1-s})^{3}}ds\leq\frac{C}{\sqrt{1-r}},\\\
&\|\bar{\tau}^{r}(\bar{x})\|\leq\int^{r}_{0}\|\bar{\omega}^{r}_{\bar{\tau}}(\bar{\gamma}^{\prime}(s))\|ds\leq\int^{r}_{0}\frac{C}{1-s}ds\leq
C|\log(1-r)|,\\\
&\|\bar{\tau}^{\perp}(\bar{x})\|\leq\int^{r}_{0}\big{(}\|\bar{\omega}^{\perp}_{\bar{\tau}}(\bar{\gamma}^{\prime}(s))\|+\|\bar{\sigma}^{\perp}(\bar{\gamma}(s))\|\big{)}ds\leq\int^{r}_{0}\left(\frac{C}{\sqrt{1-s}}+\frac{C}{\sqrt{1-s}}\right)ds\leq
C.\\\ \end{split}$

∎

###  3.4. A deformation vector field on
$\partial\bar{\Omega}\setminus\bar{\Lambda}$

Recall that $s$ is the deformation section constructed in Step 2 of the proof
of Lemma 3.13. Let $u$ denote the deformation vector field on
$\Omega\setminus\Lambda$ integrated from the representative one-form
$\omega=d^{D}s$. We denote by $\bar{u}$ the image under the infinitesimal
Pogorelov map $\Upsilon$ of $u$ on $\Omega\cap{\mathbb{H}}^{3}$, which is then
a deformation vector field on $\bar{\Omega}\cap{\mathbb{D}}^{3}$ and thus
defines a deformation vector field on $\partial_{j}\bar{\Omega}$ and
$\partial_{k}\bar{\Omega}$ for all $j\in\mathcal{J}$ and $k\in\mathcal{K}$
respectively. We now extend the definition of $\bar{u}$ to
$\partial_{i}\bar{\Omega}$ and $\partial_{k}\bar{\Omega}^{*}$ for all
$i\in\mathcal{I}$ and $k\in\mathcal{K}$.

For the extension to $\partial_{i}\bar{\Omega}$ of $\bar{u}$, note that by
construction, the deformation section $s$, which corresponds to the canonical
lift of $u$, is constant along each geodesic ray $r_{x}$ with
$x\in\partial_{i}\tilde{C}_{M}$. Then $\bar{s}:=\Psi(s)$ is constant along the
image $\iota(r_{x})$. By definition,

$\bar{u}(\bar{y})=\bar{s}_{\bar{y}}(\bar{y})=\bar{s}_{\bar{x}}(\bar{y}),$

for all $\bar{x}:=\iota(x)$, $\bar{y}:=\iota(y)$ with $y\in r_{x}$. So
$\bar{u}$ has a natural continuous extension to $\partial_{i}\bar{\Omega}$,
defined as $\bar{s}_{\bar{x}}(\bar{z})$, for each
$\bar{z}\in\partial_{i}\bar{\Omega}$.

For the extension to $\partial_{k}\bar{\Omega}^{*}$ of $\bar{u}$, we first
extend $u$ to $\partial_{k}\Omega^{*}$, through the duality between a couple
$(x,P)$ in ${\mathbb{H}}^{3}$ and a couple $(x^{*},P^{*})$ in ${{\rm
d}\mathbb{S}}^{3}$, where $P$ is a plane containing $x$, while $x^{*}\in{{\rm
d}\mathbb{S}}^{3}$ is the point dual to $P$, and $P^{*}\subset{{\rm
d}\mathbb{S}}^{3}$ is the plane dual to $x$. Indeed, for each
$x\in\partial_{k}\Omega$ and the tangent plane $P$ to $\partial_{k}\Omega$ at
$x$, the first-order deformation of the couple $(x,P)$ under the Killing field
$s_{x}$ induces, through the duality (see Section 2.6), a first-order
deformation of $x^{*}\in\partial_{k}\bar{\Omega}^{*}$, so we obtain a
deformation vector at $x^{*}$, denoted by $u(x^{*})$, which is equal to
$s_{x}(x^{*})$, since $s_{x}\in sl_{2}({\mathbb{C}})\cong so(3,1)$ and the
isometries in $SO(3,1)$ preserve the duality relation (see e.g. [40, Remark
1.7]). In this way, $u$ has an extension to $\partial_{k}\Omega^{*}$ and under
the infinitesimal Pogorelov map $\Upsilon$, $\bar{u}$ has an extension to
$\partial_{k}\bar{\Omega}^{*}$ for all $k\in\mathcal{K}$. In particular, there
is a natural correspondence between a pair of points
$\bar{x}:=\iota(x)\in\partial_{k}\bar{\Omega}$ and
$\bar{x}^{*}:=\iota(x^{*})\in\partial_{k}\bar{\Omega}^{*}$.

For simplicity, we still denote by $\bar{u}$ the extended deformation vector
field on
$(\bar{\Omega}\cup(\cup_{k\in\mathcal{K}}\partial_{k}\bar{\Omega}^{*}))\setminus\bar{\Lambda}$.
Recall that $(\bar{\tau},\bar{\sigma})$ denotes the translation and rotation
components of the corresponding deformation section $\bar{s}:=\Psi(s)$ on
$\bar{\Omega}\cap{\mathbb{D}}^{3}$.

######  Lemma 3.19.

For each $\bar{x}\in\partial_{k}\bar{\Omega}$ and
$\bar{x}^{*}\in\partial_{k}\bar{\Omega}^{*}$, we have

$\bar{u}(\bar{x}^{*})=\bar{\tau}(\bar{x})+\bar{\sigma}(\bar{x})\times(\bar{x}^{*}-\bar{x}).$

###### Proof.

By definition, $u(x^{*})=s_{x}(x^{*})$. Combined with Lemma 2.2 and the
definition of $\bar{s}$, under the infinitesimal Pogorelov map $\Upsilon$, we
have

$\bar{s}_{\bar{x}}(\bar{x}^{*}):=\Upsilon(s_{x}(x^{*}))=\Upsilon(u(x^{*}))=\bar{u}(\bar{x}^{*}).$

By the formulation of infinitesimal euclidean isometries,

$\bar{u}(\bar{x}^{*})=\bar{s}_{\bar{x}}(\bar{x}^{*})=\bar{\tau}(\bar{x})+\bar{\sigma}(\bar{x})\times(\bar{x}^{*}-\bar{x}).$

∎

Let $v$ (resp. $\bar{v}$) denote the orthogonal projection to
$(\partial\Omega\cup(\cup_{k\in\mathcal{K}}\partial_{k}\Omega^{*}))\setminus\Lambda$
(resp.
$(\partial\bar{\Omega}\cup(\cup_{k\in\mathcal{K}}\partial_{k}\bar{\Omega}^{*}))\setminus\bar{\Lambda}$)
of $u$ (resp. $\bar{u}$), called the _tangent component_ of $u$ (resp.
$\bar{u}$).

######  Proposition 3.20.

$\bar{v}$ satisfies equation (E) on
$\partial\bar{\Omega}^{*}\setminus\bar{\Lambda}$. In particular, $\bar{v}$ is
holomorphic on $\partial_{i}\bar{\Omega}$ for all $i\in\mathcal{I}$.

###### Proof.

Note that
$\partial\bar{\Omega}^{*}\setminus\bar{\Lambda}=(\cup_{i\in\mathcal{I}}\partial_{i}\bar{\Omega})\cup(\cup_{j\in\mathcal{J}}\partial_{j}\bar{\Omega})\cup(\cup_{k\in\mathcal{K}}\partial_{k}\bar{\Omega}^{*})$.
We first show that $\bar{u}$ satisfies equation (E) on
$(\cup_{j\in\mathcal{J}}\partial_{j}\bar{\Omega})\cup(\cup_{k\in\mathcal{K}}\partial_{k}\bar{\Omega}^{*})$.
By assumption, $\dot{g}$ preserves the induced metric on $S_{j}$ and the third
fundamental form on $S_{k}$ for all $j\in\mathcal{J}$ and $k\in\mathcal{K}$.
By Statement (3) of Proposition 2.3, $\dot{g}$ preserves the induced metric on
$\partial_{k}\Omega^{*}$. Therefore, the restriction of $u$ to
$\partial_{j}\Omega$ and $\partial_{k}\Omega^{*}$ are both isometric for all
$j\in\mathcal{J}$ and $k\in\mathcal{K}$. Combined with Statement (2) of Lemma
2.2, the restriction of $\bar{u}$ to $\partial_{j}\bar{\Omega}$ and
$\partial_{k}\bar{\Omega}^{*}$ are also isometric for all $j\in\mathcal{J}$
and $k\in\mathcal{K}$. By Lemma 3.8, $\bar{v}$ satisfies equation (E) on
$(\cup_{j\in\mathcal{J}}\partial_{j}\bar{\Omega})\cup(\cup_{k\in\mathcal{K}}\partial_{k}\bar{\Omega}^{*})$.

Since $\partial_{i}\bar{\Omega}\subset\partial{\mathbb{D}}^{3}$, the shape
operator $\bar{B}$ of $\partial_{i}\bar{\Omega}$ is the identity and thus
$\bar{I}=\bar{II}=\bar{III}$. Then for each $\bar{X}\in
T\partial_{i}\bar{\Omega}$,

$\partial_{\bar{III}}(\bar{B}^{-1}\bar{v})(\bar{X})=\partial_{\bar{III}}\bar{v}(\bar{X})={\nabla}^{\bar{I}}_{\bar{X}}\bar{v}+\bar{J}{\nabla}^{\bar{I}}_{\bar{J}\bar{X}}\bar{v},$

which implies that $\bar{v}$ is holomorphic on
$\cup_{i\in\mathcal{I}}\partial_{i}\bar{\Omega}$ if and only if $\bar{v}$
satisfies equation (E) on $\cup_{i\in\mathcal{I}}\partial_{i}\bar{\Omega}$. We
just need to show that $\bar{v}$ is holomorphic on
$\cup_{i\in\mathcal{I}}\partial_{i}\bar{\Omega}$. Since $\dot{g}$ preserves
the conformal structure at infinity on $\partial_{i}\Omega$ for all
$i\in\mathcal{I}$, $u$ is a holomorphic vector field on
$\cup_{i\in\mathcal{I}}\partial_{i}\Omega$.

By definition, $\bar{u}$ continuously extends to $\partial_{i}\bar{\Omega}$,
and therefore the sequence $(\bar{u}^{\perp}(\bar{x}_{n}))$ of lateral
components of $\bar{u}(\bar{x}_{n})$ converges to the lateral component
$\bar{u}^{\perp}(\bar{x})$, as $\bar{x}_{n}$ tends to
$\bar{x}\in\partial_{i}\bar{\Omega}$ along the geodesic ray $\iota(r_{x})$,
with $r_{x}$ starting from $\tilde{S}^{i}_{1}$ orthogonally. Observe that
$\partial_{i}\bar{\Omega}\subset\partial{\mathbb{D}}^{3}$, the lateral
component of $\bar{u}$ coincides with its tangent component $\bar{v}$ when
restricted to $\cup_{i\in\mathcal{I}}\partial_{i}\bar{\Omega}$, so
$\bar{v}(\bar{x})=\bar{u}^{\perp}(\bar{x})$ is the limit of
$\bar{u}^{\perp}(\bar{x}_{n})$. By the definition of the infinitesimal
Pogorelov map $\Upsilon$ (see Section 2.5), it sends lateral directions to
lateral directions. More precisely,

$\bar{u}^{\perp}(\bar{x}_{n})=\Upsilon(u^{\perp}(x_{n}))=d\iota(u^{\perp}(x_{n})).$

Note that $u(x_{n})$ tends to $u(x)$ as $x_{n}$ tends to
$x\in\partial_{i}\Omega$ along the above geodesic ray $r_{x}$ and $u(x)$ is
tangent to $\partial_{i}\Omega$. Combined with the fact that
$\iota:U\supset{\mathbb{H}}^{3}\rightarrow{\mathbb{R}}^{3}$ is an inclusion,
then for each $\bar{x}\in\partial_{i}\bar{\Omega}$,

$\bar{v}(\bar{x})=\bar{u}^{\perp}(\bar{x})=\lim\limits_{n\rightarrow\infty}\bar{u}^{\perp}(\bar{x}_{n})=\lim\limits_{n\rightarrow\infty}d\iota(u^{\perp}(x_{n}))=u^{\perp}(x)=u(x).$

Here the limits are with respect to the euclidean metric on
$T\overline{{\mathbb{H}}^{3}}$ or $T\overline{{\mathbb{D}}^{3}}$. The second-
to-last equality holds with $\partial_{i}\Omega$ identified with
$\partial_{i}\bar{\Omega}$ under $\iota$. As a consequence, $\bar{v}$ is also
holomorphic on $\cup_{i\in\mathcal{I}}\partial_{i}\Omega$. The lemma follows.

∎

###  3.5. Hölder estimate on $\bar{v}$

In this subsection, we aim to show the following statement about the Hölder
continuity of the tangent component $\bar{v}$ of the deformation vector field
$\bar{u}$.

######  Proposition 3.21.

$\bar{v}$ is $C^{\alpha}$ Hölder continuous on $\partial\bar{\Omega}^{*}$ for
all $\alpha\in(0,1)$.

Let $\bar{x}\not=\bar{y}\in\partial\bar{\Omega}\setminus\bar{\Lambda}$, and
let $\bar{z}$ be the (unique) point on the geodesic segment
$[\bar{x},\bar{y}]$ which is nearest to the center $\bar{O}$. Note that
$\bar{z}$ might coincide with $\bar{x}$ (resp. $\bar{y}$), in which case the
angle between $[\bar{O},\bar{x}]$ and $[\bar{x},\bar{y}]$ (resp. between
$[\bar{O},\bar{y}]$ and $[\bar{y},\bar{x}]$) is at least $\pi/2$. Otherwise,
$\bar{z}\in(\bar{x},\bar{y})$ and $[\bar{O},\bar{z}]$ is orthogonal to
$[\bar{x},\bar{y}]$. However, one can check that $\bar{z}$ must lie in the
interior of the segment $[\bar{x},\bar{y}]$ in some specific cases, for
instance, if both $\bar{x}$ and $\bar{y}$ lie on
$\partial_{i}\bar{\Omega}\subset\partial{\mathbb{D}}^{3}$ for some
$i\in\mathcal{I}$.

Before giving the proof of Proposition 3.21, we first consider the following
estimates.

####  3.5.1. Distance estimates on
$\partial\bar{\Omega}\setminus\bar{\Lambda}$ in ${\mathbb{R}}^{3}$

We give some comparisons among the distances $\|\bar{x}-\bar{z}\|$,
$\|\bar{y}-\bar{z}\|$, $\|\bar{x}-\bar{y}\|$ and $\sqrt{1-\|\bar{z}\|}$ in the
following claim, which will be used in the proof of Lemma 3.29 later.

######  Claim 3.22.

For $\bar{x}\not=\bar{y}\in\partial\bar{\Omega}\setminus\bar{\Lambda}$, we
have

(17) $\|\bar{x}-\bar{y}\|\leq
2\max\\{\|\bar{x}-\bar{z}\|,\|\bar{y}-\bar{z}\|\\}\leq
2\sqrt{2}\cdot\sqrt{1-\|\bar{z}\|}.$

###### Proof.

By the triangle inequality,

$\|\bar{x}-\bar{y}\|\leq\|\bar{x}-\bar{z}\|+\|\bar{y}-\bar{z}\|\leq
2\max\\{\|\bar{x}-\bar{z}\|,\|\bar{y}-\bar{z}\|\\}.$

By definition, $\bar{z}\in{\mathbb{D}}^{3}$ and
$\|\bar{z}\|\leq\min\\{\|\bar{x}\|,\|\bar{y}\|\\}$. Using the cosine formula,
we have

$\|\bar{x}\|^{2}=\|\bar{z}\|^{2}+\|\bar{x}-\bar{z}\|^{2}-2\|\bar{z}\|\cdot\|\bar{x}-\bar{z}\|\cdot\cos\angle\bar{x}\bar{z}\bar{O}.$

Combined with the facts that $\|\bar{x}\|\leq 1$ and that the angle between
$[\bar{O},\bar{z}]$ and $[\bar{x},\bar{y}]$ is at least $\pi/2$, we obtain
that

(18)
$\|\bar{x}-\bar{z}\|^{2}=\|\bar{x}\|^{2}-\|\bar{z}\|^{2}+2\|\bar{z}\|\cdot\|\bar{x}-\bar{z}\|\cdot\cos\angle\bar{x}\bar{z}\bar{O}\leq
1-\|\bar{z}\|^{2}\leq 2(1-\|\bar{z}\|).$

Similarly, we can show that

$\|\bar{y}-\bar{z}\|^{2}\leq 2(1-\|\bar{z}\|)~{}.$

Therefore,

$\max\\{\|\bar{x}-\bar{z}\|,\|\bar{y}-\bar{z}\|\\}\leq\sqrt{2}\sqrt{1-\|\bar{z}\|}.$

Combined with the above result, this shows the inequality (17). ∎

####  3.5.2. Estimates on $(\bar{\tau},\bar{\sigma})$

Recall that $\bar{\tau}$ and $\bar{\sigma}$ denote the translation and
rotation components of the deformation section $\bar{s}:=\Psi(s)$ on
$\bar{\Omega}\cap{\mathbb{D}}^{3}$. We set

$\delta\bar{\tau}=\bar{\tau}-\bar{\tau}_{0},\quad\quad\delta\bar{\sigma}=\bar{\sigma}-\bar{\sigma}_{0},$

where $\bar{\tau}_{0}\in\Gamma(T\bar{\Omega})$ is a Killing vector field and
$\bar{\sigma}_{0}\in\Gamma(T\bar{\Omega})$ is one half of the curl vector
field of $\bar{\tau}_{0}$, with $\bar{\tau}_{0}(\bar{z})=\bar{\tau}(\bar{z})$
and $\bar{\sigma}_{0}(\bar{z})=\bar{\sigma}(\bar{z})$. Also,
$(\bar{\tau}_{0},\bar{\sigma}_{0})$ is identified with a constant section
$\bar{\kappa}_{0}\in\Gamma(\bar{E})$ (see Section 2.4). In particular,
$(\bar{\kappa}_{0})_{\bar{x}}(\bar{y})=\bar{\tau}_{0}(\bar{y})$ for all
$\bar{x},\bar{y}\in\bar{\Omega}$.

Let $\bar{x}\in\partial\bar{\Omega}\setminus\bar{\Lambda}$. Recall that the
deformation one form $\bar{\omega}:=\Psi_{*}(\omega)$ is defined on
$\bar{\Omega}\subset{\mathbb{D}}^{3}$. For
$\bar{x}\in(\partial\bar{\Omega}\setminus\bar{\Lambda})\cap{\mathbb{D}}^{3}$,
by (6), we have
$\bar{\omega}_{\bar{\tau}}(\bar{X})=\bar{\nabla}_{\bar{X}}\bar{\tau}-\bar{\sigma}(\bar{x})\times\bar{X}$
and $\bar{\omega}_{\bar{\sigma}}(\bar{X})=\bar{\nabla}_{\bar{X}}\bar{\sigma}$.
Observe that $\bar{z}$ (defined at the beginning of Section 3.5) can be
connected to $\bar{x}$ along first the geodesic segment (with respect to the
spherical metric) on the sphere of radius $\|\bar{z}\|$ centered at $\bar{O}$
with endpoint (say $\bar{\xi}$) lying on the line segment $[\bar{O},\bar{x}]$
and then along the radial segment from $\bar{\xi}$ to $\bar{x}$ (see Figure
1(a)). As shown in [40, Propositions 4.4 and 4.5], by considering the
integration of $\|\bar{\omega}_{\bar{\tau}}\|$ (resp.
$\|\bar{\omega}_{\bar{\sigma}}\|$) along these two paths and using the
estimates (see Lemma 3.17) of the radial and lateral components of the one-
forms $\bar{\omega}_{\bar{\tau}}$ and $\bar{\omega}_{\bar{\sigma}}$, the
following estimates on $\|\delta\bar{\tau}(\bar{x})\|$,
$\|\delta\bar{\tau}^{\perp}(\bar{x})\|$, $\|\delta\bar{\sigma}(\bar{x})\|$ and
$\|\delta\bar{\sigma}^{r}(\bar{x})\|$ are obtained.

Note that
$(\partial\bar{\Omega}\setminus\bar{\Lambda})\cap{\mathbb{D}}^{3}=(\cup_{j\in\mathcal{J}}\partial_{j}\bar{\Omega})\cup(\cup_{k\in\mathcal{K}}\partial_{k}\bar{\Omega})$,
while
$(\partial\bar{\Omega}\setminus\bar{\Lambda})\cap\partial{\mathbb{D}}^{3}=\cup_{i\in\mathcal{I}}\partial_{i}\bar{\Omega}$.

######  Lemma 3.23.

For
$\bar{x}\in(\cup_{j\in\mathcal{J}}\partial_{j}\bar{\Omega})\cup(\cup_{k\in\mathcal{K}}\partial_{k}\bar{\Omega})$,
we have the following estimates:

$\begin{split}&\|\delta\bar{\tau}(\bar{x})\|\leq
C\left|\log\left(\frac{1-\|\bar{x}\|}{1-\|\bar{z}\|}\right)\right|,\\\
&\|\delta\bar{\tau}^{\perp}(\bar{x})\|\leq
C\|\bar{x}-\bar{z}\|+C\big{(}\sqrt{1-\|\bar{z}\|}-\sqrt{1-\|\bar{x}\|}\big{)};\\\
&\|\delta\bar{\sigma}(\bar{x})\|\leq\frac{C\|\bar{x}-\bar{z}\|}{1-\|\bar{z}\|}+C(\frac{1}{\sqrt{1-\|\bar{x}\|}}-\frac{1}{\sqrt{1-\|\bar{z}\|}}),\\\
&\|\delta\bar{\sigma}^{r}(\bar{x})\|\leq
C\left|\log\left(\frac{1-\|\bar{x}\|}{1-\|\bar{z}\|}\right)\right|.\\\
\end{split}$

In particular, if $\|\bar{x}-\bar{z}\|$ is small enough, then

(19) $\begin{split}\sqrt{1-\|\bar{x}\|}\cdot\|\delta\bar{\tau}(\bar{x})\|&\leq
C\sqrt{1-\|\bar{x}\|}\cdot\left|\log\left(\frac{1-\|\bar{x}\|}{1-\|\bar{z}\|}\right)\right|\leq
C\|\bar{x}-\bar{z}\|\cdot\big{|}\log(\|\bar{x}-\bar{z}\|)\big{|}~{}.\\\
\|\delta\bar{\tau}^{\perp}(\bar{x})\|&\leq
C\|\bar{x}-\bar{z}\|+C\big{(}\sqrt{1-\|\bar{z}\|}-\sqrt{1-\|\bar{x}\|}\big{)}\leq
C\|\bar{x}-\bar{z}\|.\end{split}$

######  Claim 3.24.

Assume that $\bar{x}\in\cup_{i\in\mathcal{I}}\partial_{i}\bar{\Omega}$ and
$\|\bar{x}-\bar{z}\|$ is small enough. Then

$\|\delta\bar{\tau}^{\perp}(\bar{x})\|\leq C\|\bar{x}-\bar{z}\|~{}.$

###### Proof.

Let $(\bar{x}_{t})_{t\in[\varepsilon_{0},1]}$ be a family of points on the
segment $[\bar{x},\bar{z}]$ with the distance $\|\bar{x}_{t}-\bar{x}\|=1-t$.
By Lemma 3.13, the one-form $\omega$ over $\Omega$ (including those points
near the boundary $\partial_{i}\Omega$) has a uniform bound. Observe that
$\bar{z}$ can be connected to $\bar{x}_{t}$ along first the geodesic segment
on the sphere of radius $\|\bar{z}\|$ centered at $\bar{O}$ with endpoint (say
$\bar{\xi}_{t}$) lying on the line segment $[\bar{O},\bar{x}_{t}]$ and then
along the segment from $\bar{\xi}_{t}$ to $\bar{x}_{t}$ (see Figure 1(b)). As
shown in [40, Proposition 4.4]), integrating $\|\bar{\omega}_{\bar{\tau}}\|$
(resp. $\|\bar{\omega}_{\bar{\sigma}}\|$) along these two paths and combining
the estimates (see Lemma 3.17) of the radial and lateral components of the
one-forms $\bar{\omega}_{\bar{\tau}}$ (resp. $\bar{\omega}_{\bar{\sigma}}$),
the results of Lemma 3.23 actually still hold with $\bar{x}$ replaced by
$\bar{x}_{t}$.

$\bar{O}$$\bar{x}$$\bar{y}$$\bar{z}$$\bar{\xi}$ (a)
$\bar{x}\in\partial\bar{\Omega}\cap{\mathbb{D}}^{3}$.

$\bar{O}$$\bar{x}_{t}$$\bar{x}$$\bar{y}$$\bar{z}$$\bar{\xi}_{t}$ (b)
$\bar{x}\in\partial\bar{\Omega}\cap\partial{\mathbb{D}}^{3}$.

Figure 1. Integral path from $\bar{z}$ to $\bar{x}$ (resp. $\bar{x}_{t}$),
where $\bar{x}_{t}$ tends to $\bar{x}$ as $t\rightarrow 1$ (the shaded regions
are two connected components of $\partial\bar{\Omega}\setminus\bar{\Lambda}$).

By assumption that $\|\bar{x}-\bar{z}\|$ is small enough,
$\|\bar{x}_{t}-\bar{z}\|$ is small enough. For $t\in(\epsilon_{0},1)$, we can
apply $\bar{x}_{t}\in{\mathbb{D}}^{3}\cap\bar{\Omega}$ to the second
inequality in Equation (19) and obtain that

$\|\delta\bar{\tau}^{\perp}(\bar{x}_{t})\|\leq C\|\bar{x}_{t}-\bar{z}\|.$

Let $t\rightarrow 1$. Then $\bar{x}_{t}$ converges to $\bar{x}$. Since the
action on the lateral vectors under the infinitesimal Pogorelov map $\Upsilon$
continuously extends to the boundary
$\cup_{i\in\mathcal{I}}\partial_{i}\bar{\Omega}$ (see e.g. the proof of
Proposition 3.20),

$\|\delta\bar{\tau}^{\perp}(\bar{x})\|\leq C\|\bar{x}-\bar{z}\|.$

The claim follows. ∎

####  3.5.3. Estimates on projection angles to $T\partial\bar{\Omega}$ and
$T\partial\bar{\Omega}^{*}$ of radial and lateral vectors

Let $\bar{x}\in\partial\bar{\Omega}\setminus\bar{\Lambda}$ and let
$P_{\bar{x}}$ denote the tangent plane to $\partial\bar{\Omega}$ at $\bar{x}$.
Let $\alpha^{r}_{\bar{x}}$ (resp. $\alpha^{\perp}_{\bar{x}}$) denote the
projection angles to the tangent planes $P_{\bar{x}}$ of a radial (resp.
lateral) vector at $\bar{z}$. Recall that $\theta_{\bar{x}}$ is the angle
between the outwards-pointing normal vector to $\partial\bar{\Omega}$ at
$\bar{x}$ and the radial direction $\bar{O}\bar{x}$ at $\bar{x}$.

######  Claim 3.25.

Let $\bar{x},\bar{y}\in\partial\bar{\Omega}\setminus\bar{\Lambda}$ with
$\bar{x}$, $\bar{y}$ close enough to $\bar{\Lambda}$, then

$\max\\{\cos\alpha^{r}_{\bar{x}},\cos\alpha^{r}_{\bar{y}}\\}\leq
C\sqrt{1-\|\bar{z}\|}~{}.$

In particular, the radial component $(\bar{x}-\bar{y})^{r}$ of
$\bar{x}-\bar{y}$ at $\bar{z}$ satisfies that

(20) $\|(\bar{x}-\bar{y})^{r}\|\leq
C\|\bar{x}-\bar{y}\|\cdot\sqrt{1-\|\bar{z}\|}~{}.$

###### Proof.

Let $Q_{\bar{x}}$ denote the plane orthogonal to the segment
$[\bar{O},\bar{x}]$ at $\bar{x}$ and let $\zeta^{r}_{\bar{x}}$ denote the
projection angle to the plane $Q_{\bar{x}}$ of a radial vector at $\bar{z}$,
while the (dihedral) angle between the tangent plane $P_{\bar{x}}$ to
$\partial\bar{\Omega}$ at $\bar{x}$ and the plane $Q_{\bar{x}}$ is
$\theta_{\bar{x}}$. As a consequence, the projection angle
$\alpha^{r}_{\bar{x}}$ satisfies

(21)
$\zeta^{r}_{\bar{x}}-\theta_{\bar{x}}\leq\alpha^{r}_{\bar{x}}\leq\zeta^{r}_{\bar{x}}+\theta_{\bar{x}}~{},$

where the equalities hold only if the plane $O\bar{x}\bar{y}$ is orthogonal to
$P_{\bar{x}}$. In particular, the left equality holds if
$\bar{z}\in(\bar{x},\bar{y})$ and the inward-pointing normal direction to
$P_{\bar{x}}$ at $\bar{x}$ lies on the boundary $\bar{O}\bar{x}$ or inside the
triangle $\bar{x}\bar{O}\bar{y}$ (as shown in Figure 2(a)), or
$\bar{z}=\bar{x}$ (as shown in Figure 2(d)), while the right equality holds if
$\bar{z}\in(\bar{x},\bar{y})$ and the inward-pointing normal direction to
$P_{\bar{x}}$ at $\bar{x}$ lies on the boundary $\bar{O}\bar{x}$ or outside
the triangle $\bar{x}\bar{O}\bar{y}$ (as shown in Figure 2(b)). In particular,
$\theta_{\bar{x}}=0$ if and only if
$\bar{x}\in\partial\bar{\Omega}\cap\partial{\mathbb{D}}^{3}$.

$\bar{O}$$P_{\bar{x}}$$Q_{\bar{x}}$$\bar{x}$$\bar{y}$$\bar{z}$ (a)
$\bar{z}\in(\bar{x},\bar{y})$,
$\alpha^{r}_{\bar{x}}={\zeta}^{r}_{\bar{x}}-\theta_{\bar{x}}=\angle{\bar{O}\bar{x}\bar{y}}-\theta_{\bar{x}}$.

$\bar{O}$$P_{\bar{x}}$$Q_{\bar{x}}$$\bar{y}$$\bar{z}$$\bar{x}$ (b)
$\bar{z}\in(\bar{x},\bar{y})$,
$\alpha^{r}_{\bar{x}}={\zeta}^{r}_{\bar{x}}+\theta_{\bar{x}}=\angle{\bar{O}\bar{x}\bar{y}}+\theta_{\bar{x}}$.

$\bar{O}$$P_{\bar{x}}$$Q_{\bar{x}}$$\bar{x}$$\bar{y}=\bar{z}$ (c)
$\bar{z}=\bar{y}$,
$\alpha^{r}_{\bar{x}}=\pi-{\zeta}^{r}_{\bar{x}}-\theta_{\bar{x}}\geq\pi/2-\theta_{\bar{x}}$.

$\bar{O}$$P_{\bar{x}}$$Q_{\bar{x}}$$\bar{y}$$\bar{x}=\bar{z}$ (d)
$\bar{z}=\bar{x}$,
$\alpha^{r}_{\bar{x}}=\pi-{\zeta}^{r}_{\bar{x}}-\theta_{\bar{x}}=\pi/2-\theta_{\bar{x}}$.

Figure 2. Examples of the projection angle $\alpha^{r}_{\bar{x}}$ (with
$\bar{x}$, $\bar{y}$ close to $\bar{\Lambda}$) in the case that the plane
$\bar{O}\bar{x}\bar{y}$ is orthogonal to the tangent plane $P_{\bar{x}}$, with
the amount of the angle $\alpha^{r}_{\bar{x}}$ marked and
$\partial\bar{\Omega}$ partially drawn.

Since $\bar{\Omega}$ is strictly convex and the center $\bar{O}$ is contained
in the interior of $\bar{\Omega}$, $\min\\{\|\bar{x}\|,\|\bar{y}\|\\}\geq
r_{0}$ for a constant $r_{0}>0$. If $\bar{z}\in(\bar{x},\bar{y})$, then
$\zeta^{r}_{\bar{x}}=\angle\bar{O}\bar{x}\bar{y}$ and
$\cos\zeta^{r}_{\bar{x}}=\cos\angle{\bar{O}\bar{x}\bar{y}}=\|\bar{x}-\bar{z}\|/\|\bar{x}\|$.
Combined with the estimate (17) and Lemma 3.6, we obtain

(22)
$\cos(\zeta^{r}_{\bar{x}}-\theta_{\bar{x}})=\cos\zeta^{r}_{\bar{x}}\cdot\cos\theta_{\bar{x}}+\sin\zeta^{r}_{\bar{x}}\cdot\sin\theta_{\bar{x}}\leq\|\bar{x}-\bar{z}\|/{r_{0}}+C\sqrt{1-\|\bar{x}\|}\leq
C\sqrt{1-\|\bar{z}\|},$

for $\bar{x}$ close enough to $\bar{\Lambda}$. Note that the constants $C$
above change accordingly, and we keep the same notation for simplicity
(similarly henceforth).

Note that the projection angle $\zeta^{r}_{\bar{x}}$ is always in
$[0,{\pi}/{2}]$ and $\theta_{\bar{x}}\geq 0$. If
$\bar{z}\in\partial[\bar{x},\bar{y}]$, then
$\alpha^{r}_{\bar{x}}=\pi-{\zeta}^{r}_{\bar{x}}-\theta_{\bar{x}}\geq\pi/2-\theta_{\bar{x}}$
(in particular, $\zeta^{r}_{\bar{x}}=\pi/2$ if $\bar{x}=\bar{z}$), as shown in
Figure 2(c) and Figure 2(d). Using Lemma 3.6, for $\bar{x}$ close enough to
$\bar{\Lambda}$,

$\cos\alpha^{r}_{\bar{x}}\leq\sin\theta_{\bar{x}}\leq
C\sqrt{1-\|\bar{x}\|}\leq C\sqrt{1-\|\bar{z}\|}.$

Combined with (21) and (22), it follows that for $\bar{x}$ close enough to
$\bar{\Lambda}$, $\alpha^{r}_{\bar{x}}$ always satisfies that

$\cos\alpha^{r}_{\bar{x}}\leq C\sqrt{1-\|\bar{z}\|}.$

Similarly for $\alpha^{r}_{\bar{y}}$.

If $\bar{z}\in(\bar{x},\bar{y})$, the vector $\bar{z}$ is orthogonal to
$\bar{x}-\bar{y}$. Then $(\bar{x}-\bar{y})^{r}=0$ and (20) naturally holds. If
$\bar{z}=\bar{x}$, since $[\bar{x},\bar{y}]\subset\bar{\Omega}$, the angle
between $\bar{x}-\bar{y}$ and the plane $Q_{\bar{x}}$ orthogonal to
$[\bar{O},\bar{x}]=[\bar{O},\bar{z}]$ is less than the angle between
$Q_{\bar{x}}$ and the tangent plane $P_{\bar{x}}$ (which is equal to
$\theta_{\bar{x}}$). Combined with Lemma 3.6, for $\bar{x}$ close enough to
$\bar{\Lambda}$,

$\|(\bar{x}-\bar{y})^{r}\|\leq\|\bar{x}-\bar{y}\|\cdot\sin\theta_{\bar{x}}\leq
C\|\bar{x}-\bar{y}\|\cdot\sqrt{1-\|\bar{z}\|}.$

If $\bar{z}=\bar{y}$, we consider the plane $Q_{\bar{y}}$ orthogonal to
$[\bar{O},\bar{y}]=[\bar{O},\bar{z}]$ and the tangent plane $P_{\bar{y}}$
instead. Similarly, for $\bar{y}$ close enough to $\bar{\Lambda}$,

$\|(\bar{x}-\bar{y})^{r}\|\leq\|\bar{x}-\bar{y}\|\cdot\sin\theta_{\bar{y}}\leq
C\|\bar{x}-\bar{y}\|\cdot\sqrt{1-\|\bar{z}\|}.$

The claim follows. ∎

Let $\bar{x}\in\partial_{k}\bar{\Omega}$ and let
$\bar{x}^{*}\in\partial_{k}\bar{\Omega}^{*}$ be the dual of the tangent plane
to $\partial_{k}\bar{\Omega}$ at $\bar{x}$. We denote by $P_{\bar{x}}$ (resp.
$P_{\bar{x}^{*}}$) the plane tangent to $\partial_{k}\bar{\Omega}$ and
$\partial_{k}\bar{\Omega}^{*}$ at $\bar{x}$ (resp. $\bar{x}^{*}$). Let
$\theta_{\bar{x}^{*}}$ denote the angle between the radial direction
$\bar{O}\bar{x}^{*}$ at $\bar{x}^{*}$ and the outward-pointing normal vector
to $\partial_{k}\bar{\Omega}^{*}$ at $\bar{x}^{*}$. Let
$(\bar{x}^{*}-\bar{x})^{r}$ and $(\bar{x}^{*}-\bar{x})^{\perp}$ denote the
radial and lateral components of $(\bar{x}^{*}-\bar{x})$ at $\bar{x}$. The
following are the estimates for these quantities (see [40, Proposition 4.6]).

######  Lemma 3.26.

There exists a constant $C>0$ such that for $\bar{x}$ close enough to
$\bar{\Lambda}$,

  1. (1)

$\|(\bar{x}^{*}-\bar{x})^{r}\|\leq C(1-\|\bar{x}\|).$

  2. (2)

$\|(\bar{x}^{*}-\bar{x})^{\perp}\|\leq C\sqrt{1-\|\bar{x}\|}$.

  3. (3)

$\theta_{\bar{x}^{*}}\leq C\sqrt{1-\|\bar{x}\|}$.

Let $\alpha^{r}_{\bar{x}^{*}}$ (resp. $\alpha^{\perp}_{\bar{x}^{*}}$) denote
the projection angles to $P_{\bar{x}^{*}}$ of a radial (resp. lateral) vector
at $\bar{z}$.

######  Claim 3.27.

Let
$\bar{x}^{*},\bar{y}^{*}\in\cup_{k\in\mathcal{K}}\partial_{k}\bar{\Omega}^{*}$
with $\bar{x}^{*}$, $\bar{y}^{*}$ close enough to $\bar{\Lambda}$, then

(23) $\max\\{\cos\alpha^{r}_{\bar{x}^{*}},\cos\alpha^{r}_{\bar{y}^{*}}\\}\leq
C\sqrt{1-\|\bar{z}\|}.$

###### Proof.

Let $\angle(P_{\bar{x}},P_{\bar{x}^{*}})$ denote the (un-oriented dihedral)
angle between $P_{\bar{x}}$ and $P_{\bar{x}^{*}}$, which is equal to the angle
between the outward-pointing normal vectors $n_{\bar{x}}$ and
$n_{\bar{x}^{*}}$ to $\partial\bar{\Omega}$ and $\partial\bar{\Omega}^{*}$ at
$\bar{x}$ and $\bar{x}^{*}$ respectively. By the duality between a totally
geodesic plane in ${\mathbb{H}}^{3}$ and a point in ${{\rm d}\mathbb{S}}^{3}$
(considered in the Klein models, see Section 2.6), the radial vector
$\bar{O}\bar{x}^{*}$ is orthogonal to the plane $P_{\bar{x}}$ (which is
tangent to $\partial\bar{\Omega}$ at $\bar{x}$). Then
$\angle(P_{\bar{x}},P_{\bar{x}^{*}})$ is equal to the angle between
$\bar{O}\bar{x}^{*}$ and $n_{\bar{x}^{*}}$, that is, $\theta_{\bar{x}^{*}}$.
Combined with the result for $\alpha^{r}_{\bar{x}}$ and $\alpha^{r}_{\bar{y}}$
in the proof of Claim 3.25, we obtain

$\begin{split}&\zeta^{r}_{\bar{x}}-(\theta_{\bar{x}}+\theta_{\bar{x}^{*}})\leq\alpha^{r}_{\bar{x}}-\angle(P_{\bar{x}},P_{\bar{x}^{*}})\leq\alpha^{r}_{\bar{x}^{*}}\leq\alpha^{r}_{\bar{x}}+\angle(P_{\bar{x}},P_{\bar{x}^{*}})=\zeta^{r}_{\bar{x}}+(\theta_{\bar{x}}+\theta_{\bar{x}^{*}})~{},\\\
&\zeta^{r}_{\bar{y}}-(\theta_{\bar{y}}+\theta_{\bar{y}^{*}})\leq\alpha^{r}_{\bar{y}}-\angle(P_{\bar{y}},P_{\bar{y}^{*}})\leq\alpha^{r}_{\bar{y}^{*}}\leq\alpha^{r}_{\bar{y}}+\angle(P_{\bar{y}},P_{\bar{y}^{*}})=\zeta^{r}_{\bar{y}}+(\theta_{\bar{y}}+\theta_{\bar{y}^{*}})~{}.\end{split}$

Applying similar estimate as in the proof of Claim 3.25 for
$\cos\alpha^{r}_{\bar{x}}$ and $\cos\alpha^{r}_{\bar{y}}$ and using Lemma 3.6,
the inequality (17) and Statement (3) of Lemma 3.26, then for $\bar{x}$ close
enough to $\bar{\Lambda}$,

$\begin{split}\cos\alpha^{r}_{\bar{x}^{*}}\leq\cos(\zeta^{r}_{\bar{x}}-(\theta_{\bar{x}}+\theta_{\bar{x}^{*}}))&=\cos\zeta^{r}_{\bar{x}}\cdot\cos(\theta_{\bar{x}}+\theta_{\bar{x}^{*}})+\sin\zeta^{r}_{\bar{x}}\cdot\sin(\theta_{\bar{x}}+\theta_{\bar{x}^{*}})\\\
&\leq\|\bar{x}-\bar{z}\|/{r_{0}}+C\sqrt{1-\|\bar{x}\|}\leq
C\sqrt{1-\|\bar{z}\|}.\end{split}$

Similarly for $\alpha^{r}_{\bar{y}^{*}}$. The claim follows. ∎

######  Claim 3.28.

There exists a constant $C>0$ such that

  1. (1)

for any $\bar{x},\bar{y}\in\partial\bar{\Omega}\setminus\bar{\Lambda}$, we
have

$\frac{1}{C}\|\bar{x}^{*}-\bar{y}^{*}\|\leq\|\bar{x}-\bar{y}\|\leq
C\|\bar{x}^{*}-\bar{y}^{*}\|~{}.$

  2. (2)

for any $\bar{x},\bar{y}\in\partial\bar{\Omega}\setminus\bar{\Lambda}$ lying
in different components of $\partial\bar{\Omega}\setminus\bar{\Lambda}$, we
have

$\|\bar{x}-\bar{y}\|\leq 2\|\bar{x}-\bar{y}^{*}\|~{}.$

In addition, if $\bar{x}$ and $\bar{y}$ are close enough to $\bar{\Lambda}$,
then

$\max\\{\sqrt{1-\|\bar{x}\|},\sqrt{1-\|\bar{y}\|}\\}\leq
C\|\bar{x}-\bar{y}\|~{},$

and

$\|\bar{x}-\bar{y}^{*}\|\leq C\|\bar{x}-\bar{y}\|~{}.$

###### Proof.

We first show Statement (1). By the definition of the duality and the strict
convexity of $\partial\bar{\Omega}$, the map
$D:\partial\bar{\Omega}\rightarrow\partial\bar{\Omega}^{*}$ (resp.
$D^{*}:\partial\bar{\Omega}^{*}\rightarrow\partial\bar{\Omega}$) that takes
$\bar{x}$ to $\bar{x}^{*}$ (resp. $\bar{x}^{*}$ to $\bar{x}$) is continuous.
Note that $D$ and $D^{*}$ are exactly the identity when restricted to the
intersection with $\partial{\mathbb{D}}^{3}$. Furthermore, $D$ is a $C^{1}$
homeomorphism on $\partial\bar{\Omega}$. Therefore, there is a uniform
constant $C>0$, such that

$\frac{1}{C}\|\bar{x}^{*}-\bar{y}^{*}\|\leq\|\bar{x}-\bar{y}\|\leq
C\|\bar{x}^{*}-\bar{y}^{*}\|,$

for all $\bar{x},\bar{y}\in\partial\bar{\Omega}$. Statement (1) follows.

We now consider Statement (2). Without loss of generality, we assume that
$\bar{x}\in(\cup_{i\in\mathcal{I}}\partial_{i}\bar{\Omega})\cup(\cup_{j\in\mathcal{J}}\partial_{j}\bar{\Omega})$,
$\bar{y}\in\cup_{k\in\mathcal{K}}\partial_{k}\bar{\Omega}$. Note that
$\partial\bar{\Omega}$ is strictly convex and by the definition of duality
between convex surfaces in ${\mathbb{H}}^{3}$ and ${{\rm d}\mathbb{S}}^{3}$,
the point $\bar{y}^{*}$ lies outside of $\partial{\mathbb{D}}^{3}$ and on the
other side (opposite to $\bar{x}$) of the tangent plane to
$\cup_{k\in\mathcal{K}}\partial_{k}\bar{\Omega}$ at $\bar{y}$. Assume that
$\bar{y}$ lies on a connected component, say $\partial^{0}_{k}\bar{\Omega}$,
of $\partial_{k}\bar{\Omega}\subset{\mathbb{D}}^{3}$ for some
$k\in\mathcal{K}$, with the boundary of $\partial^{0}_{k}\bar{\Omega}$ a
Jordan curve say ${\mathcal{C}}^{0}_{k}$ contained in $\bar{\Lambda}$. Observe
that the distance between $\bar{y}^{*}$ and $\bar{y}$ is less than the
distance between $\bar{y}^{*}$ and any point
$\bar{p}\in\partial\bar{\Omega}\setminus(\partial^{0}_{k}\bar{\Omega}\cup{\mathcal{C}}^{0}_{k})$,
as shown in Figure 3. Thus,

$\|\bar{x}-\bar{y}\|\leq\|\bar{x}-\bar{y}^{*}\|+\|\bar{y}-\bar{y}^{*}\|\leq
2\|\bar{x}-\bar{y}^{*}\|.$

By Statement (1) of Lemma 3.6, for $\bar{x}$ close enough to $\bar{\Lambda}$,

$\sqrt{1-\|\bar{x}\|}\leq
C\delta_{\partial\bar{\Omega}}(\bar{x},\bar{\Lambda})~{}.$

Since the principal curvatures of $\partial\bar{\Omega}\setminus\bar{\Lambda}$
are uniformly bounded between two positive constants (see Lemma 3.5) and
$\bar{x}$, $\bar{y}$ lie in different components of
$\partial\bar{\Omega}\setminus\bar{\Lambda}$, then there exists a constant
$C>0$ such that

$\delta_{\partial\bar{\Omega}}(\bar{x},\bar{\Lambda})\leq
C\|\bar{x}-\bar{y}\|.$

So we have $\sqrt{1-\|\bar{x}\|}\leq C\|\bar{x}-\bar{y}\|.$ Similarly, for
$\bar{y}$ close enough to $\bar{\Lambda}$, we can show that
$\sqrt{1-\|\bar{y}\|}\leq C\|\bar{x}-\bar{y}\|$. This implies that for
$\bar{x}$, $\bar{y}$ close enough to $\bar{\Lambda}$,

(24) $\max\\{\sqrt{1-\|\bar{x}\|},\sqrt{1-\|\bar{y}\|}\\}\leq
C\|\bar{x}-\bar{y}\|~{}.$

We stress that (24) only works for $\bar{x},\bar{y}$ lying in different
components of $\partial\bar{\Omega}\setminus\bar{\Lambda}$. If $\bar{x}$ and
$\bar{y}$ lie on the same component of
$\partial\bar{\Omega}\setminus\bar{\Lambda}$, (24) does not hold since we can
take $\|\bar{x}-\bar{y}\|$ arbitrarily small.

Observe that
$\|\bar{x}-\bar{y}^{*}\|\leq\|\bar{x}-\bar{y}\|+\|\bar{y}-\bar{y}^{*}\|$. By
Statements (1)-(2) of Lemma 3.26,

$\|\bar{y}-\bar{y}^{*}\|\leq C\sqrt{1-\|\bar{y}\|}~{}.$

Using (24), for $\bar{x}$, $\bar{y}$ close enough to $\bar{\Lambda}$,

$\|\bar{x}-\bar{y}^{*}\|\leq C\|\bar{x}-\bar{y}\|~{}.$

Statement (2) follows.

$\bar{x}\in\partial_{i}\bar{\Omega}$$\bar{O}$$\bar{x}\in\partial_{j}\bar{\Omega}$$\bar{y}\in\partial^{0}_{k}\bar{\Omega}$$\bar{y}^{*}\in(\partial^{0}_{k}\bar{\Omega})^{*}$
Figure 3. $\|\bar{y}^{*}-\bar{y}\|<\|\bar{y}^{*}-\bar{x}\|$ for all
$\bar{x}\in\partial\bar{\Omega}\setminus(\partial^{0}_{k}\bar{\Omega}\cup{\mathcal{C}}^{0}_{k})$
(where $\partial\bar{\Omega}$ is partially drawn).

∎

We are now ready to show the following key lemma.

######  Lemma 3.29.

$\bar{v}$ is $C^{\alpha}$ Hölder continuous on
$\partial\bar{\Omega}^{*}\setminus\bar{\Lambda}$ for all $\alpha\in(0,1)$.
More precisely, for any $\alpha\in(0,1)$, there exists a constant
$C_{\alpha}>0$ (depending on $\alpha$) such that for all
$\bar{x},\bar{y}\in\partial\bar{\Omega}^{*}\setminus\bar{\Lambda}$,

$\|\bar{v}(\bar{x})-\bar{v}(\bar{y})\|\leq
C_{\alpha}\|\bar{x}-\bar{y}\|^{\alpha}.$

###### Proof.

Since $\bar{v}$ is smooth on $\partial\bar{\Omega}^{*}\setminus\bar{\Lambda}$
(see Remark 3.4), it is a Lipschitz vector field when restricted to a compact
subset of $\partial\bar{\Omega}\setminus\bar{\Lambda}$. We fix a constant
$\epsilon>0$ (to be made precise below) and denote by

$\bar{K}_{1}=\\{\bar{x}\in\partial\bar{\Omega}~{}|~{}d(\bar{x},\bar{\Lambda})\geq\epsilon\\}~{}.$

Note that $\bar{K}_{1}$ is a compact subset of
$\partial\bar{\Omega}\setminus\bar{\Lambda}$. We claim that it suffices to
discuss the Hölder continuity of $\bar{v}$ for the set
$\bar{K}_{2}:=\partial\bar{\Omega}^{*}\setminus({\rm
int}\bar{K}_{1}\cup\bar{\Lambda})$. Indeed, assume that Lemma 3.29 holds
respectively for $\bar{K}_{1}$ and $\bar{K}_{2}$. For any
$\bar{x}\in\bar{K}_{1}$ and $\bar{y}\in\bar{K_{2}}$, there exist finitely many
points $\bar{x}_{1},\dots,\bar{x}_{i},\dots,\bar{x}_{k}$ with the first $i$
points lying in $\bar{K}_{1}$, $\bar{x}_{i}\in\partial\bar{K}_{1}$ and the
other points lying in $\bar{K}_{2}$ with $\bar{x}_{k}=\bar{y}$, and moreover
$\max_{1\leq j\leq
k-1}\\{\|\bar{x}_{j}-\bar{x}_{j+1}\|\\}\leq\min\\{1/2,\|\bar{x}-\bar{y}\|\\}$.
Then

$\begin{split}\|\bar{v}(\bar{x})-\bar{v}(\bar{y})\|&\leq\|\bar{v}(\bar{x})-\bar{v}(\bar{x}_{1})\|+\cdots+\|\bar{v}(\bar{x}_{k-1})-\bar{v}(\bar{y})\|\\\
&\leq
C_{\alpha}(\|\bar{x}-\bar{x}_{1}\|^{\alpha}+\cdots+\|\bar{x}_{k-1}-\bar{y}\|^{\alpha})\\\
&\leq C_{\alpha}k\|\bar{x}-\bar{y}\|^{\alpha}\leq
C^{\prime}_{\alpha}\|\bar{x}-\bar{y}\|^{\alpha}~{},\end{split}$

where the last inequality follows from the observation that $k$ can be chosen
to be uniformly bounded for any $\bar{x}\in\bar{K}_{1}$ and
$\bar{y}\in\bar{K_{2}}$. The claim follows.

We now show the Hölder continuity of $\bar{v}$ restricted to $\bar{K}_{2}$.
Let $\bar{x},\bar{y}\in\bar{K}_{2}$. Recall that
$[\bar{x},\bar{y}]\subset{\mathbb{D}}^{3}$ denotes the geodesic segment
through $\bar{x}$ and $\bar{y}$, while $\bar{z}\in[\bar{x},\bar{y}]$ denotes
the point which is nearest to the center $\bar{O}$.

Recall that in Subsection 3.5.2,

$\delta\bar{\tau}=\bar{\tau}-\bar{\tau}_{0},\quad\quad\delta\bar{\sigma}=\bar{\sigma}-\bar{\sigma}_{0}~{},$

where
$(\bar{\tau}_{0},\bar{\sigma}_{0})\in\Gamma(T\bar{\Omega})\times\Gamma(T\bar{\Omega})$
satisfies that $\bar{\tau}_{0}(\bar{z})=\bar{\tau}(\bar{z})$ and
$\bar{\sigma}_{0}(\bar{z})=\bar{\sigma}(\bar{z})$, which determines a flat
section $\bar{\kappa}_{0}\in\Gamma(\bar{E})$:
$(\bar{\kappa}_{0})_{\bar{x}}=(\bar{\kappa}_{0})_{\bar{y}}$ for all
$\bar{x},\bar{y}\in\bar{\Omega}$. It follows from the definition of
$\bar{\sigma}$ and $\bar{\tau}$ as the rotation and translation components of
an infinitesimal isometry that

$(\bar{\kappa}_{0})_{\bar{y}}(\bar{x})=\bar{\tau}_{0}(\bar{x})=\bar{\sigma}(\bar{z})\times(\bar{x}-\bar{z})+\bar{\tau}(\bar{z}),$

for all $\bar{x},\bar{y}\in\bar{\Omega}$. Note that $\bar{v}=\Pi\bar{u}$,
where $\bar{u}$ is the extended deformation vector field given in Section 3.4,
while $\Pi$ is the orthogonal projection to $\partial\bar{\Omega}^{*}$, which
behaves at $\bar{x}\in\partial\bar{\Omega}^{*}$, as a morphism
$\Pi_{\bar{x}}:T_{\bar{x}}{\mathbb{R}}^{3}\rightarrow
T_{\bar{x}}\partial\bar{\Omega}^{*}$. Note that
$\bar{u}(\bar{x})=\bar{\tau}(\bar{x})$.

Observe that it suffices to discuss the case that $\|\bar{x}-\bar{y}\|\leq
1/2$. Otherwise, we can always take the constant $C_{\alpha}$ big enough for
each $\alpha\in(0,1)$.

If
$\bar{x},\bar{y}\in(\cup_{i\in\mathcal{I}}\partial_{i}\bar{\Omega})\cup(\cup_{j\in\mathcal{J}}\partial_{j}\bar{\Omega})$,
we have

$\begin{split}\|\bar{v}(\bar{x})-\bar{v}(\bar{y})\|&=\|\Pi_{\bar{x}}(\bar{\tau}(\bar{x}))-\Pi_{\bar{y}}(\bar{\tau}(\bar{y}))\|\\\
&=\|\Pi_{\bar{x}}(\delta\bar{\tau}+\bar{\tau}_{0})(\bar{x})-\Pi_{\bar{y}}(\delta\bar{\tau}+\bar{\tau}_{0})(\bar{y})\|\\\