# Convex surfaces with prescribed induced metrics in anti-de Sitter spacetimes

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: v0, )

###### Abstract.

Let $S$ be a closed surface of genus at least $2$, let $h$ be a smooth metric
of curvature $K<-1$ on $S$, and let $h_{0}$ be a hyperbolic metric on $S$. We
show that there exists a unique quasifuchsian AdS spacetime with left metric
isotopic to $h_{0}$, containing a past-convex Cauchy surface with induced
metric isotopic to $h$.

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

J.-M. S. was partially supported by FNR project O20/14766753. J.-M.S. would
like to thank IHES and MPIM Bonn, where part of this work was completed.

###### Contents

  1. 1 Introduction
    1. 1.1 Main result
    2. 1.2 Background and motivations
    3. 1.3 Outline of the paper
  2. 2 Preliminaries
    1. 2.1 The 3-dimensional anti-de Sitter geometry
    2. 2.2 Quasifuchsian AdS spacetimes
    3. 2.3 The linearized Gauss and Codazzi equations
    4. 2.4 The left and right metrics on a spacelike surface
  3. 3 Proof of the main result
    1. 3.1 Local rigidity
    2. 3.2 Proof of Theorem 1.1
  4. 4 Application

##  1\. Introduction

###  1.1. Main result

We consider here 3-dimensional maximal globally hyperbolic and spatially
compact (MGHC) anti-de Sitter (AdS) spacetimes. If $M$ is a MGHC AdS
spacetime, it contains closed Cauchy surfaces, which are all homeomorphic to a
fixed closed surface $S$ which we will assume to be of genus at least $2$.

Those spacetimes are known since the work of Mess [Mes07] to have many
remarkable similarities with quasifuchsian hyperbolic manifolds, and are for
this reason often called “quasifuchsian AdS spacetimes”, a terminology we will
follow here. The identity component of the isometry group of the 3-dimensional
AdS space $\mathbb{A}\rm{d}\mathbb{S}^{3}$ identifies (up to finite index)
with ${\rm PSL}(2,{\mathbb{R}})\times{\rm PSL}(2,{\mathbb{R}})$, and, if $M$
is a quasifuchsian AdS spacetime, its holonomy representation
$\rho:\pi_{1}M=\pi_{1}(S)\to{\rm PSL}(2,{\mathbb{R}})\times{\rm
PSL}(2,{\mathbb{R}})$ is the product $\rho=(\rho_{L},\rho_{R})$ of two
Fuchsian representations, called its left and right representations. Those
representations $\rho_{L}$ and $\rho_{R}$ are therefore the holonomy
representations of two hyperbolic metrics $h_{L}$ and $h_{R}$ on $S$, called
the left and right metrics of $M$, see [Mes07, ABB+07].

We denote by ${\mathcal{T}}_{S}$ the Teichmüller space of $S$, which is the
space of hyperbolic metrics on $S$ (considered up to isotopy) and denote by
$\mathcal{QF}_{AdS}(S)$ the deformation space of quasifuchsian AdS spacetimes
with Cauchy surfaces homeomorphic to $S$.

The main result here is the following statement, which gives an affirmative
answer to Question 6.11 in [Sch20].

######  Theorem 1.1.

Let $h$ be a complete Riemannian metric of curvature $K<-1$ on $S$, and let
$h_{0}\in{\mathcal{T}}_{S}$ be a hyperbolic metric on $S$. There is a unique
quasifuchsian AdS spacetime $M\in\mathcal{QF}_{AdS}(S)$ with left metric
isotopic to $h_{0}$ and containing a past-convex spacelike surface with
induced metric isotopic to $h$.

Tamburelli [Tam18, Proposition 7.1] showed that the existence holds, based on
a topological intersection argument. To show the uniqueness, the key point is
to prove the infinitesimal rigidity of quasifuchsian AdS metrics with respect
to the left metric and the induced metric on a past-convex Cauchy surface (see
Proposition 3.1).

When $h$ has constant curvature, Theorem 1.1 follows from the “Landslide
Theorem” in [BMS13, Theorem 1.15]. Indeed, for each constant $K<-1$, a
quasifuchsian AdS spacetime $M$ contains a unique past-convex Cauchy surface
$S_{K}$ with induced metric of constant curvature $K$ (see [BBZ11]), and the
left (resp. right) hyperbolic metric of $M$ is obtained by a landslide of
parameter $\theta$ (resp. $-\theta$) with $\theta\in(0,\pi)$ and
$K=-1/\cos^{2}(\theta/2)$, determined by the induced metric and third
fundamental form on $S_{K}$, see [BMS13, Lemma 1.9]. Since there is a unique
landslide map with parameter $\theta\in(0,2\pi)$ (resp. $\theta\in(-2\pi,0)$)
taking a given hyperbolic metric to another one (see [BMS13, Theorem 1.15]),
there is a unique choice of $M$ such that the induced metric on $S_{K}$ and
the left (resp. right) hyperbolic metric are prescribed. This argument
actually extends to the case where $h$ has constant curvature $-1$, with
Thurston’s Earthquake Theorem [Thu06, Mes07] used instead of the result on
landslide.

It follows from Theorem 1.1, using an elementary symmetry argument, that there
also exists a unique quasifuchsian AdS spacetime having $h_{0}$ as right
hyperbolic metric, and containing a past-convex surface with induced metric
$h$. Another direct symmetry argument shows that one can prescribe the induced
metric on a future-convex, rather than past-convex, spacelike surface.

There is a well-defined duality between locally strongly convex spacelike
surfaces in quasifuchsian AdS spacetimes (see [BBZ11, Section 11.1] or [BMS15,
Section 2.6]), where by “strongly convex” we mean that the second fundamental
form is everywhere positive definite. This duality associates to every past-
convex spacelike surface $\Sigma$ a surface $\Sigma^{*}$, defined as the space
of points dual to the oriented tangent planes of $\Sigma$. This construction
defines a duality map from $\Sigma$ to $\Sigma^{*}$. If $\Sigma$ is spacelike
and strongly past-convex, then $\Sigma^{*}$ is spacelike and strongly future-
convex, and the pull-back by the duality map of the induced metric on one is
the third fundamental form on the other. Moreover, the curvature $K^{*}$ of
the induced metric on $\Sigma^{*}$ at the point $x^{*}$ which is the image
under the duality map of a point $x\in\Sigma$ is equal to $K^{*}=-K/(1+K)$,
where $K$ is the curvature of the induced metric on $\Sigma$ at $x$. Through
this duality, Theorem 1.1 also implies that given a hyperbolic metric $h_{0}$
and another metric $h$ on $S$ with curvature $K<-1$, there is a unique
quasifuchsian AdS spacetime with left (or right) hyperbolic metric $h_{0}$,
and containing a future-convex (or past-convex) spacelike surface with third
fundamental form $h$.

###  1.2. Background and motivations

Our main motivation comes from hyperbolic geometry. As already mentioned,
quasifuchsian (that is, MGHC) AdS spacetimes have deep analogies with
quasifuchsian hyperbolic manifolds. There are several questions one can
consider in a quasifuchsian hyperbolic manifold, for instance whether one can
find a (unique) quasifuchsian hyperbolic manifold containing a (unique)
geodesically convex subset such that the induced metric (or third fundamental
form) on the boundary is prescribed, see [Sch20]. Here we are motivated by a
slightly different type of question, in the setting of hyperbolic ends. A
prime example of hyperbolic end is a connected component of the complement of
the convex core in a convex co-compact hyperbolic manifold, but the following
definition is somewhat more general.

######  Definition 1.2.

A hyperbolic end is a non-complete 3-dimensional hyperbolic manifold $(E,g)$,
where $E=S\times(0,\infty)$, complete on the side corresponding to $\infty$,
and having as metric completion a pleated concave surface at the side
corresponding to $0$.

Given a hyperbolic end $E$, we denote by $\partial_{0}E$ the pleated surface
boundary component corresponding to $0$, and by $\partial_{\infty}E$ the
asymptotic boundary component corresponding to $\infty$.

The asymptotic boundary $\partial_{\infty}E$ of $E$ is naturally equipped with
a conformal structure, and more precisely with a complex projective structure.
The pleated boundary $\partial_{0}E$ is equpped with a hyperbolic metric and
with a measured geodesic lamination measuring its “pleating”.

Such a hyperbolic end contains many embedded surfaces $S\subset E$ such that
$\partial_{0}E$ and $S$ bound together a relatively compact geodesically
convex subset $\Omega$ of $E$ – we call such a surface a locally convex
surface in $E$.

Given such a smooth, locally convex surface $S\subset E$, the hyperbolic Gauss
map – which sends a point $x\in S$ to the endpoint at infinity of the geodesic
ray starting from $x$, normal to $S$, towards the concave side – is a
homeomorphism from $S$ to $\partial_{\infty}E$.

We are interested in the following question and its dual, see also [Sch20].

######  Question 1.3.

Let $S$ be a closed surface of genus at least $2$, let $h$ be a smooth metric
of curvature $K>-1$ on $S$, and let $c\in{\mathcal{T}}_{S}$ be a conformal
structure on $S$. Is there a unique hyperbolic end $E$ containing a locally
convex surface $\Sigma$ homeomorphic to $S$, with induced metric isotopic to
$h$ and such that the pull-back by the hyperbolic Gauss map of the conformal
structure at infinity of $E$ is isotopic to $c$?

######  Question 1.4.

Let $S$ be a closed surface of genus at least $2$, let $h^{*}$ be a smooth
metric of curvature $K^{*}<1$ on $S$ such that all closed, contractible
geodesics of $(S,h^{*})$ have length strictly larger than $2\pi$, and let
$c\in{\mathcal{T}}_{S}$ be a conformal structure on $S$. Is there a unique
hyperbolic end $E$ containing a locally convex surface $\Sigma$ homeomorphic
to $S$, with third fundamental form isotopic to $h^{*}$ and such that the
pull-back by the hyperbolic Gauss map of the conformal structure at infinity
of $E$ is isotopic to $c$?

The answer to Question 1.3 is positive when $h$ has constant curvature $-1$,
this corresponds to the main result in [SW02]. When $h$ (resp. $h^{*}$) has
constant curvature $K\in(-1,0)$ (resp. $K^{*}\in(-\infty,0)$), Questions 1.3
and 1.4 correspond to Questions 9.1 and 9.2 in [BMS13]. The existence part is
proved for both questions in Theorem 1.9 and Theorem 1.10 of [BMS15].

Question 1.4 also has a positive answer in the “limit” case where $h^{*}$ is
replaced by a measured lamination, this is the main result of [DW08]. In
another limit case where $h^{*}$ corresponds to the pleating pattern
associated to a circle packing or circle pattern, Question 1.4 corresponds to
a well-known conjecture of Kojima, Mizushima and Tan, see [KMT03, KMT06b,
KMT06a, SY18].

In the analogy discovered by Mess [Mes07, ABB+07], the conformal metrics at
infinity of a quasifuchsian hyperbolic manifold correspond to the left and
right hyperbolic metrics of a quasifuchsian (or MGHC) AdS spacetime. In this
sense, Theorem 1.1 provides a positive answer to the AdS analog of Question
1.3, and also, through the duality between strongly convex spacelike surfaces
in quasifuchsian AdS spacetimes mentioned above, to the AdS analog of Question
1.4.

###  1.3. Outline of the paper

Section 2 contains background definition and results on AdS geometry, on
quasifuchsian AdS spacetimes, as well as on geometric data on immersed
surfaces in AdS spacetimes. The proof of the main result is in Section 3,
while Section 4 singles out an application.

##  2\. Preliminaries

###  2.1. The 3-dimensional anti-de Sitter geometry

Let ${\mathbb{R}}^{2,2}$ denote the real 4-dimensional vector space
${\mathbb{R}}^{4}$ equipped with a symmetric bilinear form
$\langle\cdot,\cdot\rangle_{2,2}$ of signature $(2,2)$, where $\langle
x,y\rangle_{2,2}:=x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3}-x_{4}y_{4}$. The quadric
model of the 3-dimensional anti-de Sitter (AdS) space is

$AdS^{3}:=\\{x\in{\mathbb{R}}^{2,2}~{}|~{}\langle x,x\rangle_{2,2}=-1\\}~{},$

equipped with a pseudo-Riemannian metric of signature $(2,1)$, which is
induced by restricting the bilinear form $\langle\cdot,\cdot\rangle_{2,2}$ to
the tangent space at each point of of $AdS^{3}$. It is a 3-dimensional
Lorentzian symmetric space of constant curvature $-1$ diffeomorphic to
${\mathbb{H}}^{2}\times\mathbb{S}^{1}$.

The _projective model_ $\mathbb{A}\rm{d}\mathbb{S}^{3}$ of 3-dimensional AdS
space is defined as the image of $AdS^{3}$ under the projection
$\pi:\mathbb{R}^{2,2}\backslash\\{0\\}\rightarrow\mathbb{RP}^{3}$, with the
metric induced from the projection. It is clear that the quadric model
$AdS^{3}$ is a double cover of the projective model
$\mathbb{A}\rm{d}\mathbb{S}^{3}$.

The boundary of $\mathbb{A}\rm{d}\mathbb{S}^{3}$, denoted by
$\partial\mathbb{A}\rm{d}\mathbb{S}^{3}$, is the image of
$\\{x\in\mathbb{R}^{2,2}:\langle x,x\rangle_{2,2}=0\\}$ under $\pi$, which is
foliated by two families of projective lines, called the left and right
leaves, respectively.

The geodesics in $\mathbb{A}\rm{d}\mathbb{S}^{3}$ are given by the
intersection of projective lines in ${\mathbb{RP}}^{3}$ with
$\mathbb{A}\rm{d}\mathbb{S}^{3}$: the spacelike geodesics correspond to the
projective lines intersecting the boundary
$\partial\mathbb{A}\rm{d}\mathbb{S}^{3}$ in two points, while lightlike
geodesics are tangent to $\partial\mathbb{A}\rm{d}\mathbb{S}^{3}$, and
timelike geodesics are disjoint from $\partial\mathbb{A}\rm{d}\mathbb{S}^{3}$.

The isometry group of $AdS^{3}$ is the indefinite orthogonal group ${\rm
O}(2,2)$ of reversible linear transformations on ${\mathbb{R}}^{2,2}$
preserving the bilinear form $\langle\cdot,\cdot\rangle_{2,2}$. The group of
orientation and time-orientation preserving isometries of
$\mathbb{A}\rm{d}\mathbb{S}^{3}$, denoted by
$\rm{Isom}_{0}(\mathbb{A}\rm{d}\mathbb{S}^{3})$, is the identity component of
${\rm PO}(2,2)={\rm O}(2,2)/\\{\pm{\rm Id}\\}$, which can be identified with
$\rm{PSL}(2,{\mathbb{R}})\times\rm{PSL}(2,{\mathbb{R}})$. One simple way to
see this is because the boundary $\partial\mathbb{A}\rm{d}\mathbb{S}^{3}$ is
foliated by two families of lines, and those two foliations are invariant
under any element of $\rm{Isom}_{0}(\mathbb{A}\rm{d}\mathbb{S}^{3})$, since
isometries act projectively in the projective model. The set of leaves of each
foliation is equipped with a real projective structure by the intersections
with any leave of the other foliation, and the action of
$\rm{Isom}_{0}(\mathbb{A}\rm{d}\mathbb{S}^{3})$ defines in this way two
elements of $\rm{PSL}(2,{\mathbb{R}})$.

We always use the projective model $\mathbb{A}\rm{d}\mathbb{S}^{3}$ throughout
the paper. It is interesting to notice that $\mathbb{A}\rm{d}\mathbb{S}^{3}$
has a Lie group model, in which $\mathbb{A}\rm{d}\mathbb{S}^{3}$ is identified
with $\rm{PSL}(2,{\mathbb{R}})$. More details about the geometry of anti-de
Sitter space can be found in e.g. [BS20, Mes07, ABB+07, BBD+12].

###  2.2. Quasifuchsian AdS spacetimes

An AdS spacetime is a Lorentzian 3-manifold locally isometric to
$\mathbb{A}\rm{d}\mathbb{S}^{3}$. An AdS spacetime $M$ is maximal globally
hyperbolic and spatially compact (MGHC) if

  * •

$M$ is globally hyperbolic and spatially compact (GHC): $M$ contains a closed
Cauchy surface (i.e. a spacelike surface intersecting each inextensible
timelike curve exactly once).

  * •

any isometric embedding of $M$ into a GHC AdS spacetime is an isometry.

It is well-known that a globally hyperbolic spacetime is topologically a
product of any of its Cauchy surfaces with an interval, see e.g. [BEE96,
Chapter 3]. MGHC AdS spacetimes have been shown by G. Mess [Mes07, ABB+07] to
present remarkable analogies with quasifuchsian hyperbolic manifolds, and they
are now often called quasifuchsian AdS spacetimes in the mathematics
literature.

It was proved by Mess [Mes07] that given any MGHC AdS spacetime with Cauchy
surfaces homeomorphic to $S$, its holonomy representation
$\rho:\pi_{1}(S)\rightarrow\rm{Isom}_{0}(\mathbb{A}\rm{d}\mathbb{S}^{3})=\rm{PSL}(2,{\mathbb{R}})\times\rm{PSL}(2,{\mathbb{R}})$
can be written as $\rho=(\rho_{L},\rho_{R})$, where
$\rho_{L},\rho_{R}:\pi_{1}(S)\rightarrow{\rm PSL}(2,{\mathbb{R}})$ have
maximal Euler number, and are therefore (by a result of Goldman [Gol88])
Fuchsian representations (i.e. the holonomy representations of hyperbolic
metrics on $S$). So the space $\mathcal{QF}_{AdS}(S)$ appearing in Section 1.2
is exactly the deformation space of MGHC AdS spacetimes with Cauchy surfaces
homeomorphic to $S$.

A quasifuchsian AdS spacetime $M\in\mathcal{QF}_{AdS}(S)$ contains a non-empty
closed subset $N$ which is geodesically convex (i.e. each geodesic segment of
$M$ connecting any two points of $N$ is contained in $N$). In particular, the
smallest non-empty geodesically convex closed subset of $M$ is called the
convex core of $M$, denoted by $C_{M}$. The boundary of $C_{M}$ is the union
of two (possibly identified) spacelike surfaces. In the Fuchsian case (i.e.
the left and right metrics are equal), $C_{M}$ degenerates to a totally
geodesic spacelike surface. In the non-Fuchsian case, each boundary component
of $C_{M}$ is a spacelike surface pleated along a measured geodesic
lamination, whose measures record the bending angles along totally geodesic
pieces. In both cases, the induced metrics on the two boundary components of
$C_{M}$ are hyperbolic.

###  2.3. The linearized Gauss and Codazzi equations

Let $(M,g)$ be a quasifuchsian AdS spacetime and let $\Sigma\subset M$ be an
(embedded) spacelike surface with induced metric $I$. The shape operator of
$\Sigma$ is the bundle morphism $B:T\Sigma\rightarrow T\Sigma$ defined by

$B(u)=-\nabla_{u}n~{},$

where $n$ is the future-directed unit normal vector field on $\Sigma$, $u$ is
a tangent vector field to $\Sigma$ and $\nabla$ is the Levi-Civita connection
of $g$. The second and third fundamental forms of $\Sigma$ are respectively
defined by

$II(u,v)=I(Bu,v)~{},\qquad III(u,v)=I(Bu,Bv)~{},$

for all $u,v\in T_{p}\Sigma$ with $p\in\Sigma$. We call the pair $(I,B)$ of
induced metric and shape operator of $\Sigma$ the embedding data of $\Sigma$.

It is well-known that $(I,B)$ satisfies the following Gauss and Codazzi
equations respectively:

$\begin{split}-1-\det(B)&=K~{},\\\ d^{D}B&=0~{},\end{split}$

where $K$ is the Gaussian curvature of $I$, $D$ is the Levi-Civita connection
of $I$, while the exterior derivative $d^{D}$ of $B$ (considered as a one-form
with values in $T\Sigma$) is defined as

$(d^{D}B)(u,v)=D_{u}(Bv)-D_{v}(Bu)-B([u,v])~{},$

for any two tangent vector fields $u$, $v$ on $\Sigma$.

Take the first-order derivatives of both sides of the Gauss equation and
assume that $B$ is non-degenerate. A direct computation shows that the shape
operator $B$ satisfies the following

${\rm tr}(B^{-1}\dot{B})=0~{},$

called the linearized Gauss equation.

A spacelike surface $\Sigma$ in $M$ is said to be strongly convex if the
determinant of the shape operator $B$ of $\Sigma$ is positive definite. In
particular, $\Sigma$ is said to be strongly future-convex (resp. strongly
past-convex) if the principal curvatures of $B$ are both negative (resp.
positive).

###  2.4. The left and right metrics on a spacelike surface

Let $\Sigma$ be an embedded (smooth) spacelike surface in a quasifuchsian AdS
spacetime $M$, with the embedding data $(I,B)$. Let $J$ be the complex
structure on $\Sigma$ defined by $I$. It was shown in [KS07, Lemma 3.16] that
the left and right metrics of $M$, denoted by $I^{\\#}_{+}$ and $I^{\\#}_{-}$,
can be defined alternatively by

$I^{\\#}_{\pm}=I((E\pm JB)\cdot,(E\pm JB)\cdot)~{},$

in the sense that if $I^{\\#}_{\pm}$ are defined in this manner as Riemannian
metrics on $\Sigma$, they are isometric to $h_{L}$ and $h_{R}$, respectively.

Moreover, this definition is independent on the choice of embedded smooth
spacelike surfaces. Since the left and right metrics play symmetric roles in
our question, without loss of generality, we only consider the left metric of
$M$ here.

##  3\. Proof of the main result

Let $(M,g)\in\mathcal{QF}_{AdS}(S)$ be a quasifuchsian AdS spacetime $M$ (with
its metric denoted by $g$) which has left metric $h_{L}$ and contains an
embedded smooth, strongly past-convex spacelike surface $\Sigma$ with induced
metric $I$ (and with the metric $I^{\\#}_{+}$ isotopic to $h_{L}$).

###  3.1. Local rigidity

In this section, we aim to show the infinitesimal rigidity of $g$ with respect
to the left metric $I^{\\#}_{+}$ and the induced metric $I$ on $\Sigma$, as
stated below.

######  Proposition 3.1.

Any first order deformation of $g\in\mathcal{QF}_{AdS}(S)$ which preserves the
left metric $I^{\\#}_{+}$ and the induced metric $I$ on the strongly past-
convex spacelike surface $\Sigma$ at first order is trivial.

We first consider the first-order deformation of the embedding data $(I,B)$ of
the strongly past-convex spacelike surface $\Sigma$.

######  Lemma 3.2.

Let $(I,B)$ be the induced metric and shape operator of a strongly past-convex
spacelike surface $\Sigma$ in a quasifuchsian AdS spacetime. Consider a first-
order deformation such that $I$ is fixed and $\dot{B}$ satisfies the Codazzi
and linearized Gauss equations and that the corresponding variation of
$I^{\\#}_{+}$ is trivial, i.e. induced by a vector field say $v$ on $\Sigma$.
Then $\dot{B}=0$.

Let $(I_{t},B_{t})_{t\in[0,\epsilon]}$ be a smooth one-parameter family of
embedding data with $(I_{0},B_{0})=(I,B)$, such that the first-order variation
$\dot{I}=I^{\prime}_{0}$ of $I_{t}$ at $t=0$ vanishes. Let $J_{t}$ denote the
complex structure on $\Sigma$ defined by $I_{t}$. It follows directly that
$J_{0}=J$ and $\dot{J}=0$. The first-order variation of the left metric
$I^{\\#}_{+}$ satisfies that

$\begin{split}\dot{I}^{\\#}_{+}&=\frac{d}{dt}|_{t=0}I_{t}((E+J_{t}B_{t})\cdot,(E+J_{t}B_{t})\cdot)\\\
&=I(J\dot{B},(E+JB)\cdot)+I((E+JB)\cdot,J\dot{B}\cdot)\\\
&=I^{\\#}_{+}((E+JB)^{-1}J\dot{B}\cdot,\cdot)+I^{\\#}_{+}(\cdot,(E+JB)^{-1}J\dot{B}\cdot)~{}.\end{split}$

######  Definition 3.3.

We define a bundle morphism $b:T\Sigma\rightarrow T\Sigma$ as

$b=(E+JB)^{-1}J\dot{B}~{}.$

Then the expression of $\dot{I}^{\\#}_{+}$ can be rewritten as

(1)
$\dot{I}^{\\#}_{+}=I^{\\#}_{+}(b\cdot,\cdot)+I^{\\#}_{+}(\cdot,b\cdot)~{}.$

######  Lemma 3.4.

Let $b$ be the bundle morphism defined in Definition 3.3, then

(2) ${\rm tr}((E+JB)b)=0~{},$ (3) ${\rm tr}((E+(JB)^{-1})b)=0~{}.$

###### Proof.

By definition, $b=(E+JB)^{-1}J\dot{B}$, so $J\dot{B}=(E+JB)b$. Note that $B$
is self-adjoint for $I$, so since $\dot{B}$ is a first-order variation of $B$
still satisfying this condition, $\dot{B}$ is also self-adjoint for $I$.
Therefore ${\rm tr}(J\dot{B})=0$, so ${\rm tr}((E+JB)b)=0$, which proves (2).

In addition, since $\Sigma$ is strongly convex, the shape operator $B$ of
$\Sigma$ is non-degenerate. Using $b=(E+JB)^{-1}J\dot{B}$, the linearized
Gauss equation ${\rm tr}(B^{-1}\dot{B})=0$ then translates as

${\rm tr}(-B^{-1}J(E+JB)b)=0~{}.$

Since $(JB)^{-1}=-B^{-1}J$, this can be written as

${\rm tr}((E+(JB)^{-1})b)=0~{},$

proving (3). ∎

######  Remark 3.5.

Since ${\rm tr}(JB)=0$, the Cayley-Hamilton theorem shows that
$(JB)^{2}+\det(JB)E=0$. Since $\det(JB)=\det(B)$ and using the Gauss equation
$-1-\det(B)=K$ (where $K$ is the Gaussian curvature of $I$), it follows that

$JB=(1+K)(JB)^{-1}~{}.$

Using this remark, equations (2) and (3) can be simplified and we arrive at
the following statement.

######  Lemma 3.6.

The system of equations (2) and (3) is equivalent to the following

(4) ${\rm tr}(b)=0~{},$ (5) ${\rm tr}(JBb)=0~{}.$

###### Proof.

We first show that the system of equations (2) and (3) implies that of (4) and
(5). Since the surface $\Sigma$ is strongly convex, we have $K=-1-\det(B)<-1$,
which ensures that $K+1<0$. Using Remark 3.5, equation (3) can be written as

(6) ${\rm tr}((E+\frac{JB}{K+1})b)=0~{}.$

Taking a linear combination of (2) and of (6) shows (4) and (5).

The other direction for the equivalence is clear using Remark 3.5. ∎

For the convenience of computation, we introduce the following lemma, see e.g.
[KS07, Proposition 3.12].

######  Lemma 3.7.

Let $\Sigma$ be a surface with a Riemann metric $g$. Let $A:T\Sigma\rightarrow
T\Sigma$ be a bundle morphism such that $A$ is everywhere invertible and
$d^{\nabla}A=0$, where $\nabla$ is the Levi-Civita connection of $g$. Let $h$
be the symmetric (0,2)-tensor defined by $h=g(A\cdot,A\cdot)$. Then the Levi-
Civita connection of $h$ is given by

$\nabla^{h}_{u}(v)=A^{-1}\nabla_{u}(Av)~{},$

and its curvature is given by

$K_{h}=\frac{K_{g}}{\det(A)}~{}.$

######  Claim 3.8.

The bundle morphism $b$ defined in Definition 3.3 satisfies the Codazzi
equation:

$d^{D^{\\#}}b=0~{},$

where $D^{\\#}$ is the Levi-Civita connection of $I^{\\#}_{+}$.

###### Proof.

By Lemma 3.7, the Levi-Civita connection of
$I^{\\#}_{+}=I((E+JB)\cdot,(E+JB)\cdot)$ is

$D^{\\#}_{u}v=(E+JB)^{-1}D_{u}((E+JB)v)~{}.$

Combined this with the definition of $d^{D^{\\#}}b$, we obtain

$\begin{split}(d^{D^{\\#}}b)(u,v)&=D^{\\#}_{u}(bv)-D^{\\#}_{v}(bu)-b([u,v])\\\
&=(E+JB)^{-1}D_{u}((E+JB)bv)-(E+JB)^{-1}D_{v}((E+JB)bu)-b([u,v])\\\
&=(E+JB)^{-1}D_{u}(J\dot{B}v)-(E+JB)^{-1}D_{v}(J\dot{B}u)-(E+JB)^{-1}J\dot{B}([u,v])\\\
&=(E+JB)^{-1}\big{(}D_{u}(J\dot{B}v)-D_{v}(J\dot{B}u)-J\dot{B}([u,v])\big{)}\\\
&=(E+JB)^{-1}J(d^{D}\dot{B})(u,v)\\\ &=0~{}.\end{split}$

The third equality uses the definition of $b$, while the last equality uses
the fact that $B_{t}$ satisfies the Codazzi equation. ∎

######  Claim 3.9.

Under the assumptions of Lemma 3.2, there exists a function
$\mu:\Sigma\to{\mathbb{R}}$ such that the bundle morphism $b$ defined in
Definition 3.3 satisfies

(7) $b=D^{\\#}v+\mu J^{\\#}~{},$

where $J^{\\#}$ is the complex structure of $I^{\\#}_{+}$.

###### Proof.

Under the assumptions of Lemma 3.2, the variation of $I^{\\#}_{+}$ is induced
by a vector field $v$ on $\Sigma$, that is,
$\dot{I}^{\\#}_{+}={\mathcal{L}}_{v}I^{\\#}_{+}$. A direct computation shows
that

${\mathcal{L}}_{v}I^{\\#}_{+}=I^{\\#}_{+}(D^{\\#}_{\cdot}v,\cdot)+I^{\\#}_{+}(\cdot,D^{\\#}_{\cdot}v)~{}.$

Combining (1):
$\dot{I}^{\\#}=I^{\\#}_{+}(b\cdot,\cdot)+I^{\\#}_{+}(\cdot,b\cdot)$, this
means that $b$ and $D^{\\#}v$ have the same self-adjoint component, or in
other terms $b-D^{\\#}v$ is anti-self-adjoint for $I^{\\#}$. So there exists a
function $\mu:\Sigma\to{\mathbb{R}}$ such that

$b-D^{\\#}v=\mu J^{\\#}~{}.$

The claim follows. ∎

The Codazzi equation applied to $b$ and the fact that $I^{\\#}_{+}$ has
constant curvature $-1$ then lead to the following lemma.

######  Lemma 3.10.

The vector field $v$ on $\Sigma$ and the function $\mu:\Sigma\to{\mathbb{R}}$
in Claim 3.9 satisfy

$v+J^{\\#}D^{\\#}\mu=0~{}.$

###### Proof.

We compute $d^{D^{\\#}}$ of each term of the right-hand side of (7). For all
$x,y\in T_{p}\Sigma$ with $p\in\Sigma$,

$\displaystyle(d^{D^{\\#}}D^{\\#}v)(x,y)$ $\displaystyle=$ $\displaystyle
D^{\\#}_{x}D^{\\#}_{y}v-D^{\\#}_{y}D^{\\#}_{x}v-D^{\\#}_{[x,y]}v$
$\displaystyle=$ $\displaystyle(-K^{\\#}J^{\\#}v)da^{\\#}(x,y)$
$\displaystyle=$ $\displaystyle(J^{\\#}v)da^{\\#}(x,y)~{},$
$\displaystyle(d^{D^{\\#}}\mu J^{\\#})(x,y)$ $\displaystyle=$ $\displaystyle
D^{\\#}_{x}(\mu J^{\\#}y)-D^{\\#}_{y}(\mu J^{\\#}x)-\mu J^{\\#}[x,y]$
$\displaystyle=$ $\displaystyle d\mu(x)J^{\\#}y-d\mu(y)J^{\\#}x~{}.$
$\displaystyle=$ $\displaystyle-(D^{\\#}\mu)da^{\\#}(x,y)~{},$

where $K^{\\#}$ is the Gaussian curvature of $I^{\\#}_{+}$ and $da^{\\#}$ is
the area element of $I^{\\#}_{+}$.

Putting the two terms together and combining with Claim 3.8 and (7), the
result follows. ∎

Replacing this in the expression (7) of $b$ then leads to the following.

######  Claim 3.11.

Under the assumptions of Lemma 3.2, the bundle morphism $b$ defined in
Definition 3.3 has the following expression:

(8) $b=J^{\\#}(-D^{\\#}D^{\\#}\mu+\mu E)~{},$

where $\mu$ is the function given in Claim 3.9.

######  Lemma 3.12.

Under the assumptions of Lemma 3.2, we have

(9) ${\rm tr}(JBJ^{\\#}(-D^{\\#}D^{\\#}\mu+\mu E))=0~{}.$

###### Proof.

This follows directly from Equation (5), with the expression of $b$ in the
previous claims. ∎

It can be noted that (8) implies that $b$ is traceless, so (4) is always
satisfied.

######  Lemma 3.13.

The operator $JBJ^{\\#}$ in (9) is self-adjoint for $I^{\\#}_{+}$ and negative
definite. More precisely, if $(e_{1},e_{2})$ is an orthonormal basis for $I$
of eigenvectors of $B$, with eigenvalues $k_{1},k_{2}$ then
$((E+JB)^{-1}(e_{1}),(E+JB)^{-1}(e_{2}))$ is an orthonormal basis of
$I^{\\#}_{+}$ of eigenvectors of $JBJ^{\\#}$, with eigenvalues
$-k_{2},-k_{1}$.

###### Proof.

Recall that $J^{\\#}=(E+JB)^{-1}J(E+JB)$ and
$I^{\\#}_{+}=I((E+JB)\cdot,(E+JB)\cdot)$. Since $B$ is self-adjoint for $I$,
and $J$ is the complex structure of $I$, a direct computation shows that

$I^{\\#}_{+}(JBJ^{\\#}\cdot,\cdot)=I^{\\#}_{+}(\cdot,JBJ^{\\#}\cdot)~{},$

which means that $JBJ^{\\#}$ is self-adjoint for $I^{\\#}_{+}$.

Let $E_{i}=(E+JB)^{-1}(e_{i}),i=1,2$. Then

$J^{\\#}E_{i}=(E+JB)^{-1}Je_{i}~{}.$

However, $JB$ commutes with $E+JB$ and therefore also with $(E+JB)^{-1}$. As a
consequence,

$JBJ^{\\#}E_{i}=JB(E+JB)^{-1}Je_{i}=(E+JB)^{-1}JBJe_{i}~{}.$

Clearly, $JBJe_{1}=-k_{2}e_{1}$ while $JBJe_{2}=-k_{1}e_{2}$. Then

$JBJ^{\\#}E_{1}=-k_{2}(E+JB)^{-1}e_{1}=-k_{2}E_{1}~{},$

and

$JBJ^{\\#}E_{2}=-k_{1}(E+JB)^{-1}e_{2}=-k_{1}E_{2}~{}.$

The result follows. ∎

######  Remark 3.14.

Another (simpler) way to obtain the result is by checking that

$JBJ^{\\#}=(E+JB)^{-1}JBJ(E+JB)~{}.$

As a consequence, the operator $\mu\mapsto{\rm
tr}(JBJ^{\\#}(-D^{\\#}D^{\\#}\mu+\mu E))$ is elliptic. This makes it possible
to apply the maximum principle and obtain the infinitesimal rigidity statement
needed.

######  Corollary 3.15.

Under the assumptions of Lemma 3.2, there is no non-zero solution of (9).

###### Proof.

Let $\mu$ be a solution of (9). Since $\mu$ is a smooth function on a closed
surface $\Sigma$, $\mu$ achieves the maximum (resp. minimum) on $\Sigma$, say
at $x_{0}$ (resp. $y_{0}$), and thus has a local maximum (resp. minimum) at
$x_{0}$ (resp. $y_{0}$). By Lemma 3.13, at the local maximum $x_{0}$ of $\mu$,
${\rm tr}(JBJ^{\\#}(-D^{\\#}D^{\\#}\mu))\leq 0$. Combined with (9), we have
${\rm tr}(JBJ^{\\#}(\mu E))\geq 0$ at $x_{0}$. Using Lemma 3.13 again, it
follows that $\mu(x_{0})\leq 0$ and so $\mu\leq 0$ everywhere.

On the other hand, at the local minimum $y_{0}$ of $\mu$, ${\rm
tr}(JBJ^{\\#}(-D^{\\#}D^{\\#}\mu))\geq 0$. Similarly as above, we can show
that $\mu(y_{0})\geq 0$ and so $\mu\geq 0$ everywhere. So $\mu=0$. ∎

###### Proof of Lemma 3.2.

Combining Corollary 3.15, Definition 3.3 and (8), we have
$J\dot{B}=(E+JB)b=0.$ Lemma 3.2 follows. ∎

Before showing Proposition 3.1, we introduce the following proposition which
ensures the existence and the uniqueness (up to isometries) of the maximal
extension of a GHC AdS spacetime (see [CBG69, Theorem 3]).

######  Proposition 3.16.

Let $(M,g)$ be a GHC AdS spacetime. There exists a unique (considered up to
isometries) MGHC AdS spacetime $(M^{\prime},g^{\prime})$ with particles,
called the maximal extension of $(M,g)$, in which $(M,g)$ can be isometrically
embedded.

We are now ready to prove Proposition 3.1.

###### Proof of Proposition 3.1.

Let $\dot{g}$ be a first-order deformation of $(M,g)\in\mathcal{QF}_{AdS}(S)$
which preserves the left metric $I^{\\#}_{+}$ and the induced metric $I$ on
the strongly past-convex spacelike surface $\Sigma$ at first order. Assume
that $\dot{g}$ is given by a one-parameter family
$(M,g_{t})_{t\in[0,\epsilon]}$ of quasifuchsian AdS spacetimes in
$\mathcal{QF}_{AdS}(S)$. The induced left metrics are denoted by
$I^{\\#}_{+,t}$, and the induced embedding data of $\Sigma$ are denoted by
$(I_{t},B_{t})$. Up to a diffeomorphism of $M$ isotopic to the identity, we
can assume that the induced metric $I_{t}$ on $\Sigma$ is fixed at first order
(i.e. $\dot{I}=0$) and the first-order variation of $I^{\\#}_{+}$ is induced
by a vector field say $v$ on $\Sigma$ (which means that
$\dot{I}^{\\#}_{+}={\mathcal{L}}_{v}I^{\\#}_{+}$). Then the assumptions of
Lemma 3.2 are satisfied and we have $\dot{B}=0$.

To show that $\dot{g}=0$, we first claim that the embedding data
$(\Sigma,I_{t},B_{t})$ uniquely determines a quasifuchsian AdS metric (which
is $g_{t}$). Indeed, consider the manifold $\Sigma\times(-\frac{\pi}{2},0]$
with the following metric:

$h_{t}=-ds^{2}+I_{t}((\cos(s)E+\sin(s)B_{t})\cdot,(\cos(s)E+\sin(s)B_{t})\cdot)~{},$

where $E$ is the identity isomorphism on $T\Sigma$ and
$s\in(-\frac{\pi}{2},0]$. Note that for each $s\in(-\frac{\pi}{2},0]$, the
surface $\Sigma\times\\{s\\}$ is the equidistant surface at distance $s$ from
the surface $\Sigma\times\\{0\\}$ on the convex side. The Lorentzian metric
$h_{t}$ is a GHC AdS metric on $\Sigma\times(-\frac{\pi}{2},0]$.

By Proposition 3.16, there exists a unique maximal extension of the AdS
spacetime $(\Sigma\times(-\frac{\pi}{2},0],h_{t})$, which is a MGHC AdS
spacetime such that the restriction of its metric to the
$\Sigma\times(-\frac{\pi}{2},0]$ (identified with a subset of $M$) is exactly
$h_{t}$. Moreover, this maximal extension is $(M,g_{t})$. Note that
$\dot{I}=\dot{B}=0$. Taking the derivative of $h_{t}$ at $t=0$ shows that
$\dot{h}=0$ and $\dot{g}$ is therefore zero when restricted to the subset
$\Sigma\times(-\frac{\pi}{2},0]$. By the uniqueness of the maximal extension,
$\dot{g}$ is therefore zero on the whole $M$. The proposition follows. ∎

###  3.2. Proof of Theorem 1.1

To show Theorem 1.1, we first construct the following map.

Let $h$ be a complete Riemannian metric of curvature $K<-1$ on $S$ and let
$\mathcal{V}(h)\subset\mathcal{QF}_{AdS}(S)$ denote the space of quasifuchsian
AdS spacetimes which contains a past-convex spacelike surface with induced
metric isotopic to $h$.

######  Definition 3.17.

Let $\Phi_{h}:\mathcal{V}(h)\rightarrow{\mathcal{T}}_{S}$ be the map which
sends a manifold $M\in\mathcal{V}(h)$ to its left metric $h_{L}$, identified
with the metric $I^{\\#}_{+}$ on $S$.

We aim to show that $\Phi_{h}$ is a homeomorphism. Note that $\mathcal{V}(h)$
can be alternatively interpreted as the space of isometric equivariant
embeddings of $(S,h)$ into $\mathbb{A}\rm{d}\mathbb{S}^{3}$ (denoted by
$\mathcal{I}(S,h)$, see e.g. [Tam18, Section 3]), which are given by a couple
$(f,\rho)$, where
$f:(\tilde{S},\tilde{h})\rightarrow\mathbb{A}\rm{d}\mathbb{S}^{3}$ is an
isometric embedding into $\mathbb{A}\rm{d}\mathbb{S}^{3}$ with
$(\tilde{S},\tilde{h})$ a universal Riemannian cover of $(S,h)$, and
$\rho:\pi_{1}(S)\rightarrow{\rm PSL(2,{\mathbb{R}})\times{\rm
PSL}(2,{\mathbb{R}})}$ is the representation such that $f(\gamma
x)=\rho(\gamma)\circ f(x)$ for all $\gamma\in\pi_{1}(S)$ and $x\in\tilde{S}$.

It was shown by Tamburelli [Tam18, Lemma 3.2] that $\mathcal{I}(S,h)$, and
thus the space $\mathcal{V}(h)$ here, is a manifold of dimension
$6\mathfrak{g}-6$, where $\mathfrak{g}$ is the genus of $S$, by identifying
$\mathcal{I}(S,h)$ with the space of solutions of Gauss-Codazzi equations and
using classical techniques of elliptic operators. Moreover, he showed that
$\Phi_{h}$ is proper [Tam18, Corollary 7.4].

###### Proof of Theorem 1.1.

Note that $\mathcal{V}(h)$ has (real) dimension $6\mathfrak{g}-6$, which is
the same as that of ${\mathcal{T}}_{S}$. Combined with 3.1, the differential
$d\Phi_{h}$ is an isomorphism. So $\Phi_{h}$ is a local homeomorphism.
Combined with the fact that $\Phi_{h}$ is proper [Tam18, Corollary 7.4],
$\Phi_{h}$ is a covering map of degree say $d(h)$. It remains to show that
$d(h)=1$.

Let $h_{0}$ be a metric with curvature $K<-1$ on $S$, and let $h_{1}$ be
another metric on $S$, of constant curvature $K<-1$. As already mentioned, it
follows from [BMS13, Theorem 1.15] that
$\Phi_{h_{1}}:\mathcal{V}(h_{1})\to{\mathcal{T}}_{S}$ is one-to-one. Let
$(h_{t})_{t\in[0,1]}$ be a smooth one-parameter family of Riemannian metrics
on $S$ with curvature less than $-1$ (considered up to isotopy) connecting
$h_{0}$ to $h_{1}$.

Let

$\mathcal{V}=\\{(t,B_{t})~{}|~{}t\in[0,1],B_{t}\in\mathcal{V}(h_{t})\\}~{},$

and let $\Phi:\mathcal{V}\to[0,1]\times{\mathcal{T}}_{S}$ be the function
defined by

$\Phi(t,B_{t})=(t,\Phi_{h_{t}}(B_{t}))~{}.$

We claim that at each point $(t,B_{t})\in\mathcal{V}$, $d\Phi$ is an
isomorphism. Indeed, we know that $d\Phi_{|\\{0\\}\times
T_{B_{t}}\mathcal{V}(h_{t})}$ is an isomorphisms from $\\{0\\}\times
T_{B_{t}}\mathcal{V}(h_{t})$ to $\\{0\\}\times
T_{\Phi_{h_{t}}(B_{t})}{\mathcal{T}}_{S}$ – this is equivalent to the fact
that $d\Phi_{h_{t}}$ is an isomorphism, as seen above. Moreover, the
projection on the first factor in ${\mathbb{R}}\times
T_{h_{t}}{\mathcal{T}}_{S}$ of the image under $d\Phi$ of any vector of the
form $(1,\dot{B}_{t})$ is $1$. It follows that, in a suitable basis, $d\Phi$
at $(t,B_{t})$ is block triangular, with one $1\times 1$ block with entry $1$,
and one $(6\mathfrak{g}-6)\times(6\mathfrak{g}-6)$ which is invertible. So
$d\Phi$ at $(t,B_{t})$ is an isomorphism.

Since each $\Phi_{h_{t}},t\in[0,1]$ is proper, the function $\Phi$ is also
proper. It is therefore a covering map. It follows that $d(h_{0})=d(h_{1})=1$.
This applies to any metric $h_{0}$ on $S$ with curvature $K<-1$, and concludes
the proof. ∎

##  4\. Application

In this subsection, we apply Theorem 1.1 to give an alternative
parametrization of the space $\mathcal{QF}_{AdS}(S)$.

It is known that the deformation space $\mathcal{QF}_{AdS}(S)$ of
quasifuchsian AdS spacetimes can be parameterized in several ways, such as the
Mess parameterization [Mes07, BS20] by
${\mathcal{T}}_{S}\times{\mathcal{T}}_{S}$ in terms of the left and right
metrics (which is closely related to Thurston’s Earthquake Theorem for
hyperbolic metrics on $S$ [Ker83]), and the parametrization by
${\mathcal{T}}_{S}\times\mathcal{ML}_{S}$ in terms of the induced metric and
the bending lamination of the past (or future) boundary of the convex core
[BS09, Proposition 5.8].

Furthermore, $\mathcal{QF}_{AdS}(S)$ can also be parameterized by the
cotangent bundle $T^{*}{\mathcal{T}}_{S}$ of ${\mathcal{T}}_{S}$, in terms of
maximal spacelike surfaces in germs of AdS spacetimes (see [KS07, Lemma 3.3]),
or by ${\mathcal{T}}_{S}\times{\mathcal{T}}_{S}$ in terms of two hyperbolic
metrics homothetic respectively to the first and third fundamental forms of
past-convex constant Gaussian curvature $K$-surfaces, where
$K\in(-\infty,-1)$, see [BMS13, Lemma 1.9]. Recently, Mazzoli and Viaggi have
provided shear-bend coordinates for $\mathcal{QF}_{AdS}(S)$ [MV23].

Using the result of Theorem 1.1, we give an alternative parametrization of
$\mathcal{QF}_{AdS}(S)$ below, which can be viewed as a mixed version of Mess
parametrization and the parametrization in terms of the first and third
fundamental forms of constant Gaussian curvature surfaces. Since the left and
right metrics (resp. past-convex and future-convex spacelike surfaces, or the
first and third fundamental forms) play symmetric roles in the
parametrization, we only state the parametrization of $\mathcal{QF}_{AdS}(S)$
here in terms of its left metric and the induced metric on the past-convex
constant Gaussian curvature $K$-surfaces.

######  Definition 4.1.

Let $K\in(-\infty,-1)$. We define a map
$\phi_{K}:\mathcal{QF}_{AdS}(S)\rightarrow{\mathcal{T}}_{S}\times{\mathcal{T}}_{S}$
by taking a quasifuchsian AdS spacetime $M\in\mathcal{QF}_{AdS}(S)$ to its
left metric and the hyperbolic metric homothetic to the induced metric on an
embedded past-convex spacelike surface in $M$ of constant curvature $K$.

The map $\phi_{K}$ is well-defined, by the fact given by Barbot, Béguin and
Zeghib that each connected component of the complement of the convex core
$C_{M}$ in a quasifuchsian AdS spacetime $M$ admits a unique foliation by
spacelike surfaces of constant curvature $K$, with $K$ monotonic along the
foliation, varying from $-1$ near the past (resp. future) boundary component
of $C_{M}$ and $-\infty$ near the initial (resp. final) singularity of $M$,
see [BBZ11].

######  Proposition 4.2.

For each $K\in(-\infty,-1)$, the map $\phi_{K}$ is a homeomorphism.

###### Proof.

Fix $K\in(-\infty,-1)$. For any couple
$(h,h^{\prime})\in{\mathcal{T}}_{S}\times{\mathcal{T}}_{S}$, by applying the
proof of Theorem 1.1 to the manifold $\mathcal{V}((1/|K|)h^{\prime})$, there
is a unique $M\in\mathcal{QF}_{AdS}(S)$ which has left metric isotopic to $h$
and contains a past-convex spacelike surface with induced metric isotopic to
$(1/|K|)h^{\prime}$. So the map $\phi_{K}$ is a bijection.

As the projection to the first factor of the holonomy representation of $M$,
the left metric depends continuously on the geometric structures of $M$. Note
that the induced metric on a past-convex spacelike surface of constant
curvature $K$ depends continuously on the AdS metric of $M$, as can be seen
from the relation between the induced metrics on past-convex surfaces of
constant curvature in quasifuchsian AdS spacetimes and their left and right
metrics, see [BMS13, section 1.7]. Moreover, the space of isometric
equivariant embeddings of $(S,(1/|K|)h^{\prime})$ into
$\mathbb{A}\rm{d}\mathbb{S}^{3}$ is identified with the space of solutions of
Gauss-Codazzi equations, which depends smoothly on the initial data
$h^{\prime}\in{\mathcal{T}}_{S}$. As a consequence, the maps $\phi_{K}$ and
$\phi^{-1}_{K}$ are both continuous. Combined with the above bijection of
$\phi_{K}$, it follows that $\phi_{K}$ is a homeomorphism. ∎

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