##  1 Globally hyperbolic AdS manifolds and hyperbolic ends

We recall in this section some key notions concerning AdS geometry and more
specifically the geometry of globally hyperbolic AdS spacetimes. Additional
results can be found e.g. in [mess, mess-notes, barbot-merigot, maximal]. We
also present hyperbolic ends.

###  1.1 The anti-de Sitter space

Here we use the hyperboloid model of the anti-de Sitter space. Let $\R^{d,2}$
denote the real vector space $\R^{d+2}$ endowed with the standard quadratic
form $\mathbf{q}$ of signature $(d,2)$:

$\mathbf{q}(x)=x_{1}^{2}+\ldots+x_{d}^{2}-x_{d+1}^{2}-x_{d+2}^{2}~{}.$

We denote by $\langle\cdot,\cdot\rangle$ the associated bilinear form.

The anti-de Sitter space of dimension $d+1$ is the quadric:

$\AdS^{d+1}=\\{x\in\R^{d,2}\mid\mathbf{q}(x)=-1\\}~{}.$

The restriction of $\mathbf{q}$ to the tangent bundle of $\AdS^{d+1}$ endows
the anti-de Sitter space with a Lorentzian metric of constant sectional
curvature $-1$, which we denote by $g_{\AdS}$. This metric is homogeneous
under the action of the group $\mathrm{O}(d,2)$ of linear transformations of
$\R^{d,2}$ preserving $\mathbf{q}$. We denote by $\SO_{\circ}(d,2)$ the
connected component of the identity in $\mathrm{O}(d,2)$. This is an index 4
subgroup consisting of those isometries of $\AdS^{d+1}$ preserving an
orientation of space and time. We call it for short the group of orientation-
preserving isometries.

The subgroup of $\SO_{\circ}(d,2)$ fixing the point
$(0,\ldots,0,1)\in\AdS^{d+1}$ is the group $\SO_{\circ}(d,1)$ embedded via

$A\mapsto\left(\matrix{A}&\\\ 1\right)~{}.$

The (space and time-oriented) anti-de Sitter space of dimension $d+1$ can thus
be identified with the coset space

$\SO_{\circ}(d,2)/\SO_{\circ}(d,1)~{}.\\\ $

_Boundary._ The space $\AdS^{d+1}$ can also be identified with an open set of
the $d+1$-dimensional sphere, seen as the double cover of $\Proj{\R^{d,2}}$,
via the map

$\begin{array}[]{ccc}\AdS^{d+1}&\to&\bS^{d+1}=(\R^{d,2}\setminus\\{0\\})/\R_{>0}\Id\\\
x&\mapsto&\R_{>0}x~{}.\end{array}$

Its boundary in $\bS^{d+1}$ is called the _Einstein space_.

The Einstein space $\Ein^{d}$ is defined as

$\Ein^{d}=\partial_{\infty}\AdS^{d+1}=\\{x\in\R^{d,2}\backslash\\{0\\}\mid\mathbf{q}(x)=0\\}/\R_{>0}\Id~{}.$

The Einstein space carries a conformally flat Lorentz metric which is
conformally invariant under the action of $\mathrm{\SO}_{\circ}(d,2)$.

_Geodesics and causality in $\AdS^{d+1}$._ The geodesics of $\AdS^{d+1}$ are
its intersections with $2$-planes $P$ in $\R^{d,2}$. These are of three kinds:

  * •

If $\mathbf{q}_{|P}$ is negative definite, then $P\cap\AdS^{d+1}$ is an
ellipse. It is a _timelike geodesic_ , i.e. the Lorentz metric is negative
along that geodesic.

  * •

If $\mathbf{q}_{|P}$ is non-positive with $1$-dimensional kernel, then
$P\cap\AdS^{d+1}$ consists of two parallel affine lines, each of which is a
_lightlike geodesic_ , i.e. the Lorentz metric vanishes along that geodesic.

  * •

If $\mathbf{q}_{|P}$ has signature $(1,1)$, then $P\cap\AdS^{d+1}$ consists of
two branches of hyperbolas, each of which is a _spacelike geodesic_ , i.e. the
Lorentz metric is positive along that geodesic.

We call two points $x$ and $y$ in $\AdS^{d+1}$ _space (resp. light, time)
related_ if they belong to the same spacelike (resp. lightlike, timelike)
geodesic. We have the following characterization:

######  Proposition 1.

Two points $x$ and $y\in\AdS^{d+1}$ are

  * •

space related if and only if $\langle x,y\rangle<-1$,

  * •

light related if and only if $\langle x,y\rangle=-1$,

  * •

time related if and only if $-1<\langle x,y\rangle<1$.

######  Remark 1.1.

If $\langle x,y\rangle\geq 1$ then $x$ and $y$ do not belong to a common
geodesic, but $x$ and $-y$ are light or space related. In the projective model
$\AdS^{d+1}/\pm\Id$, any two points are either space, light, or time related.

_Photons and causality in $\Ein^{d}$._ A _photon_ in $\Ein^{d}$ is the
projectivisation of a totally isotropic $2$-plane in $\R^{d,2}$. We call two
points $[x]$ and $[y]\in\Ein^{d}$ _light related_ if they belong to the same
photon and _space related_ if they are the endpoints of a spacelike geodesic
in $\AdS^{d+1}$. We have again a characterization in terms of scalar products:

######  Proposition 2.

Two points $[x]$ and $[y]\in\Ein^{d}$ are

  * •

space related if and only if $\langle x,y\rangle<0$,

  * •

light related if and only if $\langle x,y\rangle=0$.

Note that $[x]$ is always space or light related to either $[y]$ or $[-y]$.
Causality thus does not make sense in the projective model
$\mathbb{E}\mathrm{in}^{d}=\Ein^{d}/\pm\Id$. A more robust notion is space
relation for triples of points.

We call a subset $S$ of $\Ein^{d}$ _acausal_ if any two points in $S$ are
space related, and _achronal_ if any two points in $S$ are space or light
related.

######  Proposition 3.

If $\\{[x],[y],[z]\\}\subset\Ein^{d}$ is acausal, then the restriction of
$\mathbf{q}$ to $\Span(x,y,z)$ has signature $(2,1)$.

Conversely, if the restriction of $\mathbf{q}$ to $\Span(x,y,z)$ has signature
$(2,1)$, then there exist unique $\epsilon_{y}$ and
$\epsilon_{z}\in\\{-1,1\\}$ such that
$\\{[x],[\epsilon_{y}y],[\epsilon_{z}z]\\}$ is acausal.

###  1.2 Spacelike hypersurfaces in $\AdS^{d+1}$

A smooth hypersurface $\mathcal{H}$ in $\AdS^{d+1}$ is called spacelike when
the restriction of the Lorentz metric to $\mathcal{H}$ is positive definite.
Here, we will construct hypersurfaces that are piecewise geodesic, and it is
thus useful to generalize this definition to a lower regularity.

Let $\mathcal{H}$ be a Lipschitz manifold of dimension $d$. Let
$d_{\mathcal{H}}$ be a Lipschitz distance on $\mathcal{H}$ (i.e. a distance
which is locally bi-Lipschitz to the Euclidean distance in local coordinates).

A map $i:\mathcal{H}\to\AdS^{d+1}$ is a _spacelike immersion_ if it is locally
Lipschitz and if every point in $\mathcal{H}$ has a neighbourhood $U$ such
that there exists a constant $c>0$ satisfying

$\langle i(p),i(p^{\prime})\rangle\leq-1-c\,d_{\mathcal{H}}^{2}(p,p^{\prime})$

for all $p,p^{\prime}\in U$.

A Lipschitz hypersurface $\mathcal{H}$ of $\AdS^{d+1}$ is called _spacelike_
if the inclusion $i:\mathcal{H}\to\AdS^{d+1}$ is a spacelike immersion.

By Proposition 1 the above condition implies that $i(p)$ is space related to
$i(p^{\prime})$ when $p,p^{\prime}\in\mathcal{H}$ are sufficiently close. The
constant $c$ prevents the hypersurface to be “tangent” to the light cone
through $p$. In particular, we have:

######  Proposition 4.

If $\mathcal{H}$ and $i$ are of class $\mathcal{C}^{1}$, then $i$ is a
spacelike immersion if and only if $i^{*}g_{\AdS}$ is positive definite at
every point.

Before we prove Proposition 4, let us interpret spacelike immersions in terms
of graphs in an appropriate model for $\AdS^{d+1}$. Consider the open
hemisphere $\bS^{d}_{+}=\\{(x_{0},\dots,x_{d})\in\bS^{d}\,|\,x_{0}>0\\}$. The
map

$\Phi:\left\\{\matrix{\bS}^{1}\times\bS^{d}_{+}&\to\AdS^{d+1}\\\
(\theta,x)\mapsto\left(\frac{x_{1}}{x_{0}},\dots,\frac{x_{d}}{x_{0}},\frac{\cos\theta}{x_{0}},\frac{\sin\theta}{x_{0}}\right)\right.$

is a diffeomorphism, and
$\Phi^{*}g_{\AdS}=x_{0}^{-2}(-d\theta^{2}+g_{\bS^{d}})$. An important feature
of this model is that $\bS^{d}_{+}$ equipped with $x_{0}^{-2}g_{\bS^{d}}$ is
isometric to the hyperbolic space $\HH^{d}$, in particular the hypersurfaces
$\theta={\rm cst}$ are totally geodesic copies of $\HH^{d}$. Another one is
that it extends to the boundary as a conformal map
$\partial\Phi:\bS^{1}\times\bS^{d-1}\to\partial\AdS^{d+1}=\Ein^{d}$.

######  Lemma 1.2.

Let $\mathcal{H}$ be a Lipschitz manifold, and $i:\mathcal{H}\to\AdS^{d+1}$ a
spacelike immersion. Write $i(p)=\Phi(\theta(p),x(p))$ for $p\in\mathcal{H}$.
Then the map $x:\mathcal{H}\to\bS^{d}_{+}$ is locally bi-Lipschitz.

######  Proof 1.3.

For $(\theta,x),(\theta^{\prime},x^{\prime})\in\bS^{1}\times\bS^{d}_{+}$ we
find ⟨Φ(θ,x),Φ(θ’,x’)⟩&= x1x’1+⋯+xdx’d-cos(θ-θ’)x0x’0
= 1-12∥x-x’∥2-cos(θ-θ’)x0x’0-1
≥-∥x-x’∥22x0x’0-1 This shows that for $p,p^{\prime}\in\mathcal{H}$, we have

$\|x(p)-x(p^{\prime})\|^{2}\geq-x_{0}(p)x_{0}(p^{\prime})(1+\langle
i(p),i(p^{\prime})\rangle$

Since $p\mapsto x_{0}(p)$ is continuous, it is locally bounded from below by
some $c^{\prime}>0$, and locally we find

$\|x(p)-x(p^{\prime})\|\geq\frac{\sqrt{c}}{c^{\prime}}d_{\mathcal{H}}(p,p^{\prime})$

Lemma 1.2 means that a Lipschitz spacelike hypersurface is locally a graph in
the conformal model $\AdS^{d+1}\approx\bS^{1}\times\bS^{d}_{+}$. Now given a
function from $\bS^{d}_{+}$ to $\bS^{1}$, we wish to know under which
condition its graph is a Lipschitz spacelike hypersurface.

######  Lemma 1.4.

Let $U\subset\bS^{d}_{+}$ be an open subset, and consider a map
$\theta:U\to\bS^{1}$. The map $i:U\to\AdS^{d+1}$ defined by
$i(x)=\Phi(\theta(x),x)$ is a spacelike immersion if and only if $\theta$ is
locally contracting (i.e. every point in $U$ has a neighbourhood on which
$\theta$ is $k$-Lipschitz for some $k\in(0,1)$).

######  Remark 1.5.

As a distance on $\bS^{d}_{+}$ we can pick either the spherical distance or
the Euclidean distance. The condition on $\theta$ being locally contracting
does not depend on this choice, since for every $\varepsilon>0$ we can find a
neighbourhood of any point on which they are $1+\varepsilon$-bi-Lipschitz. We
will use the Euclidean distance in the proof.

######  Proof 1.6 (Proof of Lemma 1.4).

For $x,x^{\prime}\in U$ we have

$\langle
i(x),i(x^{\prime})\rangle=\frac{1-\frac{1}{2}\|x-x^{\prime}\|^{2}-\cos(\theta(x)-\theta(x^{\prime}))}{x_{0}x^{\prime}_{0}}-1.$

If $\theta$ is $k$-Lipschitz for some $k\in(0,1)$, then up to shrinking $U$ we
obtain

$1-\cos(\theta(x)-\theta(x^{\prime}))\leq\frac{k}{2}\|x-x^{\prime}\|^{2}$

Since $x_{0},x^{\prime}_{0}\leq 1$, we find

$\langle
i(x),i(x^{\prime})\rangle\leq-1-\frac{1-k}{2}\|x-x^{\prime}\|^{2}~{},$

hence $i$ is a spacelike immersion.

Now assume that $i$ is a spacelike immersion, and let $c\in(0,\frac{1}{2})$ be
a constant given by the definition (note that $c$ can always be replaced by a
smaller constant). The map $\theta$ is continuous, so we can shrink $U$ in
order to have

$1-\cos(\theta(x)-\theta(x^{\prime}))\geq\frac{1-c}{2}|\theta(x)-\theta(x^{\prime})|^{2}$

for all $x,x^{\prime}\in U$. This in turn leads to

$|\theta(x)-\theta(x^{\prime})|^{2}\leq\underbrace{\frac{\frac{1}{2}-c}{\frac{1}{2}-\frac{c}{2}}}_{<1}\|x-x^{\prime}\|^{2}~{}.$

######  Proof 1.7 (Proof of Proposition 4).

If $i$ is $\mathcal{C}^{1}$, then the map $x:\mathcal{H}\to\bS^{d}_{+}$ from
Lemma 1.2 is bi-Lipschitz and $\mathcal{C}^{1}$, hence a local diffeomorphism.
So we may assume that $\mathcal{H}$ is an open subset of $\bS^{d}_{+}$ and
that $i(x)=\Phi(\theta(x),x)$ where $\theta:\mathcal{H}\to\bS^{1}$ is
$\mathcal{C}^{1}$. Now Lemma 1.4 shows that $i$ is a spacelike immersion if
and only if $\theta$ is locally contracting, which is equivalent to
$\|d\theta\|<1$.
But $i^{*}g_{\AdS}$ is in the same conformal class as
$-d\theta^{2}+g_{S^{d}}$, so it is positive definite if and only if
$\|d\theta\|<1$.

More generally, if $i:\mathcal{H}\to\AdS^{d+1}$ is a spacelike Lipschitz
immersion and $\gamma:[0,1]\to\mathcal{H}$ is a Lipschitz path, then
$i\circ\gamma$ is Lipschitz hence differentiable at Lebesgue almost every
point. The derivative of $i\circ\gamma$ is never timelike, so one can then
define the _length_ of $i(\gamma)$ as

$L_{\AdS}(i(\gamma))=\int_{0}^{1}\sqrt{g_{\AdS}((i\circ\gamma)^{\prime}(t),(i\circ\gamma)^{\prime}(t))}\mathrm{d}t~{}.$

Finally, for $x,y\in\mathcal{H}$, set

$i^{*}d_{\AdS}(x,y)=\inf_{\gamma(0)=x,\gamma(1)=y}L_{\AdS}(i\circ\gamma)~{}.$

The spacelike immersion property of $i$ implies that $i^{*}d_{\AdS}$ is a
Lipschitz distance on $\mathcal{H}$ (the usual proof for smooth Riemannian
metrics can be easily adapted thanks to the graph description of Lemma 1.2).
When $\mathcal{H}$ and $i$ are $\mathcal{C}^{1}$, it is simply the Riemannian
distance associated to $i^{*}g_{\AdS}$.

The spacelike immersion $i$ is called _complete_ when the distance
$i^{*}d_{\AdS}$ is complete.
A spacelike hypersurface $\mathcal{H}\subset\AdS^{d+1}$ is called _complete_
if the inclusion $i:\mathcal{H}\to\AdS^{d+1}$ is a complete spacelike
immersion.

The following proposition brings together a number of key properties of
spacelike hypersurfaces in $\AdS^{d+1}$.

######  Proposition 5.

Let $\mathcal{H}$ be a connected Lipschitz manifold of dimension $d$ and
$i:\mathcal{H}\to\AdS^{d+1}$ a complete spacelike immersion. Then

  * (1)

$\mathcal{H}$ is homeomorphic to the open ball $\mathrm{B}^{d}$ of dimension
$d$,

  * (2)

$i$ is an embedding,

  * (3)

$i(x)$ and $i(y)$ are space related for any $x\neq y\in\mathcal{H}$,

  * (4)

$i$ extends continuously to an achronal embedding

$\partial i:\bS^{d-1}=\partial\mathrm{B}^{d}\to\Ein^{d}~{}.\\\ $

######  Proof 1.8.

Write $i(p)=\Phi(\theta(p),x(p))$ for $p\in\mathcal{H}$. We have seen in Lemma
1.2 that $x:\mathcal{H}\to\bS^{d}_{+}$ is locally bi-Lipschitz. We can use it
to define another distance on $\mathcal{H}$ by defining the length of a
Lipschitz curve $\gamma:[0,1]\to\mathcal{H}$ as the hyperbolic length
$L_{\HH^{d}}(x\circ\gamma)$, and the distance $x^{*}d_{\HH^{d}}$ as the
infimumum of lengths of paths joining two points.
Note that $L_{\AdS}(i\circ\gamma)\leq L_{\HH^{d}}(x\circ\gamma)$, so
$i^{*}d_{\AdS}\leq x^{*}d_{\HH^{d}}$. Completeness of $i$ and the Hopf–Rinow
Theorem for length spaces (see [gromov_book, Theorem 1.9] or
[bridson_haefliger, Proposition I.3.7]) imply that closed balls for
$i^{*}d_{\AdS}$ are compact, so $x^{*}d_{\HH^{d}}$ is also complete. This
shows that $x$ is a local isometry between length spaces, the source being
complete, so it is a covering map (see [bridson_haefliger, Proposition I.3.28]
for a proof in the context of length spaces). This implies that the hyperbolic
metric on $x(\mathcal{H})$ is complete, so $x$ is onto. Since it is a
covering, it must be a homeomorphism. This proves $(1)$ and $(2)$.
Now consider a lift $\widetilde{i}:\mathcal{H}\to\widetilde{\AdS}^{d+1}$ to
the universal cover. Lemma 1.2 shows that
$\widetilde{i}(\widetilde{\mathcal{H}})$ is weakly spacelike as defined in
[maximal, Section 3.2], so property $(3)$ follow from [maximal, Proposition
3.5]. This in turn implies that $\mathcal{H}$ is (globally) the graph of a
$1$-Lipschitz function $\theta:\bS^{d}_{+}\to\R$ (for the spherical distance),
so $(4)$ follows from the extendability of Lipschitz functions.

As a consequence of this proof we get the following description of complete
spacelike hypersurfaces in $\AdS^{d+1}$.

######  Corollary 6.

Let $\mathcal{H}\subset\AdS^{d+1}$ be a complete spacelike hypersurface. There
is a distance decreasing function $\theta:\bS^{d}_{+}\to\bS^{1}$ (i.e.
$|\theta(x)-\theta(x^{\prime})|<d_{\bS^{d}}(x,x^{\prime})$ whenever $x\neq
x^{\prime}$) such that $\mathcal{H}=\\{\Phi(\theta(x),x)|x\in S^{d}_{+}\\}$.

######  Remark 1.9.

Here it is important to use the spherical distance on $\bS^{d}_{+}$ rather
than the Euclidean distance.

###  1.3 Second fundamental form

Here we recall the classical notion of second fundamental form in a setting
which applies both to spacelike hypersurfaces in $\AdS^{d+1}$ and to
$\AdS^{d+1}$ itself inside the flat pseudo-Riemannian space $\R^{d,2}$.

Let $(M,g)$ be a smooth oriented pseudo-Riemannian manifold of signature
$(p,q)$, and let $\mathcal{H}\subset M$ be an oriented hypersurface of class
$\mathcal{C}^{2}$ such that the restriction of $g$ to $\mathcal{H}$ has
signature $(p,q-1)$. Let us denote by $TM_{|\mathcal{H}}$ the pull-back of the
tangent bundle $TM$ by the inclusion $\mathcal{H}\hookrightarrow M$. The
tangent bundle $T\mathcal{H}$ is a sub-bundle of $TM_{|\mathcal{H}}$ and
$TM_{|\mathcal{H}}$ splits orthogonally (with respect to $g$) as

$TM_{|\mathcal{H}}=T\mathcal{H}\oplus\R N~{},$

where $N$ is the unit normal to $\mathcal{H}$ (i.e. $g(N,N)\equiv-1$ and the
orientation of $N$ is compatible with those of $M$ and $\mathcal{H}$).

The Levi–Civita connection $\nabla^{M}$ of $M$ restricts to a connection on
$TM_{|\mathcal{H}}$ (that we still denote $\nabla^{M}$). Since $g(N,N)$ is
constant, $\nabla^{M}_{X}N$ is orthogonal to $N$ and thus tangent to
$\mathcal{H}$ for every vector $X$ tangent to $\mathcal{H}$.

The _second fundamental form_ of $\mathcal{H}$ is the bilinear form on
$T\mathcal{H}$ defined by

$\secondFF_{\mathcal{H}}(X,Y)=g(\nabla^{M}_{X}N,Y)~{}.$

The second fundamental form relates the Levi–Civita connection
$\nabla^{\mathcal{H}}$ of $(\mathcal{H},g_{|\mathcal{H}})$ to the ambient
connection $\nabla^{M}$:

######  Proposition 7.

For every vector fields $X$ and $Y$ on $\mathcal{H}$, we have

$\nabla^{M}_{X}Y=\nabla^{\mathcal{H}}_{X}Y+\secondFF_{\mathcal{H}}(X,Y)N~{}.$

######  Remark 1.10.

The reader familiar with Riemannian geometry will notice a sign difference in
the definition of the second fundamental form. This is due to the fact that we
assume here that $g$ is negative in the normal direction.

###  1.4 Convexity in $\AdS^{d+1}$

Let $V$ be a hyperplane in $\R^{d,2}$ in restriction to which the quadratic
form $\mathbf{q}$ has signature $(d,1)$. Then $V\cap\AdS^{d+1}$ is a two-
sheeted hyperboloid of dimension $d$, each connected component of which is a
totally geodesic spacelike hypersurface. We call such a connected component a
_spacelike hyperplane_. They are the totally geodesic copies of $\HH^{d}$ in
$\AdS^{d+1}$. For any $\theta_{0}\in\bS^{1}$ the set
$\Phi(\\{\theta_{0}\\}\times\bS^{d}_{+})$, where
$\Phi:\bS^{1}\times\bS^{d}_{+}\to\AdS^{d+1}$ is the diffeomorphism defined
above, is a spacelike hyperplane.

If $W\subset V\cap\AdS^{d+1}$ is a spacelike hyperplane, we denote by
$\overline{W}$ the other component of $V\cap\AdS^{d+1}$, i.e. the image of $W$
by $x\mapsto-x$. We say that a point $x\in\AdS^{d+1}\backslash\overline{W}$ is
_in the past_ of $W$ if there exists a future-oriented timelike geodesic
segment from $x$ to a point of $W$ which does not intersect $\overline{W}$.

Let $\mathcal{H}\subset\AdS^{d+1}$ be a Lipschitz spacelike hypersurface. We
say that $\mathcal{H}$ is _convex_ if for every $x\in\mathcal{H}$, there
exists a spacelike hyperplane $W$ containing $x$ such that $\mathcal{H}$ is
contained in the past of $W$.

The hypersurface $\mathcal{H}$ is _locally convex_ if every point
$x\in\mathcal{H}$ has an open neighbourhood $U$ such that $\mathcal{H}\cap U$
is convex.

For complete hypersurfaces, the local convexity property globalizes.

######  Proposition 8.

Let $\mathcal{H}$ be a complete Lipschitz spacelike hypersurface. Then
$\mathcal{H}$ is convex if and only if it is locally convex.

######  Proof 1.11.

Let $p\in\mathcal{H}$ and consider a spacelike hyperplane $W$ containing $p$
such that a neighbourhood of $p$ in $\mathcal{H}$ is in the past of $W$.

Up to the action of $\SO_{\circ}(d,2)$, we may assume that
$W=\Phi(\\{0\\}\times S^{d}_{+})$, and write $p=\Phi(0,x)$. We have seen in
the proof of Proposition 5 that $\mathcal{H}$ is the graph of a function
$\theta:\bS^{d}_{+}\to\bS^{1}$ in the conformal model
$\AdS^{d+1}\approx\bS^{1}\times\bS^{d}_{+}$. Since $\theta$ is distance
decreasing, it is not onto and we may consider $\theta$ as a real valued map.

Assume by contradiction that $\mathcal{H}$ is not in the past of $W$, i.e.
$\theta>0$ at some point $y\in\bS^{d}_{+}$. Along the spherical geodesic
$\gamma:[0,1]\to\bS^{d}_{+}$ joining $x$ and $y$, the function $\theta$
possesses a local minimum. If $z=\gamma(t_{0})$ is such a point of
$\bS^{d}_{+}$, consider a spacelike hyperplane $W^{\prime}$ containing
$\Phi(\theta(z),z)$ such that a neighbourhood of $\Phi(\theta(z),z)$ in
$\mathcal{H}$ is in the past of $W^{\prime}$. Note that $W^{\prime}$ is the
graph of a function $\alpha:\bS^{d}_{+}\to S^{1}$. For $t$ near $t_{0}$, we
have $\alpha\circ\gamma(t)\geq\theta\circ\gamma(t)\geq\theta(z)$, with
equality at $t=t_{0}$. Hence $(\alpha\circ\gamma)^{\prime}(t_{0})=0$, and the
geodesic $t\mapsto\Phi(\alpha\circ\gamma(t),\gamma(t))$ is tangent to
$\Phi(\\{\theta(z)\\},\bS^{d}_{+})$ at $\Phi(\theta(z),z)$, therefore
$\alpha\circ\gamma$ is constant. It implies that $\theta\circ\gamma$ is
constant on a neighbourhood of every point where a local minimum is obtained,
hence $\theta\circ\gamma\geq 0$ which is a contradiction.

For hypersurfaces of class $\mathcal{C}^{2}$, convexity is characterized by
the sign of the second fundamental form:

######  Proposition 9.

Let $\mathcal{H}$ be a spacelike hypersurface of class $\mathcal{C}^{2}$. Then
$\mathcal{H}$ is locally convex if and only if its second fundamental form is
non-negative.

This motivates the following strengthenings:

A complete spacelike hypersurface $\mathcal{H}$ of class $\mathcal{C}^{2}$ is
called _strongly convex_ if its second fundamental form
$\secondFF_{\mathcal{H}}$ is positive definite, and _uniformly strongly
convex_ if there exists a constant $c>0$ such that

$\secondFF_{\mathcal{H}}\geq c\,g_{|\mathcal{H}}~{}.$

###  1.5 Globally hyperbolic spacetimes

Let $N$ be a Lorentzian manifold. A $\mathcal{C}^{1}$ curve on $N$ is called
_causal_ if its tangent direction is nowhere spacelike. Such a curve is called
_inextensible_ if it is maximal among causal curves (for the inclusion).

A _Cauchy hypersurface_ in $N$ is a topological hypersurface intersecting
every inextensible causal curve at exactly one point. A Lorentzian manifold
admitting a Cauchy hypersurface is called _globally hyperbolic_.

A globally hyperbolic Lorentzian manifold $N$ always admits a _temporal
function_ , i.e. a function to $\R$ with no critical points whose level sets
are spacelike hypersurfaces, see e.g. [bernal-sanchez]. Moreover, all smooth
Cauchy hypersurfaces are diffeomorphic, and if $M$ is a smooth Cauchy
hypersurface, then $N$ is diffeomorphic to $M\times\R$.

A globally hyperbolic Lorentzian manifold is _Cauchy compact_ if its Cauchy
hypersurfaces are compact.

A globally hyperbolic Cauchy compact $\AdS$ manifold is _maximal_ if it is not
isometric to a proper subset of another globally hyperbolic $\AdS$ manifold.

From now on, we abreviate “globally hyperbolic Cauchy compact” into “GHC”, and
“globally hyperbolic maximal Cauchy compact” into “GHMC”.

We recall here a description of GHMC AdS spacetimes, due to Mess [mess, mess-
notes]. (It is only stated in dimension $2+1$ in [mess], but the argument
works in higher dimension as pointed in [barbot_causal]. For other proofs see
[barbot_causal, Corollary 11.2] and [barbot-merigot, Proposition 4.8].)

Let $\Lambda$ be a closed subset of $\Ein^{d}$. The _domain of dependence_ of
$\lambda$ is the open set

$\Omega(\Lambda)=\\{x\in\AdS^{d+1}\mid\langle x,y\rangle<0\textrm{ for all
}[y]\in\Lambda\\}~{}.$

The following theorem describes GHMC AdS manifolds as quotients of the domain
of dependence of the boundary at infinity of a Cauchy hypersurface. Il was
proved by Mess in dimension 2+1 and extended by Barbot in higher dimensions.

######  Theorem 1.12 (Mess).

$\Gamma$ be a subgroup of $\SO_{\circ}(d,2)$ acting freely, properly
discontinuously and cocompactly on a spacelike hypersurface $\mathcal{H}$ and
let $\partial_{\infty}\mathcal{H}$ be the boundary of $\mathcal{H}$ in
$\Ein^{d}$. Then $\Gamma$ acts properly discontinuously on
$\Omega(\partial_{\infty}\mathcal{H})$, the quotient
$N=\Gamma\backslash\Omega(\partial_{\infty}\mathcal{H})$ is a GHMC AdS
manifold and $\Gamma\backslash\mathcal{H}\subset N$ is a Cauchy hypersurface.

Conversely, let $N$ be a GHMC AdS manifold of dimension $d+1$. Then there
exists a discrete subgroup $\Gamma$ of $\SO_{\circ}(d,2)$ and a
$\Gamma$-invariant complete spacelike hypersurface
$\mathcal{H}\subset\AdS^{d+1}$ such that $\Gamma$ acts properly
discontinuously and cocompactly on $\mathcal{H}$ and $N$ is isometric to
$\Gamma\backslash\Omega(\partial_{\infty}\mathcal{H})$.

The following statement combines results from Barbot–Mérigot [barbot-merigot],
Barbot [barbot_deformations] and Danciger–Guéritaud–Kassel [DGK:convex].

######  Theorem 1.13 (Barbot–Mérigot, Barbot, Danciger–Guéritaud–Kassel).

Let $N=\Gamma\backslash\Omega$ be a GHMC $\AdS$ manifold. The following
properties are equivalent:

  * $(i)$

The group $\Gamma$ is Gromov hyperbolic,

  * $(ii)$

The limit set $\Lambda_{\Gamma}$ is acausal,

  * $(iii)$

The manifold $N$ contains a convex Cauchy hypersurface,

  * $(iv)$

The manifold $N$ contains a strongly convex Cauchy hypersurface,

  * $(v)$

The group $\Gamma$ acts convex-cocompactly on $\Omega$.

We call such a GHMC $\AdS$ manifold _quasifuchsian_. It is called _Fuchsian_
if it possesses a totally geodesic Cauchy hypersurface.

######  Proof 1.14 (Sketch of the proof).

$(v)\Rightarrow(iv)$ is Lemma 6.4 in [DGK:convex].
$(iv)\Rightarrow(iii)$ is straightforward.
$(iii)\Rightarrow(i)$ follows from Proposition 8.3 in [barbot-merigot].
$(i)\Rightarrow(ii)$ is Theorem 1.4 in [barbot_deformations].
Finally, the main result of [barbot-merigot] is that $(ii)$ is equivalent to
$\Gamma$ being $P_{1}$-Anosov, and the latter is equivalent to $(v)$ according
to Theorem 1.7 in [DGK:convex].

###  1.6 Spacelike $\AdS$ structures

In this section we introduce a notion _spacelike $\AdS$ structure_ on a
manifold $M$, in a way that emulates the notion of $(G,X)$-structure. The
“developing map” of such a structure is an equivariant spacelike immersion of
the universal cover of $M$, from which one obtains a GHMC $\AdS$ manifold
homeomorphic to $M\times\R$. This will allow us to reduce the construction of
GHMC manifolds with prescribed Cauchy hypersurfaces to the construction of a
spacelike AdS structure on a manifold one dimension lower.

Let $M$ be a Lipschitz manifold of dimension $d$.

A _spacelike $\AdS$ atlas_ on $M$ is the data of an atlas
$(U_{i},\phi_{i})_{i\in I}$ where $(U_{i})_{i\in I}$ is an open cover of $M$
and $\phi_{i}:U_{i}\to\AdS^{d+1}$ is a Lipschitz spacelike immersion, such
that for all $i,j\in I$ and all $x\in U_{i}\cap U_{j}$, there exists an
orientation preserving isometry $g$ of $\AdS^{d+1}$ such that

$\phi_{j}=g\circ\phi_{i}$

in a neighbourhood of $x$.

A _spacelike $\AdS$ structure_ on $M$ is a maximal spacelike $\AdS$ atlas.

Two spacelike $\AdS$ atlases $(U_{i},\phi_{i})_{i\in I}$ and
$(V_{j},\psi_{j})_{j\in J}$ on $M$ are _equivalent_ if for every $(i,j)\in
I\times J$ and every $x\in U_{i}\cap V_{j}$, there exists an orientation
preserving isometry $g$ of $\AdS^{d+1}$ such that

$\psi_{j}=g\circ\phi_{i}$

in a neighbourhood of $x$.

Note that this definition is almost identical to that of a $(G,X)$-structure,
except that the charts are required to be spacelike immersions instead of
local homeomorphisms.

Emulating the theory of $(G,X)$-structures, we want to “patch together” the
local charts into an equivariant spacelike immersion of the universal cover.
In order to do so, remark first that there is a unique way to patch together
two local charts:

######  Lemma 1.15.

Let $U$ be a non empty open subset of $M$, $\phi:U\to\AdS^{d+1}$ a spacelike
immersion, and $g$ an orientation preserving isometry of $\AdS^{d+1}$ such
that $g\circ\phi=\phi$. Then $g=\Id$.

######  Proof 1.16.

Consider a point $x\in\phi(U)$ at which $\phi(U)$ is differentiable (it exists
because of Rademacher’s Theorem). The tangent space $T_{x}\phi(U)$ is
spacelike because of Lemma 1.4. Now $g$ fixes every point of the spacelike
hyperplane tangent to $\phi(U)$ at $x$, hence $g=\Id$.

As a consequence, we can still define the holonomy representation and the
developing map of a spacelike $\AdS$ structure.

######  Corollary 10.

Let $(U_{i},\phi_{i})_{i\in I}$ be a spacelike $\AdS$ atlas on $M$. Then there
exists a representation $\rho:\pi_{1}(M)\to\SO_{\circ}(d,2)$ and a
$\rho$-equivariant spacelike immersion $\phi:\tilde{M}\to\AdS^{d+1}$ such
that, for all $x\in\tilde{M}$ and all $U_{i}$ containing $\pi(x)$, there
exists $g\in\SO_{\circ}(d,2)$ such that

$\phi_{i}\circ\pi=g\circ\phi$

in a neighbourhood of $x$. (Here, $\pi:\tilde{M}\to M$ denotes the covering
map.)

Moreover, if another pair $(\rho^{\prime},\phi^{\prime})$ satisfies the same
properties, then there is a unique $g\in\SO_{\circ}(d,2)$ such that
$\rho^{\prime}=g\rho g^{-1}$ and $\phi^{\prime}=g\circ\phi$.

######  Proof 1.17.

The proof is the same as for $(G,X)$-structures. Let us write
$G=\SO_{\circ}(d,2)$ and $H=\SO_{\circ}(d,1)$, so that $\AdS^{d+1}=G/H$.
Given $i,j\in I$ and $x\in U_{i}\cap U_{j}$, write $g_{ij}(x)\in G$ the
element such that $\phi_{i}=g_{ij}(x)\cap\phi_{j}$ on a neighbourhood of $x$
(it is unique thanks to Lemma 1.15). Because of its uniqueness, it satisfies
the cocycle rule $g_{ik}(x)=g_{ij}(x)g_{jk}(x)$ whenever $x\in U_{i}\cap
U_{j}\cap U_{k}$, and we can consider the $G$-principal bundle $P$ over $M$
with transitions $g_{ij}$.
Another consequence of Lemma 1.15 is that the map $x\mapsto g_{ij}(x)$ is
locally constant, so $P$ inherits a flat connection, and we can consider its
holonomy representation $\rho:\pi_{1}(M)\to G$.
Now let $E$ be the associated $G/H$-bundle over $M$. The relation
$\phi_{i}=g_{ij}\circ\phi_{j}$ shows that there is a section $\sigma$ of $E$
that locally reads as $\phi_{i}$. This section lifts to a $\rho$-equivariant
map $\phi:\widetilde{M}\to G/H=\AdS^{d+1}$ with the required properties.
If another pair $(\rho^{\prime},\phi^{\prime})$ satisfies the same properties,
then for any $x\in\widetilde{M}$ there is an element $g(x)\in G$ such that
$\phi^{\prime}=g(x)\circ\phi$ on a neighbourhood of $x$. Using Lemma 1.15 once
again, we see that the map $x\mapsto g(x)$ is locally constant, hence
constant. The equivariance implies that $\rho^{\prime}=g\rho g^{-1}$.

Conversely, a representation $\rho:\pi_{1}(M)\to\SO_{\circ}(d,2)$ and an
equivariant spacelike immersion $\phi:\tilde{M}\to\AdS^{d+1}$ define a
spacelike AdS structure on $M$ by choosing an open cover $(U_{i})_{i\in I}$ of
$M$ so that $\pi:\tilde{M}\to M$ is invertible on each $U_{i}$, with
$\phi_{i}:U_{i}\to\AdS^{d+1}$ defined as $\phi\circ\pi^{-1}$ on $U_{i}$.

Let now $\phi:\tilde{M}\to\AdS^{d+1}$ be a $\rho$-equivariant spacelike
immersion. Note that the pulled-back distance $\phi^{*}d_{\AdS}$ (as defined
in Section 1.2) is $\pi_{1}(M)$-invariant. If $M$ is moreover compact, then it
is complete, and $\phi$ is thus an embedding (see Proposition 5). This implies
that $\rho$ is discrete and faithful and $\rho(\pi_{1}(M))$ acts properly
discontinuously on $\phi(\tilde{M})$. By Theorem 1.12, $\rho$ is the thus the
holonomy of a GHMC AdS spacetime $N$ and $\phi$ embeds $M$ as a Cauchy
hypersurface in $N$. We thus obtain the following:

######  Theorem 1.18.

Let $M$ be a closed manifold of dimension $d$, let $\rho$ be a representation
of $\pi_{1}(M)$ into $\SO_{\circ}(d,2)$ and let $\phi:\tilde{M}\to\AdS^{d+1}$
be a $\rho$-equivariant spacelike embedding. Then there exists a unique
$\rho$-invariant open domain $\Omega_{\rho}\subset\AdS^{d+1}$ such that:

  1. 1.

$\pi_{1}(M)$ acts properly discontinuously on $\Omega_{\rho}$ via $\rho$,

  2. 2.

the quotient $N_{\rho}=\rho(\pi_{1}(M))\backslash\Omega_{\rho}$ is a GHMC
$\AdS$ manifold,

  3. 3.

the map $\phi$ factors to an embedding of $M$ into $N_{\rho}$ whose image is a
Cauchy hypersurface.

###  1.7 Convex ruled spacelike AdS structures

Let $N$ be a quasifuchsian AdS spacetime of dimension $d+1$. A Lipschitz
hypersurface $M\subset N$ is called _spacelike_ if its lift $\tilde{M}$ to the
universal cover of $N$ (which is an open subset of $\AdS^{d+1}$ by Mess’s
Theorem) is a Lipschitz spacelike hypersurface.
We say that $M$ is _past-convex_ if $\tilde{M}$ is past-convex.
We say that $M$ is ruled if each $x\in M$ lies in the relative interior of a
geodesic segment of $N$ which is contained in $M$.

######  Lemma 1.19.

Let $N$ be a quasifuchsian AdS spacetime. Then $N$ contains a unique past-
convex ruled Cauchy hypersurface.

######  Proof 1.20.

For the existence, write $N=\Gamma\backslash\Omega$, and let
$\Lambda=\partial\Omega\cap\partial_{\infty}\AdS^{d+1}$ be its limit set. We
denote by $\Conv(\Lambda)\subset\AdS^{d+1}\cup\partial_{\infty}\AdS^{d+1}$ the
convex hull of $\Lambda$, and
$C(N)=\Gamma\backslash(\Conv(\Lambda)\setminus\Lambda)\subset N$ the convex
core of $N$. Its boundary $\partial N$ has two connected components, and we
consider the future component $\partial_{+}C(N)$. It is a Cauchy hypersurface
in $N$ [barbot-merigot, Lemma 4.9], and moreover a spacelike hypersurface
thanks to [barbot-merigot, Lemma 3.16] and Lemma 1.4. It is past-convex by
definition. Let $x\in\partial_{+}C(N)$, and consider a lift
$\tilde{x}\in\partial\Conv(\Lambda)$. This lift does not belong to $\Lambda$,
so it cannot be an extreme point of the convex set $\Conv(\Lambda)$, it
therefore lies in the relative interior of a geodesic segment of $\AdS^{d+1}$
contained in $\Conv(\Lambda)$. This geodesic segment is spacelike because
$\partial_{+}C(N)$ is a spacelike hypersurface. It must lie in
$\partial\Conv(\Lambda)$ because the interior of $\Conv(\Lambda)$ is convex.
Now for the uniqueness, let $S\subset N$ be a ruled past convex Cauchy
hypersurface, and $\tilde{S}\subset\Omega$ its lift. We then have
$\partial\tilde{S}\subset\partial_{\infty}\AdS^{d+1}=\Lambda$ thanks to
[maximal, Corollary 3.8]. Let
$\mathcal{C}\subset\AdS^{d+1}\cup\partial_{\infty}\AdS^{d+1}$ be the closed
convex hull of $\tilde{S}$. It is compact and convex, so by the Krein–Milman
Theorem it is the convex hull of its set of extreme points
$E\subset\tilde{S}\cup\Lambda\subset\AdS^{d+1}\cup\partial\AdS^{d+1}$. Since
$S$ is ruled, we have $\tilde{S}\cap E=\emptyset$, so
$\tilde{S}\subset\mathcal{C}=\Conv(\Lambda)$. Since $S$ is past-convex, we
find that $S=\partial_{+}C(N)$.

A spacelike $\AdS$ structure $(U_{i},\phi_{i})_{i\in I}$ on a manifold $M$ is
convex ruled if for all $i\in I$, $\phi_{i}(U_{i})$ is contained in a past-
convex ruled hypersurface.

######  Lemma 1.21.

Let $M$ be a closed manifold of dimension $d$. There is a one-to-one
correspondence between convex ruled spacelike $\AdS$ structures on $M$ and
quasifuchsian $\AdS$ spacetimes homeomorphic to $M\times\R$.

######  Proof 1.22.

It follows from Lemma 1.19, Theorem 1.18, and characterization $(iii)$ in
Theorem 1.13.

###  1.8 Convex ruled hyperbolic embedding structures and hyperbolic ends

In this section we outline the analog in hyperbolic manifolds of the notion of
spacelike $\AdS$ structures developed in Sections 1.6 and 1.7.

Let $\mathcal{H}$ be an oriented Lipschitz manifold. A map
$\phi:\mathcal{H}\to\HH^{d+1}$ is called a _Lipschitz embedding_ if
$(x,y)\mapsto d_{\HH^{d+1}}(\phi(x),\phi(y))$ is a Lipschitz distance on
$\mathcal{H}$. It is called a _Lipschitz immersion_ if every point in
$\mathcal{H}$ has a neighbourhood $U$ such that the restriction of $\phi$ to
$U$ is a Lipschitz embedding.

A Lipschitz immersion $\phi:\mathcal{H}\to\HH^{d+1}$ is called _ruled_ if for
every point $x\in\mathcal{H}$ there is an injective continuous curve
$\gamma:(-\varepsilon,\varepsilon)\to\mathcal{H}$ such that $\gamma(0)=x$ and
$\phi\circ\gamma$ is a geodesic segment in $\HH^{d+1}$.

A Lipschitz embedding $\phi:\mathcal{H}\to\HH^{d+1}$ is called _convex_ if
there is an open convex set $V\subset\HH^{d+1}$ such that $\phi(\mathcal{H})$
is an open subset of $\partial V$ and the orientations induced on $\partial V$
from $V$ and $\mathcal{H}$ coincide. A Lipschitz immersion
$\phi:\mathcal{H}\to\HH^{d+1}$ is called _locally convex_ if every point in
$\mathcal{H}$ has a neighbourhood $U$ such that the restriction of $\phi$ to
$U$ is a convex Lipschitz embedding.

We can use convex ruled Lipschitz embeddings to define convex ruled hyperbolic
embedding structures, the Riemannian counterpart of convex ruled spacelike
$\AdS$ structures.

Let $M$ be a $d$-dimensional oriented manifold. A hyperbolic embedding atlas
on $M$ is the data of an atlas $(U_{i},\phi_{i})_{i\in I}$ where
$(U_{i})_{i\in I}$ is an open cover of $M$ and $\phi_{i}:U_{i}\to\HH^{d+1}$ is
a Lipschitz embedding, such that for all $i,j\in I$ and all $x\in U_{i}\cap
U_{j}$, there exists an orientation preserving isometry $g$ of $\HH^{d+1}$
such that

$\phi_{j}=g\circ\phi_{i}$

in a neighbourhood of $x$.

A _hyperbolic embedding structure_ is a maximal hyperbolic embedding atlas. It
is called _convex ruled_ if the local charts $\phi_{i}$ are locally convex and
ruled.

Two hyperbolic embedding structures $(U_{i},\phi_{i})_{i\in I}$ and
$(V_{j},\psi_{j})_{j\in J}$ on $M$ are _isomorphic_ if for every $(i,j)\in
I\times J$ and every $x\in U_{i}\cap V_{j}$, there exists an orientation
preserving isometry $g$ of $\HH^{d+1}$ such that

$\psi_{j}=g\circ\phi_{i}$

in a neighbourhood of $x$.

The method used for Corollary 10 also provides a developing map and a holonomy
representation for convex ruled hyperbolic embedding structures.

######  Lemma 1.23.

Let $(U_{i},\phi_{i})$ be a hyperbolic embedding atlas on $M$. Then there
exists a representation $\rho:\pi_{1}(M)\to\SO_{\circ}(d+1,1)$ and a
$\rho$-equivariant Lipschitz immersion $\rho:\tilde{M}\to\HH^{d+1}$ such that,
for all $x\in\tilde{M}$ and all $U_{i}$ containing $\pi(x)$, there exists
$g\in\SO_{\circ}(d+1,1)$ such that

$\phi_{i}\circ\pi=g\circ\phi$

in a neighbourhood of $x$. (Here $\pi:\tilde{M}\to M$ denotes the universal
covering map.)

Moreover, if another pair $(\rho^{\prime},\phi^{\prime})$ satisfies the same
properties, then there is a unique $g\in\SO_{\circ}(d+1,1)$ such that
$\rho^{\prime}=g\rho g^{-1}$ and $\phi^{\prime}=g\circ\phi$.

Just as in the $\AdS$ case, the converse also holds: a pair $(\rho,\phi)$
where $\rho:\pi_{1}(\tilde{M})\to\SO_{\circ}(d+1,1)$ is a representation and
$\phi:\tilde{M}\to\HH^{d+1}$ is a $\rho$-equivariant Lipschitz immersion
determines a hyperbolic embedding structure on $M$. Finally, the map $\phi$ is
convex ruled if and only if the corresponding hyperbolic embedding structure
is convex ruled.

The hyperbolic manifolds associated to convex ruled hyperbolic embedding
structures, hyperbolic ends, require a precise definition. In simple terms, a
hyperbolic end is a hyperbolic manifold with compact concave boundary, which
is maximal in the sense of inclusion for this condition. In order to give a
precise definition, we need to define what we mean by a hyperbolic manifold
with concave boundary.

Let $\phi:\mathcal{H}\to\HH^{d+1}$ be a convex Lipschitz embedding, and
$V\subset\HH^{d+1}$ a convex open set such that $\phi(\mathcal{H})$ is an open
subset of $\partial V$. We define the set $\mathcal{N}(\phi)$ of _normal
vectors_ to $\phi$ to be the set of pairs $(x,v)$ where $x\in\mathcal{H}$ and
$v\in T_{\phi(x)}\HH^{d}$ is a unit vector pointing outside of $V$ such that
$v^{\perp}$ is a support hyperplane of $V$.

The _concave development_ of a convex Lipschitz embedding
$\phi:\mathcal{H}\to\HH^{d+1}$ is the set

$\mathcal{W}(\phi)=\set{\exp_{\phi(x)}(tv)}{(x,v)\in\mathcal{N}(\phi),t\geq
0}.$

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(10.851041043154133,2.115913139466372); [line width=1.pt,dash pattern=on 5pt
off 5pt,color=yqyqyq] (9.14,-2.54)– (11.325637763451418,1.7098512067110894);
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(8.88,-2.6) node[anchor=north west] $\phi(\mathcal{H})$; (7.62,4.58)
node[anchor=north west] $\partial_{\infty}\mathcal{W}(\phi)$; (2.7,1.7)
node[anchor=north west] $\mathcal{W}(\phi)$;

Figure 1: The concave development

A _concave hyperbolic atlas_ on a topological space $X$ is the data of an
atlas $(U_{i},\phi)_{i\in I}$ where $(U_{i})_{i\in I}$ is an open cover of $X$
and the maps $\phi_{i}:U_{i}\to\HH^{d+1}$ have the following properties:

  1. 1.

Each $\phi_{i}$ is a homeomorphism from $U_{i}$ to $\phi_{i}(U_{i})$,

  2. 2.

For each $i\in I$ there is a convex Lipschitz embedding
$\psi_{i}:\mathcal{H}_{i}\to\HH^{d+1}$ such that
$\phi_{i}(U_{i})=\mathcal{W}(\psi_{i})$,

  3. 3.

For every pair $i,j\in I$ there is an element $g_{ij}\in\SO_{\circ}(d+1,1)$
such that $\phi_{j}=g_{ij}\circ\phi_{i}$ on $U_{i}\cap U_{j}$.

A _concave hyperbolic manifold_ is the data of a connected Hausdorff second
countable topological space $X$ and a maximal concave hyperbolic atlas on $X$.

Note that a concave hyperbolic manifold is always a topological manifold with
boundary, the interior being a smooth hyperbolic manifold.

A _hyperbolic end_ of dimension $d+1$ is a concave hyperbolic manifold with
non empty compact boundary, and which is maximal (in the sense of inclusion)
under this condition.

Examples of hyperbolic ends arise naturally as the components of the
complement of the convex core in a convex cocompact hyperbolic manifold. Let
us focus on the quasifuchsian case.

A _quasifuchsian hyperbolic manifold_ of dimension $d+1$ is a complete
hyperbolic manifold $N$ homeomorphic to $M\times\R$, with $M$ a closed
manifold of dimension $d$, and which contains a non-empty compact subset
$K\subset N$ with the following convexity property: for any $x,y\in K$, any
geodesic of $N$ joining $x$ and $y$ is contained in $K$.
The smallest such compact set $K$ is called the _convex core_ of $N$, denoted
by $C(N)$. A quasifuchsian hyperbolic manifold is called _Fuchsian_ if its
convex core is a totally geodesic hypersurface.

If $N$ is a quasifuchsian hyperbolic manifold, then $N\setminus C(N)$ has two
connected components, and the closure of each component is a hyperbolic end.
When $d=1$, a hyperbolic end is just a funnel (the quotient by a hyperbolic
isometry of a half hyperbolic plane bounded by the axis of this isometry).
When $d\geq 2$, a hyperbolic end cannot always be obtained as an end of a
quasifuchsian hyperbolic manifold. Indeed, a hyperbolic end with boundary
homeomorphic to $M$ induces a holonomy representation
$\rho:\pi_{1}(M)\to\SO_{\circ}(d+1,1)$, which is faithful and discrete for a
quasifuchsian manifold, but may fail to be either for a hyperbolic end.

A hyperbolic end $E$ actually comes with two boundaries. The first one, called
the _pleated boundary_ of $E$ and denoted by $\partial_{0}E$, is the one
already mentioned after Definition 1.8, obtained when $E$ is seen as a
manifold with boundary. We will discuss its geometry further in the next
section.

The other boundary is the ideal boundary $\partial_{\infty}E$. It is
homeomorphic to $M$ and locally modelled on
$\partial_{\infty}\HH^{d+1}=\mathbb{S}^{d}$ with the action of
$\SO_{\circ}(d+1,1)$ by Möbius transformations. In other words,
$\partial_{\infty}E$ is equipped with a conformally flat structure. Moreover,
this correspondence between hyperbolic ends homeomorphic to
$M\times[0,+\infty)$ and conformally flat structures on $M$ is a bijection,
according to the following:

######  Theorem 1.24 (Kulkarni–Pinkall [kulkarni-pinkall]).

Let $M$ be an oriented closed manifold of dimension $d$ whose fundamental
group is not virtually abelian.111This topological condition ensures that
conformally flat structures on $M$ are _hyperbolic_ as defined in [kulkarni-
pinkall], and is satisfied by closed hyperbolic manifolds as well as
Gromov–Thurston manifolds. Then every conformally flat structure on $M$ is the
ideal boundary of a unique hyperbolic end with interior homeomorphic to
$M\times(0,+\infty)$.

When $d=2$, the group $\SO_{\circ}(3,1)$ is isomorphic to ${\rm PSL}(2,\C)$,
and a theorem of Gallo–Kapovich–Marden [gallo-kapovich-marden] states that any
non elementary representation of the fundamental group of a closed surface
into $\mathrm{SL}(2,\C)$ descends to the holonomy representation of a
conformally flat structure, thus providing an abundance of non quasifuchsian
hyperbolic ends.

The bridge between convex ruled hyperbolic embedding structures and hyperbolic
ends is the pleated boundary. The following lemma can be inferred from
[kulkarni-pinkall, Theorems 5.9 and 8.6] (see also [smith_moduli]), but we can
also give a direct proof without going through conformally flat structures.

######  Lemma 1.25.

If $E$ is a hyperbolic end, then the pleated boundary $\partial_{0}E$ carries
a convex ruled hyperbolic embedding structure.

######  Proof 1.26.

Following Definition 1.8, we see that the restriction of charts of a concave
hyperbolic atlas to $\partial_{0}E$ are convex Lipschitz embeddings. The fact
that $\partial_{0}E$ is ruled comes from the maximality of an end: if
$\partial_{0}E$ were strictly concave, we could “push” it to obtain a larger
concave hyperbolic manifold (see Figure 2). More precisely, consider a point
$x\in\partial_{0}E$ at which the property fails, and a concave hyperbolic
atlas for which $x$ is in a unique chart $(U_{i},\phi_{i})$. Then consider a
support hyperplane $H\subset\HH^{d+1}$ to $V_{i}$ at $\phi_{i}(x)$ (following
the notations of Definition 1.8). The failure of the ruling at $x$ means that
one can push $H$ slightly to a hyperplane $H^{\prime}$ that cuts the image of
$\phi$ in an arbitrarily small neighborhood of $\phi_{i}(x)$. In particular,
we can assume that this neighborhood is not contained in any other chart.
Replacing $V_{i}$ with its intersection with the half-space bounded by
$H^{\prime}$ that does not contain $\phi_{i}(x)$ and keeping the other charts,
we get a concave hyperbolic atlas on a larger manifold, thus contradicting the
maximality of $E$.

[line cap=round,line join=round,¿=triangle 45,x=1.0cm,y=1.0cm,scale=0.5]
(-4.3,-6.1) rectangle (28.24,4); [line width=1.pt] (2.04,-3.94) – (8.7,-6.02)
– (13.88,-3.42) – (4.58,3.24) – cycle; [line width=1.pt] (2.04,-3.94)–
(8.7,-6.02); [line width=1.pt] (8.7,-6.02)– (13.88,-3.42); [line width=1.pt]
(13.88,-3.42)– (4.58,3.24); [line width=1.pt] (4.58,3.24)– (2.04,-3.94); [line
width=1.pt] (13.88,-3.42)– (6.02,-0.78); [line width=1.pt] (6.02,-0.78)–
(2.04,-3.94); (5,0.5) node[anchor=north west] $\phi_{i}(x)$; (6.46,-3.42)
node[anchor=north west] $V_{i}$; (4.4,2.52) node[anchor=north west]
$V^{\prime}_{i}$; (9.7,1.38) node[anchor=north west] $H$; (10.88,0.02)
node[anchor=north west] $H^{\prime}$; [line width=1.pt] (-0.64,-2.74)–
(12.492680072701319,1.124872814188377); [line width=1.pt] (-0.48,-3.46)–
(12.694908583901139,0.41730042409102674);

Figure 2: Proof of Lemma 1.25.

We now wish to see how any convex ruled hyperbolic embedding structure can be
obtained as the pleated boundary of a hyperbolic end.

######  Lemma 1.27.

Any convex ruled hyperbolic embedding structure can be obtained as the pleated
boundary of a hyperbolic end, unique up to isometry.

This could be obtained by [kulkarni-pinkall, Theorem 10.6], but we provide
here a direct construction without going through conformally flat structures,
inspired by [smith_moduli, Section 4].

######  Proof 1.28 (Proof of Lemma 1.27).

Let $(U_{i},\phi_{i})_{i\in I}$ be a convex ruled hyperbolic embedding atlas
on a compact manifold $M$. We obtain a concave hyperbolic manifold with
pleated boundary isometric to $M$ by considering:

$E=\bigsqcup_{i\in I}\mathcal{W}(\phi_{i})/\sim$

where we identify $\mathcal{W}(\phi_{i}|_{U_{i}\cap U_{j}})$ with
$\mathcal{W}(\phi_{j}|_{U_{i}\cap U_{j}})$ through the element
$\gamma_{ij}\in\SO_{\circ}(d+1,1)$ such that $\phi_{i}|_{U_{i}\cap
U_{j}}=\gamma_{ij}\circ\phi_{j}|_{U_{i}\cap U_{j}}$.

Consider a concave hyperbolic manifold with compact pleated boundary
$E^{\prime}\supset E$, and let
$(x,x^{\prime})\in\partial_{0}E\times\partial_{0}E^{\prime}$ be maximising the
distance. In a chart, we find that $\partial_{0}E^{\prime}$ must be (locally
around $x^{\prime}$) included in the $r$-neighbourhood of a geodesic where
$r=d(x,x^{\prime})$ (because $\partial_{0}E$ is ruled). But for $r>0$, the
$r$-neighbourhood of a geodesic in $\HH^{d+1}$ is strictly convex, thus $r=0$,
i.e. $E^{\prime}=E$ and $E$ is maximal.

Now consider another hyperbolic end $E^{\prime}$ inducing the same convex
ruled hyperbolic embedding structure on the pleated boundary
$\partial_{0}E^{\prime}\approx M$. If $(V_{j},\psi_{j})_{j\in J}$ is a concave
hyperbolic atlas on $E^{\prime}$, then whenever
$V_{j}\cap\partial_{0}E^{\prime}\neq\emptyset$, $\psi_{j}(V_{j})$ contains a
neighbourhood of $\psi_{j}(V_{j}\cap\partial_{0}E^{\prime})$ in
$\mathcal{W}(\psi_{j}|_{V_{j}\cap\partial_{0}E^{\prime}})$. So $E^{\prime}$
contains a neighbourhood of $\partial_{0}E$ in $E$. By maximality of
$E^{\prime}$, we find that $E\subset E^{\prime}$, and by maximality of $E$ we
must have $E=E^{\prime}$, hence the uniqueness.

Note that we can also obtain the flat conformal structure from this
construction. If $\phi:\mathcal{H}\to\HH^{d+1}$ is a convex Lipschitz
embedding, we can consider the ideal boundary
$\partial_{\infty}\mathcal{W}(\phi)$ of its concave development:

$\partial_{\infty}\mathcal{W}(\phi)=\overline{\mathcal{W}(\phi)}\cap\partial_{\infty}\HH^{d+1}=\set{\lim_{t\to+\infty}\exp_{\phi(x)}(tv)}{(x,v)\in\mathcal{N}(\phi)}\subset\partial_{\infty}\HH^{d+1}=S^{d}.$

One gets a conformally flat manifold by setting:

$\partial_{\infty}E=\bigsqcup\partial_{\infty}\mathcal{W}(\phi_{i})/\sim$

where we identify $\partial_{\infty}\mathcal{W}(\phi_{i}|_{U_{i}\cap U_{j}})$
with $\partial_{\infty}\mathcal{W}(\phi_{j}|_{U_{i}\cap U_{j}})$ through the
element $\gamma_{ij}\in\SO_{\circ}(d+1,1)$ such that $\phi_{i}|_{U_{i}\cap
U_{j}}=\gamma_{ij}\circ\phi_{j}|_{U_{i}\cap U_{j}}$.

As a consequence of Lemma 1.25 and Lemma 1.27, we get the following hyperbolic
version of Lemma 1.21:

######  Lemma 1.29.

Let $M$ be a closed manifold. There is a one-to-one correspondence between
convex ruled hyperbolic embedding structures on $M$ up to equivalence and
hyperbolic ends $E$ with pleated boundary $\partial_{0}E\approx M$ up to
isometry.