# Rigidity of Totally Geodesic Hypersurfaces in Negative Curvature

Ben Lowe

###### Abstract.

Let $M$ be a closed hyperbolic manifold containing a totally geodesic
hypersurface $S$, and let $N$ be a closed Riemannian manifold homotopy
equivalent to $M$ with sectional curvature bounded above by $-1$. Then it
follows from the work of Besson-Courtois-Gallot that $\pi_{1}(S)$ can be
represented by a hypersurface $S^{\prime}$ in $N$ with volume less than or
equal to that of $S$. We study the equality case: if $\pi_{1}(S)$ cannot be
represented by a hypersurface $S^{\prime}$ in $N$ with volume strictly smaller
than that of $S$, then must $N$ be isometric to $M$? We show that many such
$S$ are rigid in the sense that the answer to this question is positive. On
the other hand, we construct examples of $S$ for which the answer is negative.

##  1\. Introduction

There is a sharp upper bound for the area of a minimal surface in a Riemannian
3-manifold with a negative upper bound on its sectional curvature. Namely, if
$(M,g)$ is a Riemannian 3-manifold containing a minimal surface $\Sigma$, and
if the sectional curvature $K_{g}$ satisfies $K_{g}\leq-1$, then

(1.1) $\text{Area}_{g}(\Sigma)\leq 2\pi|\chi(\Sigma)|.$

It is simple to check this inequality using the Gauss equation and the Gauss
Bonnet formula, but the question of what happens in the case of equality is
more interesting. Equality in (1.1) implies that $\Sigma$ is totally geodesic
and hyperbolic (constant curvature $-1$) in its induced metric. The
straightforward way for this to happen is if $g$ itself is hyperbolic, but is
this the only way?

V. Lima showed that the answer to this question is negative by constructing
non-hyperbolic examples of metrics on $\Sigma\times\mathbb{R}$ with sectional
curvature at most $-1$ for which equality in (1.1) is attained [Lim19]. In
section 6 we construct closed non-hyperbolic Riemannian 3-manifolds with
sectional curvature at most $-1$ that contain totally geodesic hyperbolic
surfaces. These give examples of metrics for which rigidity in the case of
equality in the area inequality (1.1) fails, even in the case when the ambient
manifold is closed. The main goal of this paper is to give conditions under
which equality in (1.1) forces the ambient metric to be hyperbolic.

In fact, our arguments will work in any dimension. For a closed hyperbolic
$n$-manifold $\Sigma$ and a closed $n+1$-manifold $N$ with sectional curvature
less than or equal to $-1$ the following holds. Suppose we have a map
$F:\Sigma\rightarrow N$ so that the induced map
$\pi_{1}(\Sigma)\rightarrow\pi_{1}(N)$ is injective. Assume also that there is
a closed hyperbolic manifold homotopy equivalent to $N$ via a homotopy
equivalence that sends $F(\Sigma)$ to the homotopy class of a totally geodesic
hypersurface. Then as we explain in Section 3.4, the next theorem is a
consequence of the work of Besson-Courtois-Gallot.

######  Theorem 1.1.

There exists a smooth map $\overline{F}$ homotopic to $F$ so that the
$n$-volume of the image $\overline{F}(\Sigma)$ of $\Sigma$ in $N$ is smaller
than the volume of $\Sigma$ in its hyperbolic metric. The map $\overline{F}$
has the property that in the case that the volume of $\overline{F}(\Sigma)$ is
equal to the volume of $\Sigma$ in its hyperbolic metric, then:

  1. (1)

The image $\overline{F}(\Sigma)$ of $\Sigma$ under $\overline{F}$ is a totally
geodesic hypersurface in $N$ with induced metric the hyperbolic metric on
$\Sigma$.

  2. (2)

The volume of any immersed hypersurface in $N$ homotopic to and distinct from
$\overline{F}(\Sigma)$ up to reparametrization is greater than that of
$\overline{F}(\Sigma)$.

######  Remark 1.2.

If $N$ is three dimensional, the previous theorem is essentially a consequence
of the fact that $\overline{\Sigma}$ is homotopic to a least area minimal
immersion [SY79] and (1.1.)

Theorem 1.1 leaves open the question of whether equality forces not just the
image of some map homotopic to $F$, but also the ambient metric $N$, to be
hyperbolic. We will show that the answer to this question depends on how the
image of $\Sigma$ sits inside $N$.

We say that a hypersurface $\Sigma$ in a Riemannian n-manifold $N$ is well-
distributed in $N$ if every point in the universal cover $\tilde{N}$ of $N$ is
contained in an embedded solid hypercube all of whose hyperfaces lie in lifts
of $\Sigma$ to $\tilde{N}$. The following is our first main theorem.

######  Theorem 1.3.

Let $N$ be a Riemannian manifold with sectional curvature at most $-1$ that
contains a totally geodesic hyperbolic hypersurface $\Sigma$. Then if $\Sigma$
is well-distributed in $N$, $N$ must be hyperbolic.

Let $M$ be a closed hyperbolic manifold that contains infinitely many totally
geodesic hypersurfaces $\Sigma_{k}$ (see [MR03], [Mil76] for examples.) Then
using Theorem 1.3, we can prove the following:

######  Theorem 1.4.

There exists $K\in\mathbb{N}$ so that the following holds. Fix $\Sigma_{k}$
for $k>K$. Let $N$ be a Riemannian manifold homotopy equivalent to $M$ with
sectional curvature at most $-1$. Assume that there is a totally geodesic
hyperbolic hypersurface in $N$ whose fundamental group injectively includes to
a subgroup conjugate to $\pi_{1}(\Sigma_{k})$ in $\pi_{1}(N)$, where we have
used the homotopy equivalence to identify $\pi_{1}(M)$ and $\pi_{1}(N)$. Then
$N$ is isometric to $M$ in its hyperbolic metric.

######  Remark 1.5.

Rather than assuming that $\Sigma$ (resp. $\Sigma_{k}$) was totally geodesic
and hyperbolic in the previous two theorems, we could instead have assumed, in
view of Theorem 1.1, that $\Sigma$ (resp. $\Sigma_{k}$) was not homotopic in
$N$ to an immersed hypersurface with volume smaller than the volume of
$\Sigma$ (resp. $\Sigma_{k}$) in its hyperbolic metric.

Although it is unclear how sharp Theorem 1.4 is, certainly some assumption on
the hypersurface in the model case is needed by the constuction in Section 6.
To the author’s knowledge, this is the first example of a rigidity phenomenon
for minimal hypersurfaces that is sensitive to how the minimal hypersurface is
distributed in the ambient space. We comment that Farrell-Jones proved that if
$N$ is a closed manifold homotopy equivalent to a closed hyperbolic manifold
$M$ of dimension above four, then $N$ and $M$ must be homeomorphic [FJ89] but
not necessarily diffeomorphic [FJ90]. The sectional curvature of such an $N$
can moreover be taken to be pinched between $-1-\epsilon$ and $-1$ for any
$\epsilon>0$. By geometrization, all closed negatively curved 3-manifolds have
hyperbolic metrics.

The results of this paper are evidence for the following conjecture.

######  Conjecture 1.6.

For every $D>0$ and $k\in\mathbb{N}$ , there exists $A=A(D,k)$ so that the
following holds. Let $N$ be a closed Riemannian manifold of dimension $k$,
sectional curvature bounded above by $-1$, and diameter at most $D$. Then if
$N$ contains a totally geodesic hyperbolic hypersurface with volume greater
than $A$, $N$ must be isometric to a hyperbolic manifold.

One could formulate a similar conjecture for totally geodesic submanifolds
$\Sigma$ of higher codimension. In this case, one would need to modify the
statement by requiring that $\Sigma$ is not contained in some higher
dimensional totally geodesic submanifold with small volume, and one would need
to allow for the possibility that the universal cover of $N$ is some other
rank one symmetric space. It might also be interesting to drop the compactness
assumption on $N$.

###  1.1. Related Work

The study of the kinds of rigidity questions considered in this paper goes
back to Calabi, who showed the following. For a Riemannian 2-sphere $S$ with
sectional curvature between 0 and 1, every simple closed geodesic must have
length at least $2\pi$ [Pog46]. When such a geodesic has length exactly
$2\pi$, then $S$ must be isometric to the unit sphere. (See [AH98], where a
proof of this result is given and attributed to Calabi.)

For $\Sigma$ a minimal sphere in a Riemannian manifold $M$ of sectional
curvature at most 1, inequality (1.1) is true with the direction of the
inequality reversed. In contrast to the negative curvature case discussed
above and the examples of V. Lima, there is a general rigidity statement in
the positive curvature three dimensional setting, due to Mazet-Rosenberg
[MR14]. Mazet recently proved a version in higher codimension [Maz20]. We also
mention that Espinar-Rosenberg obtained rigidity statements for surfaces of
constant mean curvature $H\in(0,1)$ that have the largest possible area given
that the ambient space has sectional curvature at most $-1$.

There are many examples of closed hyperbolic manifolds that contain closed
totally geodesic hypersurfaces (see [MR03] for the three dimensional case, and
[Mil76] for examples coming from so-called arithmetic hyperbolic manifolds of
simplest type.) An arithmetic hyperbolic manifold that contains one totally
geodesic hypersurface must contain infinitely many. In the other direction,
Bader-Fisher-Miller-Stover [BFMS21] and Margulis-Mohamaddi (the latter in
dimension 3) [MM22] proved that a finite volume hyperbolic manifold with
infinitely many totally geodesic hypersurfaces must be arithmetic. There are
examples of closed hyperbolic 3-manifolds that contain no totally geodesic
surfaces [MR03]. It is an open problem whether there are closed hyperbolic
manifolds in dimension above three that contain no totally geodesic
hypersurfaces.

Let $M$ be a closed hyperbolizable 3-manifold. Calegari-Marques-Neves
introduced an entropy functional on metrics $g$ defined by asymptotic counts
of minimal surfaces in $(M,g)$, and proved it to be uniquely minimized at the
hyperbolic metric over all metrics with sectional curvature at most $-1$
[CMN22]. The way we move information obtained via homogenous dynamics from
constant curvature to variable curvature in the proof of Theorem 1.4 is
inspired by their ideas.

Results that assume a negative sectional curvature upper bound often have
analogues that instead assume a negative Ricci or scalar curvature lower
bound. For example, in our joint work with Neves [LN21] we prove the following
analogue of inequality (1.1). Let $(M,g_{hyp})$ be a closed hyperbolic
3-manifold containing a closed totally geodesic surface $\Sigma$. Then for any
metric $g$ on $M$ with scalar curvature at least $-6$, the area of any surface
homotopic to $\Sigma$ must be at least

(1.2) $Area_{g_{hyp}}(\Sigma)=-2\pi\chi(\Sigma).$

For the latter theorem, rigidity in the case of equality holds, in that
equality in (1.2) implies that $g$ must be hyperbolic. In the setting of our
paper, however, whether equality in (1.1) implies that $M$ must be hyperbolic
depends on the homotopy class of $\Sigma$. Thus, perhaps surprisingly, a
sectional curvature upper bound is in this context a weaker constraint than a
scalar curvature lower bound.

###  1.2. Outline

This paper has three main parts. The reader is encouraged to assume that $M$
and $N$ are three dimensional on a first reading.

In the first part, we prove Theorem 1.3. Let $N$ be a negatively curved
manifold containing a totally geodesic hyperbolic hypersurface $\Sigma$ that
satisfies the well-distribution condition. Every point in the universal cover
of $N$ is contained in some solid embedded cube $\Box$ whose hyperfaces lie in
lifts $\tilde{\Sigma}$ of $\Sigma$ to the universal cover. The crucial point
for our argument is that there is an isometric embedding $\Phi$ of the
boundary of $\Box$ in $\mathbb{H}^{n}$. After constructing $\Phi$, we define a
new metric on $\mathbb{H}^{n}$ by gluing the unbounded connected component of
$\mathbb{H}^{n}-\Phi(\partial\Box)$ to $\Box$. Using the Rauch comparison
theorem, we are able to argue that this metric must have been isometric to
$\mathbb{H}^{n}$. We thus conclude that the metric on the interior of $\Box$
has constant curvature $-1$. Since the point we chose was arbitrary, this
shows that $N$ has constant curvature $-1$.

In the second part, we find examples of totally geodesic hypersurfaces
$\Sigma$ in closed hyperbolic manifolds $M$ that satisfy a slightly stronger
version of the well distribution condition. Here we rely on work by Mozes-Shah
that implies uniform distribution statements for totally geodesic
hypersurfaces in closed hyperbolic manifolds. The strong well distribution
condition will imply that for any Riemannian manifold $N$ homotopy equivalent
to $M$ with sectional curvature at most $-1$, any totally geodesic
hypersurface in $N$ corresponding to $\Sigma$ will satisfy the well
distribution condition. This allows us to apply Theorem 1.3 to prove Theorem
1.4.

In the third part, we construct examples of closed Riemannian manifolds $M$
with sectional curvature at most $-1$ that contain totally geodesic hyperbolic
hypersurfaces without themselves being hyperbolic.

##  2\. Acknowledgements

We are grateful to Laurent Mazet for explaining to us the construction in
Proposition 6.2. We also thank Antoine Song and Matthew Stover. The author was
supported by NSF grant DMS-2202830.

##  3\. Background

In this section we collect some facts that we will need in the paper.

### 3.1.

Informally speaking, the Rauch comparison theorem states that if one
Riemannian manifold is more negatively curved than the other, then its Jacobi
fields will grow faster than those of the other. We will need the following
corollary of that theorem (see [CE08][Cor. 1.35, Remark 1.37].)

######  Corollary 3.1 (Corollary of Rauch Comparison Theorem).

Let $(M,g)$ be a Riemannian manifold of dimension $n$ and let $m\in M$. Assume
the sectional curvature $K_{g}$ of $(M,g)$ satisfies $K_{g}\leq-1$. Choose a
point $m_{0}\in\mathbb{H}^{n}$, and let $I:TM_{m}\rightarrow
T\mathbb{H}^{n}_{m_{0}}$ be a linear isometry between the tangent spaces. Let
$c:[0,1]\rightarrow M$ be a geodesic segment in $(M,g)$ so that $exp_{m}$ is a
nonsingular embedding on $s\cdot exp_{m}^{-1}(c(t))$ for $0\leq s\leq 1$ and
$0\leq t\leq 1$. Then we have

(3.1) $L[c]\geq L[exp_{m_{0}}\circ I\circ exp_{m}^{-1}(c)].$

Moreover, in the case of equality the image of the map

$(s,t)\mapsto exp_{m}(s\cdot exp_{m}^{-1}(c(t))),\hskip 5.69054pt0\leq s\leq
1,\hskip 2.84526pt0\leq t\leq 1$

is a solid totally geodesic triangle, and every 2-plane tangent to the image
has constant sectional curvature $-1$.

### 3.2.

Let $M$ be a closed hyperbolic n-manifold. Then $M$ is the quotient of
$\mathbb{H}^{n}$ by a discrete group isomorphic to $\pi_{1}(M)$ acting
properly discontinuously and by isometries. If $N$ is closed negatively curved
Riemannian manifold, then $M$ is homotopy equivalent to $N$ if and only if
$\pi_{1}(M)$ is isomorphic to $\pi_{1}(N)$. Assume this is the case, and fix a
homotopy equivalence $F:M\rightarrow N$. We can lift $F$ to a map
$\tilde{F}:\tilde{M}\rightarrow\tilde{N}$ that commutes with the actions of
$\pi_{1}(M)$ and $\pi_{1}(N)$ by deck transformations.

The boundary at infinity $\partial_{\infty}(X)$ of a simply connected
negatively curved manifold $X$ is the set of geodesic rays in $X$ up to the
equivalence relation of remaining at finite distance for all time when
parametrized by arc-length. For any $x\in X$, the exponential map defines a
bijection between the unit sphere in the tangent space to $X$ and
$\partial_{\infty}(X)$. $\partial_{\infty}(X)$ is topologized so that this map
is a homeomorphism for all $x$. Every geodesic in $X$ is uniquely determined
by its two endpoints in $\partial_{\infty}(X)$, and conversely any two
distinct points in $\partial_{\infty}(X)$ determine a geodesic.

The fundamental groups $\pi_{1}(M)$ and $\pi_{1}(N)$ act by homeomorphisms on
$\partial_{\infty}(\tilde{M})$ and $\partial_{\infty}(\tilde{N})$. The map
$\tilde{F}$ induces a homeomorphism
$\tilde{F}_{\infty}:\partial_{\infty}\tilde{M}\rightarrow\partial_{\infty}\tilde{N}$
between the boundaries at infinity of $\tilde{M}$ and $\tilde{N}$, that
commutes with the actions of $\pi_{1}(M)$ and $\pi_{1}(N)$.

Suppose that $\Sigma$ and $\Sigma_{N}$ are closed totally geodesic hyperbolic
hypersurfaces of $M$ and $N$ so that $F_{*}\pi_{1}(\Sigma)$ is conjugate to
$\pi_{1}(\Sigma_{N})$ in $\pi_{1}(N)$. Let $\tilde{\Sigma}$ be a lift of
$\Sigma$ to $\tilde{M}$, and suppose that the intersection

$G=\cap_{i=1}^{n-k}\gamma_{i}\tilde{\Sigma}$

is transverse for some $\gamma_{1},..,\gamma_{n-k}\in\pi_{1}(M)$. Then $G$ is
a $k$-dimensional totally geodesic subspace in $\tilde{M}\cong\mathbb{H}^{n}$.
The equatorial $n-2$-sphere $\tilde{F}_{\infty}(\gamma_{i}\tilde{\Sigma})$ in
$\partial_{\infty}(N)$ bounds a unique totally geodesic hyperplane that lifts
$\Sigma_{N}$, which by an abuse of notation we call
$\gamma_{i}\tilde{\Sigma}_{n}$. It will be important for us that

$\cap_{i=1}^{n-k}\gamma_{i}\tilde{\Sigma}_{N}$

is a totally geodesic k-plane with boundary at infinity
$\tilde{F}_{\infty}(\partial_{\infty}G)$. This implies, for example, that if
$\Box$ is an embedded hypercube in $\tilde{M}$ whose hyperfaces are totally
geodesic and extend to totally geodesic hyperplanes containing lifts
$\gamma\tilde{\Sigma}$ of $\Sigma$ to $\tilde{M}$, then the region $\Box_{N}$
bounded by the corresponding lifts of $\Sigma_{N}$ in $\tilde{N}$ will also be
a hypercube.

### 3.3.

We state the result from homogeneous dynamics that we will need. Let
$\Sigma_{k}$ be a sequence of distinct totally geodesic hypersurfaces in a
closed hyperbolic n-manifold $M$. Denote by $\hat{\Sigma}_{k}$ their lifts to
the Grassmann bundle $Gr_{n-1}(M)$ of unoriented tangent $n-1$ planes to $M$.
The $\hat{\Sigma}_{k}$ define probability measures $\mu_{k}$ on $Gr_{n-1}(M)$
by

$\mu_{k}(f):=\frac{1}{Vol(\hat{\Sigma}_{k})}\int_{\hat{\Sigma}_{k}}fd\hat{\Sigma}_{k}.$

for $f$ a continuous function on $Gr_{n-1}(M)$, and where $d\hat{\Sigma}_{k}$
is the volume form for the hyperbolic metric on $\Sigma_{k}$. The hyperbolic
metric induces a metric on $Gr_{n-1}(M)$ on which we denote the volume
measure, normalized to have unit volume, by $\mu_{Leb}$. It follows from
Ratner’s measure classification theorem that any weak-$*$ limit of the
$\mu_{k}$ is equal to a convex combination of measures supported on totally
geodesic hypersurfaces and $\mu_{Leb}$. That we can rule out ergodic
components that are supported on totally geodesic hypersurfaces is a
consequence of work by Mozes-Shah [MS95][Theorem 1.1]:

######  Theorem 3.2.

The $\mu_{k}$ weak-$*$ converge to $\mu_{Leb}$.

In fact, we will only require a weaker statement. We say that a surface
$\Sigma$ is $\epsilon$-dense if every tangent plane in $Gr_{n-1}(M)$ is at a
distance of at most $\epsilon$ from some tangent plane to $\Sigma$, where the
distance is measured in the natural metric on $Gr_{n-1}(M)$. The corollary
below follows from Theorem (3.2).

######  Corollary 3.3.

For every $\epsilon>0$ there is $K$ so that $\Sigma_{k}$ is $\epsilon$-dense
if $k>K$.

### 3.4.

Finally, we explain how Theorem 1.1 follows from the work of Besson-Courtois-
Gallot. That $\overline{F}$ is homotopic to a smooth map with Jacobian
pointwise smaller than 1 is a direct consequence of [BCG99][Theorem 1.10],
taking the manifold $X$ in the statement of that theorem to be the universal
cover $\tilde{N}$ of $N$, and the representation to correspond to the action
of of $\pi_{1}(\Sigma)\subset\pi_{1}(N)$ on $\tilde{N}$ by deck
transformations. That the action of $\pi_{1}(\Sigma)$ is convex cocompact
follows from the fact that $N$ is homotopy equivalent to a closed hyperbolic
manifold via a homotopy equivalence that sends $F(\Sigma)$ to a totally
geodesic hypersurface. For the equality case, if no immersion homotopic to
$\overline{F}$ has smaller volume than the volume of $\Sigma$ in its
hyperbolic metric, then the volume of $\overline{F}(\Sigma)$ is equal to the
volume of $\Sigma$ in the hyperbolic metric, and the Jacobian of
$\overline{F}(\Sigma)$ must be everywhere equal to 1. In that case
[BCG99][Theorem 1.2] implies that the differential of $\overline{F}$ is at
every point an isometry onto its image. The image $\overline{F}(\Sigma)$ is
therefore isometric to the hyperbolic metric on $\Sigma$ in its induced
metric.

Since $\overline{F}(\Sigma)$ minimizes volume over all hypersurfaces homotopic
to it, we know that $\overline{F}(\Sigma)$ is a minimal hypersurface. Thus at
every point the principal curvatures $\lambda_{1},..,\lambda_{n}$ sum to zero.
If any were nonzero, then we could find $\lambda_{i}>0$ and $\lambda_{j}<0$.
The tangent plane to $\overline{F}(\Sigma)$ spanned by the corresponding
principal directions would then have sectional curvature strictly less than
one by the Gauss equation and the fact that the sectional curvature of $N$ is
less than or equal to $-1$, which contradicts the fact that
$\overline{F}(\Sigma)$ is hyperbolic in its induced metric. If there were some
other immersed hypersurface in the homotopy class of $\overline{F}(\Sigma)$
with the same volume, then one could argue as in [BCG08][Proof of Theorem 1.2]
to show that it is equal to $\overline{F}(\Sigma)$.

##  4\. Proof of Theorem 1.4

In this section we give the proof of Theorem 1.3. Again we recommend that the
reader assume $N$ is three dimensional on a first reading.

Assume that $\Sigma$ is a closed totally geodesic hypersurface in $N$ that
satisfies the well-distribution condition. This means that every point
$p\in\tilde{N}$ is contained in an embedded solid hypercube $\Box$ whose
hyperfaces are contained in lifts of $\Sigma$ to $\tilde{N}$. We claim that
there is an isometric embedding $\Phi$ of the boundary $\partial\Box$ of
$\Box$ in $\mathbb{H}^{n}$.

### 4.1.

To illustrate some of the ideas, we first give a proof of a similar statement
with the boundary of the cube $\partial\Box$ replaced by the boundary $\Delta$
of a solid embedded tetrahedron $\Delta\subset\tilde{N}$ whose faces are
contained in lifts of $\Sigma$. We assume also that $N$ is three dimensional.
We originally tried to work with tetrahedra rather than cubes, but it does not
seem possible to follow the approach of Section 5 using tetrahedra.

We claim that $\Delta$ isometrically embeds in $\mathbb{H}^{3}$. Label the
vertices of $\Delta$ as $A,B,C,D$. Note that the faces of $\Delta$ meet at
constant angles, since the faces are totally geodesic– i.e., for any edge of
$\Delta$, the angle that the two faces meeting at that edge make at any point
on the edge is the same.

Then since the face $ABC$ is contained in a totally geodesic plane isometric
to $\mathbb{H}^{2}$, we can map $ABC$ isometrically into some copy of
$\mathbb{H}^{2}$ contained in $\mathbb{H}^{3}$. Call this map $\Phi$.

Note also that the choice of $\Phi$ together with a choice of orientation for
$\mathbb{H}^{3}$ and for the triangle $ABC$ uniquely determines linear
isometries between the tangent spaces to $A$, $B$, and $C$ and respectively
$\Phi(A)$, $\Phi(B)$, and $\Phi(C)$– fix such a linear isometry

$d\Phi_{A}:T_{A}(M)\rightarrow T_{\Phi(A)}(\mathbb{H}^{3}),$

and similarly for $B$ and $C$. We can then extend $\Phi$ to the line segment
$AD$ by sending it to the geodesic ray beginning at $\Phi(A)$ with tangent
vector at $\Phi(A)$ equal to the image under $d\Phi_{A}$ of the tangent vector
to $AD$ at $A$, and similarly for $B$ and $C$.

By the fact that the faces of $\Delta$ meet along constant angles, we know
that $\Phi(AD)$ and $\Phi(BD)$ intersect, and that triangle $\Phi(ABD)$ is
isometric to triangle $ABD$, and similarly for triangles $ACD$ and $BCD$. We
can thus extend $\Phi$ to an isometry on all of $\Delta$ as desired.

### 4.2.

In the cube case, it is possible by a similar but more complicated argument to
define an isometric embedding $\Phi$ of $\partial\Box$ in $\mathbb{H}^{n}$ “by
hand.” Instead, we give a short proof that this is possible using a developing
map argument.

Note that every point $p$ on a hyperface of $\Box$ has a neighborhood in
$\partial\Box$ that isometrically embeds in $\mathbb{H}^{n}$. If $p$ is
contained in the interior of a hyperface then this is immediate. Otherwise we
can first define an isometric embedding $\Phi^{\prime}$ on the intersection of
a neighborhood of $p$ with a hyperface $H$ containing $p$. Then the fact that
hyperfaces of $\Box$ meet along constant angles, since they are contained in
totally geodesic hypersurfaces, allows us to extend $\Phi^{\prime}$ to a
neighborhood of $p$ in $\partial\Box$.

To define $\Phi$, we fix a point $p$ in $\partial\Box$, and define $\Phi$ on a
neighborhood of $p$ in $\partial\Box$ to be some isometric embedding of that
neighborhood in $\mathbb{H}^{n}$. Then for any $p^{\prime}\in\partial\Box$ and
any path $\gamma$ in $\partial\Box$ joining $p$ to $p^{\prime}$, we can use
the fact that each point has a neighborhood in $\partial\Box$ that
isometrically embeds in $\mathbb{H}^{n}$ to extend $\Phi$ along the path
$\gamma$ and define $\Phi(p^{\prime})$. Since $\partial\Box$ is simply
connected, $\Phi(p^{\prime})$ is well-defined independent of the choice of
$\gamma$. Therefore the map $\Phi$ is a well-defined local isometry onto its
image from $\partial\Box$ into $\mathbb{H}^{n}$. Note that $\Phi$ is an
isometry restricted to each hyperface of $\partial\Box$ and that the images of
opposite hyperfaces under $\Phi$ are disjoint. Therefore $\Phi$ is an
isometric embedding onto its image.

### 4.3.

We define a new Riemannian manifold $H$ as follows. First, we take the closure
of the non-compact connected component of the complement of $\Phi(\Box)$ in
$\mathbb{H}^{n}$ equipped with the hyperbolic metric. We then “fill in” the
boundary $\Phi(\Box)$ by gluing in the region $R_{\Box}$ bounded by $\Box$ in
$\tilde{N}$ to obtain $H$. Since $\Phi$ is an isometry, this defines a
Riemannian manifold which we denote by $H$. Note that apriori the metric
tensor of $H$ is only continuous.

We claim that $H$ must be isometric to $\mathbb{H}^{n}$. Our proof is similar
to some of the arguments in [SZ90] (see also the recent paper [GS22] and the
references therein.)

Every totally geodesic two-dimensional plane $S$ in $\mathbb{H}^{n}$ that
intersects $\Phi(\Box)$ transversely corresponds to a totally geodesic surface
with boundary $S^{\prime}$ in $H$, whose polygonal boundary is contained in
$\Phi(\Box)$. The surface $S^{\prime}$ is isometric to $S$ with the compact
region bounded by some $m$-sided polygon $P^{\prime}$ removed. Note that
$P^{\prime}$ is isometric to a polygon $P$ in $S\cong\mathbb{H}^{2}$, and
label the vertices of $P$ (resp. $P^{\prime}$) as $v_{1},..,v_{\ell}$ (resp.
$v_{1}^{\prime},..,v_{\ell}^{\prime}$.)

First consider the case that $P$ is a triangle. Note that $P$ has the same
angles and side-lengths as a geodesic triangle in $\mathbb{H}^{2}$. Then since
$R_{\Box}$ is geodesically convex and has curvature bounded above by $-1$, the
equality case of Corollary 3.1 implies that $P$ bounds a totally geodesic
hyperbolic solid triangle in $R_{\Box}$, and $S^{\prime}$ thus extends to an
embedded hyperbolic plane in $H$.

To prove the general case, we will show that for any $i,j$ with $1\leq
i,j\leq\ell$ and $\ell-|i-j|$ or $|i-j|$ less than $k$, the length of
$v_{i}^{\prime}v_{j}^{\prime}$ is greater than or equal to that of
$v_{i}v_{j}$. Take this as the inductive hypothesis, assume that it holds for
$k$, and choose $i$ and $j$ with $|i-j|=k-1$. Relabeling if necessary we can
assume $i=1$ and $j=k$.

First we claim that for any $m$ such that $1<m<k$, the angle
$<v_{m-1}^{\prime}v_{k}^{\prime}v_{m}^{\prime}$ is less than or equal to
$<v_{m-1}v_{k}v_{m}$. To see this, by the inductive hypothesis
$v_{m-1}^{\prime}v_{k}^{\prime}$ and $v_{m}^{\prime}v_{k}^{\prime}$ have
length greater than or equal to respectively $v_{m-1}v_{k}$ and $v_{m}v_{k}$,
and the length of $v_{m-1}^{\prime}v_{m}^{\prime}$ is the same as that of
$v_{m-1}v_{m}$. The claim then follows from Corollary 3.1.

Figure 1.

Second, we claim that angle $>v_{1}^{\prime}v_{k}^{\prime}v_{k+1}^{\prime}$ is
greater than or equal to angle $>v_{1}v_{k}v_{k+1}$. Note that the sum of
$>v_{1}^{\prime}v_{k}^{\prime}v_{k+1}^{\prime}$ and

(4.1) $\sum_{p=2}^{k-1}>v_{p-1}^{\prime}v_{k}^{\prime}v_{p}^{\prime}$

is equal to $>v_{k-1}^{\prime}v_{k}^{\prime}v_{k+1}^{\prime}$, which is equal
to $>v_{k-1}v_{k}v_{k+1}$. Therefore since (4.1) is smaller than or equal to
the same sum with the $v_{i}^{\prime}$ replaced by $v_{i}$, this implies that
$>v_{1}^{\prime}v_{k}^{\prime}v_{k+1}^{\prime}$ is greater than or equal to
$>v_{1}v_{k}v_{k+1}$. Using Corollary 3.1 we can then conclude that the length
of $v_{1}^{\prime}v_{k+1}^{\prime}$ is greater than or equal to the length of
$v_{1}v_{k+1}$. In the case of equality, we could then use the equality case
of Corollary 3.1 to conclude that $v_{1}^{\prime}v_{k}^{\prime}$ has length
equal to $v_{1}v_{k}$ and $v_{1}^{\prime}v_{k+1}^{\prime}$ has length equal to
$v_{1}v_{k+1}$, and that $\Delta v_{1}^{\prime}v_{k+1}^{\prime}v_{k}^{\prime}$
can be filled in by a totally geodesic hyperbolic triangle. Continuing in this
way we see that we must have had equality at every previous stage of the
induction, and we finally get that the length of
$v_{1}^{\prime}v_{3}^{\prime}$ equals the length of $v_{1}v_{3}$, and so
$\Delta v_{1}^{\prime}v_{2}^{\prime}v_{3}^{\prime}$ can be filled in by a
hyperbolic triangle. Removing $\Delta
v_{1}^{\prime}v_{2}^{\prime}v_{3}^{\prime}$ we obtain a new polygon with one
fewer side on which we can repeat the same argument. If we never have
equality, then we can conclude the finite induction to get that
$v_{1}^{\prime}v_{\ell}^{\prime}$ has length strictly greater than
$v_{1}v_{\ell}$, which is a contradiction since the two have the same length.
We have thus shown that $S^{\prime}$ extends to an embedded totally geodesic
hyperbolic plane.

### 4.4.

We explain why this implies that $H$ has constant curvature $-1$. Take a large
metric sphere $B$ in $\mathbb{H}^{n}$ that contains $\Phi(\Box)$, and let
$\mathcal{C}$ be the set of round (but not necessarily great) circles
contained in the metric sphere $\partial B$. There is a natural disk bundle
$\mathcal{D}$ over $\mathcal{C}$ whose fiber over $C\in\mathcal{C}$ is equal
to the totally geodesic disc in $B$ bounded by $C$. We define a map $f$ from
$\mathcal{D}$ to the Grassmann bundle $Gr_{2}(\overline{R_{\Box}})$ of
unoriented tangent 2-planes to the closure of $R_{\Box}$ as follows. Here
$Gr_{2}(\overline{R_{\Box}})$ is defined by viewing $\overline{R_{\Box}}$ as a
subset of $N$ and pulling back $Gr_{2}(N)$ to $\overline{R_{\Box}}$.

Suppose $d\in\mathcal{D}$ corresponds to a circle $C\in\mathcal{C}$ that
bounds a totally geodesic disc $D$ containing $d$. We take the unique totally
geodesic tangent plane $S$ that intersects $\partial B$ in $D$, which by what
we have shown above corresponds to a totally geodesic plane $S^{\prime}$ in
$H$. This $S^{\prime}$ contains an isometric copy $D^{\prime}$ of the
hyperbolic disc $S\cap B$. We define $f(d)$ to be the tangent plane
$\Pi^{\prime}$ to the point $d^{\prime}$ on $D^{\prime}$ corresponding to $d$
if it is contained in $\overline{R_{\Box}}$, or else the parallel transport of
$\Pi^{\prime}$ along the geodesic joining $d^{\prime}$ to its nearest point
projection to $\partial\overline{R_{\Box}}$.

Note that $f$ is continuous, and $f$ is injective near every point that maps
into the interior of $R_{\Box}$. Since $\mathcal{D}$ and the interior of
$Gr_{2}(\overline{R_{\Box}})$ both have dimension $3n-4$, the invariance of
domain theorem implies that $f$ is a local homeomorphism onto its image near
any such point $d$. Since the image of $f$ is both open and closed, the fact
that $Gr_{2}(\overline{R_{\Box}})$ is connected implies that $f$ is
surjective. It follows that $R_{\Box}$ has constant curvature $-1$, which
implies that $H$ is isometric to $\mathbb{H}^{n}$.

##  5\. Proof of Theorem 1.5

Let $M$ be a closed hyperbolic manifold, and let $N$ be a Riemannian manifold
homotopy equivalent to $N$ with sectional curvature at most $-1$. Fix a
homotopy equivalence $F:M\rightarrow N$. Let $\Sigma$ be a totally geodesic
hypersurface in $M$, and assume that $N$ contains a totally geodesic
hypersurface $\Sigma_{N}$ whose fundamental group includes to
$\pi_{1}(\Sigma)$ up to conjugacy, where we’ve used $F$ to identify
$\pi_{1}(M)$ and $\pi_{1}(N)$. To show that $\Sigma_{N}$ satisfies the well-
distribution property, we will actually need $\Sigma$ to satisfy a stronger
condition, which we define now.

######  Definition 5.1.

We say that $\Sigma$ satisfies the strong well-distribution property if the
following holds. Let $\mathcal{B}_{\Sigma}$ be the set of solid embedded
hypercubes in $\mathbb{H}^{n}=\tilde{M}$ each of whose boundary hyperfaces is
contained in a lift of $\Sigma$ to the universal cover $\mathbb{H}^{n}$. Then
for each geodesic $\gamma$ in the universal cover
$\tilde{M}\cong\mathbb{H}^{n}$ of $M$, we can find hypercubes
$\Box_{i}\in\mathcal{B}_{\Sigma}$, $i\in\mathbb{Z}$, so that the following
holds. This sequence of hypercubes is then said to enclose $\gamma.$

  1. (1)

Exactly two hyperfaces $H_{i,top}$ and $H_{i,bottom}$ of $\Box_{i}$ intersect
$\gamma$

  2. (2)

The totally geodesic hyperplanes that contain the hyperfaces of $\Box_{i}$,
excluding $H_{i,top}$ and $H_{i,bottom}$, do not intersect $\gamma$.

  3. (3)

$\Box_{i-1}$ and $\Box_{i}$ satisfy the interlocking property: the totally
geodesic hyperplane that contains $H_{i-1,top}$ is contained in the region
between those of $H_{i,top}$ and $H_{i,bottom}$.

  4. (4)

Each point on $\gamma$ is contained in some $\Box_{i}$.

Now let $\Sigma_{k}$ be an infinite sequence of distinct closed totally
geodesic hypersurfaces in $M$. We claim that for $k$ large enough,
$\Sigma_{k}$ satisfies the strong well-distribution property. By Corollary 3.3
we know that for every $\epsilon>0$ the $\Sigma_{k}$ are $\epsilon$-dense in
$Gr_{n-1}(M)$ for $k$ large enough.

For $\epsilon>0$ sufficiently small, we define the following hypercube
$\Box_{model}(\epsilon)$ with totally geodesic faces in $\mathbb{H}^{n}$. Take
some point $P$ in $\mathbb{H}^{n}$, and let $E_{1},..,E_{n}$ be an orthonormal
basis for the tangent space at that point. For each $i$, let $H_{i}^{+}$ and
$H_{i}^{-}$ be the totally geodesic hyperplanes orthogonal to the geodesic
through $P$ in the direction of $E_{i}$, and at signed distance of
respectively $\epsilon$ and $-\epsilon$ from $P$. Let $U_{i}$ be the connected
component of the complement of $H_{i}^{+}\cup H_{i}^{-}$ in $\mathbb{H}^{n}$
containing $P$. We define $\Box_{model}(\epsilon)$ to be the closure of the
intersection $\cap_{i=1}^{n}U_{i}$, where we have taken $\epsilon$ small
enough that this intersection is a solid embedded hypercube.

We say that two hypercubes $\Box$ and $\Box^{\prime}$ with totally geodesic
hyperbolic faces are $\delta$-close if there is a labeling of the vertices
$V_{1},..,V_{\ell}$ of $\Box$ and the vertices
$V_{1}^{\prime},..,V_{\ell}^{\prime}$ of $\Box^{\prime}$ such that

  1. (1)

For all $i,j$ such that $V_{i}$ and $V_{j}$ are adjacent the length of
$V_{i}V_{j}$ and the length of $V_{i}^{\prime}V_{j}^{\prime}$ differ by less
than $\delta$.

  2. (2)

For all $i,j,p$ such that $V_{i}$ and $V_{p}$ are adjacent to $V_{j}$ the
angle $>V_{i}V_{j}V_{p}$ differs from the angle
$>V_{i}^{\prime}V_{j}^{\prime}V_{p}^{\prime}$ by less than $\delta$.

Denote by $\mathcal{B}_{\Sigma_{k}}^{\epsilon}$ the set of hypercubes $\Box$
in $\mathcal{B}_{\Sigma_{k}}$ that are $\delta$-close to
$\Box_{model}(\epsilon)$, for some small $\delta$ be be specified later. The
reason for defining $\mathcal{B}_{\Sigma_{k}}^{\epsilon}$ is that having lots
of elements of $\mathcal{B}_{\Sigma_{k}}$ of controlled shape will be useful
for enclosing $\gamma$.

Fix $\epsilon>0$ small enough that $\Box_{model}(\epsilon)$ is defined. For
every geodesic $\gamma$ in the universal cover, we claim that we can find a
collection of solid hypercubes $\Box$ contained in
$\mathcal{B}_{\Sigma_{k}}^{\epsilon}$ that enclose $\gamma$, provided that $k$
was chosen sufficiently large and $\delta$ was chosen sufficiently small at
the start. Let $\mathcal{B}_{\Sigma_{k}}^{\epsilon}(\gamma)$ be the set of
hypercubes $\Box$ in the universal cover so that the following holds:

  1. (1)

$\Box$ is the lift of some element of $\mathcal{B}_{\Sigma_{k}}^{\epsilon}$ to
the universal cover

  2. (2)

$\gamma$ intersects the boundary of $\Box$ in exactly two points $p_{1}$ and
$p_{2}$ on opposite hyperfaces $H_{top}$ and $H_{bottom}$ of $\Box$

  3. (3)

$p_{1}$ and $p_{2}$ are at a distance of less than $\delta$ from the centers
of $H_{top}$ and $H_{bottom}$ respectively. Here the centers of $H_{top}$ and
$H_{bottom}$ are the endpoints of the geodesic segment of shortest length
beginning at $H_{bottom}$ and ending at $H_{top}$.

  4. (4)

$\gamma$ makes an angle between $\pi/2-\delta$ and $\pi/2+\delta$ with
$H_{top}$ and $H_{bottom}$ at respectively $p_{1}$ and $p_{2}$

Here $\delta$ is chosen small enough that for every $\gamma$ and $\Box$
satisfying the above four conditions, $\gamma$ will be disjoint from all of
the totally geodesic hyperplanes that contain hyperfaces of $\Box$ different
from $H_{top}$ and $H_{bottom}$. We emphasize that all of the requirements we
impose on $\delta$ and $k$ will be independent of $\gamma$. That they can be
satisfied follows from the fact that given any $\delta^{\prime}$, any embedded
copy of $\Box_{model}(\epsilon)$ in $\mathbb{H}^{n}$ is
$\delta^{\prime}$-close to some element of
$\mathcal{B}_{\Sigma_{k}}^{\epsilon}(\gamma)$, provided $\delta$ was chosen
small enough and $k$ was chosen large enough at the start.

If $k$ was chosen large enough, then every point on $\gamma$ will be contained
in some element of $\mathcal{B}_{\Sigma_{k}}^{\epsilon}(\gamma)$. Suppose that
$p\in\gamma$ is contained in some
$\Box_{p}\in\mathcal{B}_{\Sigma_{k}}^{\epsilon}(\gamma)$, whose boundary
intersects $\gamma$ at $p_{1}$ and $p_{2}$. Then provided $\delta$ was chosen
small enough and $k$ was chosen large enough, we can find some
$\Box_{p_{1}}\in\mathcal{B}_{\Sigma_{k}}^{\epsilon}(\gamma)$ that contains
$p_{1}$, and whose boundary intersects $\gamma$ at points $p^{\prime}$ and
$p^{\prime\prime}$ so that

$\min(d(p^{\prime},p_{1}),d(p^{\prime\prime},p_{1})>\epsilon/4.$

In this case, again provided $\delta$ was chosen small enough, $\Box_{p_{1}}$
and $\Box_{p}$ satisfy the interlocking property, and in a similar way we can
find a $\Box_{p_{2}}$ containing $p_{2}$ in its interior so that
$\Box_{p_{2}}$ and $\Box_{p}$ satisfy the interlocking property. Continuing in
this way, we can find a collection of
$\Box\in\mathcal{B}_{\Sigma_{k}}^{\epsilon}(\gamma)$ that enclose $\gamma$.
This shows that $\Sigma_{k}$ satisfies the strong well-distribution property
for large enough $k$.

### 5.1.

Suppose that $\Sigma$ satisfies the strong well-distribution property. Then we
claim that $\Sigma_{N}$ is well-distributed in $N$. First, note that each
hypercube $\Box$ bounded by lifts of $\Sigma$ to the universal cover
corresponds to a hypercube $\Box^{\prime}$ in $N$ bounded by lifts of
$\Sigma_{N}$. This can be seen as follows. Suppose that $\Box$ is bounded by
$2n+2$ totally geodesic hyperplanes $H_{1},..,H_{2n+2}$ that are lifts of
$\Sigma$.

Recall the discussion in 3.2. The map $F$ defines a homeomorphism between the
boundary at infinity of $\mathbb{H}^{n}$ and the boundary at infinity of the
universal cover $\tilde{N}$ of $N$. Taking the images of the boundaries at
infinity of the $H_{i}$ we obtain $2n+2$ $n-2$-discs in
$\partial_{\infty}\tilde{N}$. These bound totally geodesic hyperplanes that
project down to $\Sigma_{N}$. We thus obtain a hypercube $\Box^{\prime}$ in
$\tilde{N}$ whose sides are contained in lifts of $\Sigma_{N}$.

We claim that $\Sigma_{N}$ satisfies the well distribution property in $N$.
Let $\gamma_{N}$ be a geodesic in the universal cover $\tilde{N}$. Let
$\gamma$ be a geodesic in $\tilde{M}=\mathbb{H}^{n}$ with the same endpoints
at infinity as $\gamma_{N}$, identifying the two boundaries at infinity as
above. Since $\Sigma$ satisfies the strong well-distribution property, we can
find a sequence of hypercubes $\Box_{i}:i\in\mathbb{Z}$ in $\mathbb{H}^{n}$
whose hyperfaces are contained in lifts of $\Sigma$ so that the following are
true:

  1. (1)

Exactly two hyperfaces $H_{i,top}$ and $H_{i,bottom}$ of $\Box_{i}$ intersect
$\gamma$

  2. (2)

$H_{i-1,top}$ is contained in the region between $H_{i,top}$ and
$H_{i,bottom}$

  3. (3)

Each point on $\gamma$ is contained in some $\Box_{i}$.

The hyperplanes corresponding to the faces of $\Box_{i}$ excluding $H_{i,top}$
and $H_{i,bottom}$ bound an infinite solid rectangular “pillar” that contains
$\gamma$.

Figure 2. Hypercubes enclosing $\gamma$

The hypercubes $\Box_{i}^{\prime}$ corresponding to the $\Box_{i}$ satisfy the
same three properties in $\tilde{N}$, but with $\gamma$ replaced by
$\gamma_{N}$. Item (1) follows from the fact that a totally geodesic
hyperplane $H$ in $\tilde{N}$ will intersect a given geodesic $\gamma$ exactly
if the two endpoints of $\gamma$ in $\partial_{\infty}\tilde{N}$ are in
separate components of the complement of $\partial_{\infty}H$ in
$\partial_{\infty}\tilde{N}$. This implies that whether $\gamma$ intersects
$H$ is determined by what happens in $\mathbb{H}^{n}$ for the corresponding
geodesic and lift of $\Sigma$. The second property follows in a similar way.

The first two properties imply that the subset of $\gamma$ of points contained
in some $\Box_{i}^{\prime}$ is both open and closed, and so must be all of
$\gamma$.

Therefore, since every point in $N$ is contained in some geodesic, every point
$p$ of $\tilde{N}$ is contained in an embedded hypercube $\Box_{p}^{\prime}$
as above. By Theorem 1.3, this implies that $N$ has constant curvature $-1$.
Since $N$ is homotopy equivalent to $M$, $N$ must be isometric to $M$ by
Mostow’s rigidity theorem.

##  6\. Examples Where Rigidity Fails

In this section we prove the following theorem. We expect that a similar
construction works in higher dimensions, but for simplicity we focus on the
three dimensional case.

######  Theorem 6.1.

There exists a closed hyperbolic 3-manifold $(M,g_{hyp})$ and a metric $g$ on
$M$ that is not isometric to $g_{hyp}$ such that the following hold

  1. (1)

$(M,g_{hyp})$ contains a closed totally geodesic surface $S$.

  2. (2)

$S$ is a totally geodesic hyperbolic surface in the metric $(M,g)$

  3. (3)

The sectional curvature of $g$ is less than or equal to $-1$.

Our construction will be based on the following proposition.

######  Proposition 6.2.

Let $\ell>1$ and let $g_{\ell}$ be a metric on $S\times\mathbb{R}$ with
constant curvature $-\ell$ so that $S\times\\{0\\}$ is a totally geodesic
surface in the metric $g_{\ell}$. There exists a smooth metric $g$ on
$S\times\mathbb{R}$ satisfying the following conditions:

  1. (1)

$g=g_{\ell}$ outside of a compact set

  2. (2)

$g$ has sectional curvature $K_{g}\leq-1$

  3. (3)

$S\times\\{0\\}$ is totally geodesic in $g$ with constant sectional curvature
$-1$.

V. Lima constructed non-hyperbolic examples as in the proposition, but with
point (1) excluded. The examples we give were explained to us by Laurent
Mazet, and we are grateful to him for allowing us to include them. Our
verifications closely follow [Lim19][section 3].

###### Proof.

Let $(S,g_{hyp})$ be a hyperbolic surface, and let $f$ be a function defined
by

$f(t):=\frac{1}{\ell}\cosh(\ell t)+\chi(t)(1-\frac{1}{\ell})$

for $\chi(t)$ a smooth non-negative function satisfying, for $\epsilon>0$ and
$M$ respectively sufficiently small and large to be specified later:

  1. (1)

$\chi(t)=1$ near $t=0$, $\chi(t)=0$ for $|t|>M$

  2. (2)

$\max(|\chi^{\prime}(t)|,|\chi^{\prime\prime}(t)|)<\epsilon$

We take $\ell>1$. Define $g$ to be the warped product metric

$g:=f^{2}(t)g_{hyp}^{2}+dt^{2}.$

Writing $g_{\ell}$ as

$g_{\ell}=\frac{1}{\ell^{2}}\cosh^{2}(\ell t)g_{hyp}^{2}+dt^{2},$

Only the second item in the statement of the proposition requires an argument.
In computing the curvature of $g$, we follow [Lim19]. For a vector field $V$
on $(S\times\mathbb{R},g)$, define the projection to $S$ by

$V^{S}:=V-g(V,\partial_{t})\partial_{t}$

The curvature tensor $R$ of $g$ can be written in terms of the curvature
tensor $R_{S}$ of $(S,g_{hyp})$ as

(6.1) $\displaystyle R(X,Y,Z)$
$\displaystyle=R_{S}(X^{S},Y^{S})Z^{S}-\left(\frac{f^{\prime}}{f}\right)^{2}(g(X,Z)Y-g(Y,Z)X)$
(6.2) $\displaystyle+(\log
f)^{\prime\prime}g(Z,\partial_{t})(g(Y,\partial_{t})X-g(X,\partial_{t})Y)$
(6.3) $\displaystyle-(\log
f)^{\prime\prime}(g(X,Z)g(Y,\partial_{t})-g(Y,Z)g(X,\partial_{t})\partial_{t},$

for $X$, $Y$, and $Z$ smooth vector fields. Given a plane $\Pi$ spanned by
unit length orthogonal vector fields $X,Y$, we have that the sectional
curvature of $\Pi$ is given by

(6.4) $\displaystyle K_{\Pi}$
$\displaystyle=\frac{K_{S}-(f^{\prime})^{2}}{f^{2}}+\left(\frac{-K_{S}+(f^{\prime})^{2}-f^{\prime\prime}f}{f^{2}}\right)(g(X,\partial_{t})^{2}+g(Y,\partial_{t})^{2})$

For $|t|$ smaller than some $t_{0}$ we have that $\chi(t)$ is equal to $1$.
The numerator of the coefficent of
$(g(X,\partial_{t})^{2}+g(Y,\partial_{t})^{2})$ is then equal to

(6.5) $\displaystyle=1+\sinh^{2}(\ell t)-\ell\cosh\ell
t(\frac{1}{\ell}\cosh(\ell t)+\chi(t)(1-\frac{1}{\ell}))$ (6.6)
$\displaystyle=(1-\ell)\cosh\ell t,$

which is less than zero, so we have that (6.4) is maximized when
$(g(X,\partial_{t})^{2}+g(Y,\partial_{t})^{2})=0$. In this case, we just need
to show that the initial term

(6.7) $\frac{K_{S}-(f^{\prime})^{2}}{f^{2}}=\frac{-1-(\sinh(\ell
t)+\chi^{\prime}(t)(1-\frac{1}{\ell}))^{2}}{\left(\frac{1}{\ell}\cosh(\ell
t)+\chi(t)(1-\frac{1}{\ell})\right)^{2}}$

is at most $-1$. This will actually be true for all $t$, not just $|t|<t_{0}$,
which we verify now. Note that this holds at $t=0$. It is then enough to show
that the quantity (6.7) is decreasing in $t$ for $t>0$ and increasing in $t$
for $t<0$. For the derivative of the quantity (6.7) to be negative, we need

(6.8)

$\displaystyle 0>$ $\displaystyle-2(\sinh(\ell
t)+\chi^{\prime}(t)(1-\frac{1}{\ell}))(\ell\cosh(\ell
t)+\chi^{\prime\prime}(t)(1-\frac{1}{\ell}))(\frac{1}{\ell}\cosh(\ell
t)+\chi(t)(1-\frac{1}{\ell}))+$ $\displaystyle(1+(\sinh(\ell
t)+\chi^{\prime}(t)(1-\frac{1}{\ell}))^{2})(\sinh(\ell
t)+\chi^{\prime}(t)(1-\frac{1}{\ell})).$

Choose $t_{0}$ so that $\chi(t)=1$ for $|t|<t_{0}$. We first do the case
$|t|<t_{0}$. Assume that $t>0$. To verify (6.8), it is enough to show that

(6.9) $0\geq\sinh(\ell t)(-2\cosh^{2}(\ell t)-2\ell\cosh(\ell
t)\chi(t)(1-\frac{1}{\ell})+1+\sinh^{2}(\ell t)),$

where the RHS of (6.9) is all of the terms of the RHS of (6.8) that don’t
contain $\chi^{\prime}(t)$ or $\chi^{\prime\prime}(t)$. Since $\sinh(\ell
t)(-2\ell\cosh(\ell t)\chi(t)(1-\frac{1}{\ell}))$ is always non-positive
provided $t>0$, for (6.9) to hold it is enough that

(6.10) $0\geq-2\cosh^{2}(\ell t)+1+\sinh^{2}(\ell t),$

which is true since the RHS equals $-\cosh^{2}(\ell t)$. Thus the quantity
(6.7) is negative if $t_{0}>t>0$, and the same reasoning shows it is positive
if $-t_{0}<t<0$.

Now assume that $t\geq t_{0}$ (the case $t\leq-t_{0}$ is similar.) Then by the
previous computations, we will be able to conclude that (6.7) is smaller than
$-1$ for all $t$ as long as $-\sinh(\ell t)\cosh^{2}(\ell t)$ is larger in
absolute value than all of the terms of the RHS of (6.8) that, when expanded
out, contain $\chi^{\prime}(t)$ or $\chi^{\prime\prime}(t)$. This will be true
provided $\epsilon$ was taken small enough at the start, since
$\max(|\chi^{\prime}(t)|,|\chi^{\prime\prime}(t)|)<\epsilon$ and no term in
(6.8) has more than three factors equal to $\sinh(\ell t)$ or $\cosh(\ell t)$.
We have thus shown that (6.7) is smaller than $-1$ for all $t$.

We now check the case that $|t|>t_{0}$. Because the initial term in (6.4) is
always less than or equal to $-1$, if the second term in (6.4) is negative
then we are done. If not, it is enough to check (6.4) in the case that
$g(X,\partial_{t})^{2}+g(Y,\partial_{t})^{2}$ assumes its maximum possible
value:

$g(X,\partial_{t})^{2}+g(Y,\partial_{t})^{2}=1.$

The quantity (6.4) is then equal to $-f^{\prime\prime}/f$. Therefore we need
to show that

$-1\geq-f^{\prime\prime}/f=-\frac{\ell^{2}\cosh(\ell
t)+\ell\chi^{\prime\prime}(t)(1-\frac{1}{\ell})}{\cosh(\ell
t)+\ell\chi(t)(1-\frac{1}{\ell})})).$

Since $\chi\leq 1$, it is enough to show that

(6.11) $1\leq\frac{\ell^{2}\cosh(\ell t)}{\cosh(\ell
t)+\ell-1}+\frac{\chi^{\prime\prime}(t)(\ell-1)}{\cosh(\ell t)+\ell-1}.$

Note that

$\ell^{2}\cosh(\ell t)\geq\ell\cosh(\ell t)+\ell(\ell-1),$

so the first term on the RHS of (6.11) is greater than or equal to

$\frac{\ell\cosh(\ell t)+\ell(\ell-1)}{\cosh(\ell t)+\ell-1}=\ell.$

We can make the second term on the RHS of (6.11) as small as desired by
choosing $\epsilon$ small enough because $|\chi^{\prime\prime}|<\epsilon$.
Therefore inequality (6.11) holds, which completes the proof of Proposition
6.2.

∎

We also need the following lemma.

######  Lemma 6.3.

Let $M$ be a closed hyperbolic 3-manifold containing an embedded two-sided
totally geodesic surface $S$. Then for any $L>0$, there exists a finite cover
$M_{0}$ of $M$ which contains a totally geodesic surface $S_{0}$ isometric to
$S$ and with normal injectivity radius greater than $L$.

Recall that $S_{0}$ has normal injectivity radius greater than $L$ in $M_{0}$
if no two distinct geodesic segments of length less than $L$ beginning normal
to $S_{0}$ intersect. This means that the normal exponential map
$S_{0}\times(-L,L)\rightarrow M_{0}$ is a diffeomorphism onto its image, for
$S_{0}\times(-L,L)$ identified with a tubular neighborhood of the zero section
of the normal bundle of $S_{0}$ in $M_{0}$ in the natural way.

###### Proof.

This is a consequence of subgroup separability for
$\pi_{1}(S,p)\subset\pi_{1}(M,p)$, for some choice of basepoint $p$ on $S$
[MR03][Lemma 5.3.6]. This means that for any finite subset
$\\{g_{1},..g_{n}\\}\subset\pi_{1}(M,p)$ disjoint from $\pi_{1}(S,p)$, we can
choose a finite index subgroup of $\pi_{1}(M,p)$ containing $\pi_{1}(S,p)$ but
none of $g_{1},..g_{n}$.

Let $F$ be the Fuchsian cover of $M$ corresponding to $\pi_{1}(S)$. Fix a
connected polyhedral fundamental domain $P$ for the action of $\pi_{1}(M)$ on
the universal cover $\mathbb{H}^{3}$ of $M$. Then a fundamental domain in
$\mathbb{H}^{3}$ for the cover $F$ is tiled by copies of $P$, and any fixed
normal neighborhood of $S$ in $F$ is contained in the projections of finitely
many such copies. The images of the projections to $F$ of the copies of $P$
correspond to cosets of $\pi_{1}(S)$ in $\pi_{1}(M)$. Choose finitely many
coset representatives $g_{1},..,g_{n}$ so that the corresponding projections
of copies of $P$ contain the $L$-neighborhood of $S$ in $F$.

Now use subgroup separability to choose a finite index subgroup of
$\pi_{1}(M)$ containing $\pi_{1}(S)$ but none of $g_{1},..,g_{n}$. The finite
cover of $M$ corresponding to this finite index subgroup is then also covered
by $F$, and the restriction of the covering map to the $L$-neighborhood of $S$
in $F$ is a diffeomorphism onto its image. This shows that we can make the
normal injectivity radius of $S$ as large as desired by passing to finite
covers. ∎

We can now prove Theorem 6.1. (Compare [Low21][Section 6], where a similar
construction appeared.) Take a closed hyperbolic 3-manifold $(M,g_{hyp})$
containing an embedded two-sided totally geodesic surface $S$. See [MR03] for
examples.

Let $F\cong S\times\mathbb{R}$ be the Fuchsian cover of $(M,g_{hyp})$
corresponding to $S$. Let $g_{\ell}$ be the homothetic scaling of the
hyperbolic metric on $F$ that has constant curvature $-\ell$ for some
$\ell>1$. We apply Proposition 6.2 to obtain a metric $g_{K}$ on $F$ that
agrees with $g_{\ell}$ outside a compact set $K$, and so that $S\times\\{0\\}$
is a totally geodesic hyperbolic surface.

Next we apply Lemma 6.3 to find a finite cover of $M^{\prime}$ so that $F$
also covers $M^{\prime}$, and the restriction of the covering map
$\rho:F\rightarrow M^{\prime}$ to $K$ is injective. We can then define a
metric on $M^{\prime}$ to be the scaled hyperbolic metric with constant
curvature $-\ell$ outside of $\rho(K)$, and to be the pushforward of $g_{K}$
under $\rho$ on $\rho(K)$. Since $g_{\ell}$ and $g_{K}$ agree outside of the
compact set $K$, this gives a smooth well-defined metric on $M^{\prime}$,
which completes the proof.

## References

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