# Entropy and stability of hyperbolic manifolds

Antoine Song  California Institute of Technology
177 Linde Hall, #1200 E. California Blvd., Pasadena, CA 91125
<EMAIL_ADDRESS>
###### Abstract.

Let $M$ be a closed oriented hyperbolic manifold of dimension at least $3$ and
let $g_{0}$ be the hyperbolic metric on $M$. Besson-Courtois-Gallot proved
that for any Riemannian metric $g$ on $M$ with same volume as $g_{0}$, its
volume entropy $h(g)$ satisfies $h(g)\geq n-1$ with equality only when $g$ is
isometric to $g_{0}$. We show that the hyperbolic metric is stable in the
following sense. If $g_{i}$ is a sequence of Riemaniann metrics on $M$ of same
volume as $g_{0}$ such that $h(g_{i})$ converges to $n-1$, then $(M,g_{i})$
converges to $(M,g_{0})$ as metric measure spaces in the Gromov-Prokhorov
topology. The proof is based on an intrinsic uniqueness result for spherical
Plateau solutions of $M$.

## Introduction

Let $M$ be a hyperbolic manifold of dimension at least $3$ with hyperbolic
metric $g_{0}$. If $g$ is a Riemannian metric on $M$, let $h(g)$ denote its
volume entropy. The fundamental volume entropy inequality, proved by Besson-
Courtois-Gallot in [BCG95, BCG96], asserts that for any Riemannian metric $g$
on $M$ of same volume as $g_{0}$, we have

(1) $h(g)\geq h(g_{0})=n-1.$

Moreover Besson-Courtois-Gallot showed that this inequality is rigid in the
sense that if equality holds, then $g$ is isometric to $g_{0}$. In this paper
we show that this inequality is in fact stable for the Gromov-Prokhorov
topology:

######  Theorem 0.1.

Let $(M,g_{0})$ be a closed oriented hyperbolic manifold of dimension at least
$3$. Let $\\{g_{i}\\}_{i\geq 1}$ be a sequence of Riemannian metrics on $M$
with $\operatorname{Vol}(M,g_{i})=\operatorname{Vol}(M,g_{0})$. If

$\lim_{i\to\infty}h(g_{i})=h(g_{0})=n-1,$

then $(M,g_{i})$ converges to $(M,g_{0})$ in the Gromov-Prokhorov topology.

Unlike other geometric stability theorems, the Gromov-Hausdorff topology is
not suitable for our problem. Indeed by adding thin and long threads to the
hyperbolic metric $g_{0}$, we get a new metric $g$ whose volume and volume
entropy are arbitrarily close to $\operatorname{Vol}(M,g_{0})$ and $n-1$
respectively. Although in this example, $(M,g)$ is far from $(M,g_{0})$ in the
Gromov-Hausdorff topology, $(M,g)$ does approximate $(M,g_{0})$ in the Gromov-
Prokhorov topology for metric measure spaces. Recall that a sequence of metric
measure spaces $(X_{i},d_{i},m_{i})$ converges to $(X,d,m)$ in the Gromov-
Prokhorov topology if there exist a separable Banach space $\mathcal{B}$ and
isometric embeddings

$j_{i}:(X_{i},d_{i})\to\mathcal{B},\quad j:(X,d)\to\mathcal{B}$

such that the push-forward measures $(j_{i})_{\sharp}m_{i}$ converge weakly to
$j_{\sharp}m$. Any closed Riemannian manifold $(M,g)$ is endowed with the
length metric of $g$ and the Riemannian volume measure
$\operatorname{dvol}_{g}$, so it can be regarded as a metric measure space.

The question of stability for the volume entropy was stated by Courtois in
[Cou98], and variants of this problem have been previously studied by
Bessières-Besson-Courtois-Gallot [BBCG12] under a lower bound on the Ricci
curvature (see also [LW11]), by Guillarmou-Lefeuvre [GL19] and Guillarmou-
Knieper-Lefeuvre [GKL22] for neighborhoods of negatively curved manifolds, and
Butt [But22] assuming uniform negative curvature bounds.

An important feature of Theorem 0.1 is that we do not assume any a priori
curvature bound. It will be clear from the proof that Theorem 0.1 holds when
replacing $(M,g_{i})$ by $(N,g_{i})$ where $N$ is a smooth manifold homotopy
equivalent (but not necessarily diffeomorphic) to $M$. The differential
rigidity result of [BBCG12] can then be seen as a corollary of Theorem 0.1: if
the Ricci curvature of $g_{i}$ is bounded from below, the limit in the Gromov-
Prokhorov sense yields the limit in the Gromov-Hausdorff sense, and a result
in the theory of Cheeger-Colding [CC97] says that if $(N,g_{i})$ is Gromov-
Hausdorff close to $(M,g_{0})$ then $N$ is diffeomorphic to $M$.

The main new input in the proof of Theorem 0.1 is to combine the theory of
integral currents in metric spaces from Geometric Measure Theory [AK00a,
Lan11, Wen11, SW11] with the original methods of Besson-Courtois-Gallot
[BCG95, BCG96]. In particular we make essential use of a compactness theorem
due to Wenger [Wen11] which is formulated in terms of the intrinsic flat
topology for integral current spaces [SW11]. There is a second important
ingredient in the proof of Theorem 0.1, which is an application of the
equidistribution theorem for geodesic spheres in closed hyperbolic manifolds.
These two parts together yield a stability result stronger than Theorem 0.1:
under the same assumptions, $(M,g_{i})$ actually converges to $(M,g_{0})$ with
respect to the intrinsic flat topology, see Theorem 3.3.

The stability of geometric inequalities has been extensively studied. In the
context of bounded Ricci curvature and the Gromov-Hausdorff topology, see
[Col96b, Col96a, Pet99, Aub05, CRX19]. For spectral isoperimetric inequalities
on surfaces and stability results with respect to certain Sobolev norms, see
[KNPS21]. For stability theorems formulated with the intrinsic flat topology,
in the context of scalar curvature problems, see [LS14, HLS17, S+21] and
references therein. It seems that the stronger version of our main result,
Theorem 3.3, gives one of the first examples of stability in the intrinsic
flat topology without any additional assumptions. For results towards
stability of tori with almost nonnegative scalar curvature, see [LNN20, All21,
CL22]. Finally, for the “restricted” Gromov-Hausdorff-Prokhorov stability of
the 3-dimensional Positive Mass Theorem, see our work with Conghan Dong
[DS23].

The first part of the proof of Theorem 0.1 is based on a certain volume
minimization problem in infinite dimension, called the spherical Plateau
problem. Let $(M,g_{0})$ be a closed oriented hyperbolic manifold with its
hyperbolic metric and let $\Gamma:=\pi_{1}(M)$. A key step in Besson-Courtois-
Gallot’s proof of the entropy inequality (1) is the following remarkable
property of hyperbolic metrics: there is a Riemannian isometric embedding of
the rescaled hyperbolic manifold $(M,\frac{(n-1)^{2}}{4n}g_{0})$ as a volume
minimizing submanifold inside a quotient of the Hilbert unit sphere
$S^{\infty}$ by an action of $\Gamma$. That result led us to introduce in
[Son23] the spherical Plateau problem for group homology classes. In the
particular case of a hyperbolic manifold $M$, the set-up we will use is the
following: consider the unit sphere $\mathcal{S}(\Gamma)$ in the Hilbert space
$\ell^{2}(\Gamma)$ and let $\Gamma$ act on $\mathcal{S}(\Gamma)$ be the
regular representation. Denote by $\mathcal{S}(\Gamma)/\Gamma$ the
corresponding quotient Hilbert manifold, endowed with the standard round
Hilbert-Riemannian metric $\mathbf{g}_{\text{Hil}}$. There is a unique
homotopy class $\mathscr{F}(M)$ of smooth immersions from $M$ to
$\mathcal{S}(\Gamma)/\Gamma$ inducing an isomorphism on the fundamental
groups. Besson-Courtois-Gallot define the _spherical volume_ of $M$ [BCG91] as
follows

$\operatorname{SphereVol}(M):=\inf\\{\operatorname{Vol}(M,\phi^{*}\mathbf{g}_{\text{Hil}});\quad\phi\in\mathscr{F}(M)\\}.$

Now consider any minimizing sequence of maps $\phi_{i}\in\mathscr{F}(M)$,
namely a sequence such that

$\lim_{i\to\infty}\operatorname{Vol}(M,\phi_{i}^{*}\mathbf{g}_{\text{Hil}})=\operatorname{SphereVol}(M).$

By Wenger’s compactness theorem [Wen11], the images $\phi_{i}(M)$
subsequentially converge as integral current spaces to an integral current
space $C_{\infty}=(X_{\infty},d_{\infty},T_{\infty})$ in the intrinsic flat
topology, in the sense of Sormani-Wenger [SW11]. Here
$(X_{\infty},d_{\infty})$ is a metric space and $T_{\infty}$ is an integral
current in the completion of $(X_{\infty},d_{\infty})$, see Subsection 1.1. We
call any such intrinsic flat limit $C_{\infty}$ a _spherical Plateau solution_
for $M$.

By the work of Besson-Courtois-Gallot [BCG95, BCG96],

$\operatorname{SphereVol}(M)=\operatorname{Vol}(M,\frac{(n-1)^{2}}{4n}g_{0}).$

Our main theorem on the spherical Plateau problem for $M$ is the following
intrinsic uniqueness result:

######  Theorem 0.2.

If $(M,g_{0})$ is a closed oriented hyperbolic manifold of dimension at least
$3$, then any spherical Plateau solution for $M$ is intrinsically isomorphic
to $(M,\frac{(n-1)^{2}}{4n}g_{0})$.

The notion of “intrinsic isomorphism” between two integral current spaces,
used in the above statement, is defined in Definition 2.5; it essentially
means that the intrinsic metrics (i.e. the length structures) of the
underlying metric spaces are isometric and the induced current structures are
the same. In [Son23], we outline the proof of a similar intrinsic uniqueness
result for spherical Plateau solutions of any closed oriented $3$-manifold and
the proof of the intrinsic asymptotic rigidity of higher dimensional analogues
of hyperbolic Dehn fillings.

### Outline of proofs

Theorem 0.2: In order to describe the proof, it is helpful to recall how
Besson-Courtois-Gallot were able to compute the spherical volume
$\operatorname{SphereVol}(M)=\operatorname{Vol}(M,\frac{(n-1)^{2}}{4n}g_{0})$.
Their main tool was the barycenter map ${\mathrm{Bar}}$. In our setting, this
is a Lipschitz map which under some technical conditions sends cycles in
$\mathcal{S}(\Gamma)/\Gamma$ of the form $\phi(M)$, where
$\phi\in\mathcal{F}(M)$, to the rescaled hyperbolic manifold
$(M,\frac{(n-1)^{2}}{4n}g_{0})$ with topological degree $1$. Roughly speaking,
the Jacobian of restriction of the barycenter map
${\mathrm{Bar}}:\phi(M)\to(M,\frac{(n-1)^{2}}{4n}g_{0})$ satisfies [BCG95,
BCG96]

(2) $|\operatorname{Jac}{\mathrm{Bar}}|\leq 1,$

which in particular implies that
$\operatorname{SphereVol}(M)\geq\operatorname{Vol}(M,\frac{(n-1)^{2}}{4n}g_{0})$.
Then the opposite inequality is checked by finding an explicit sequence of
embeddings $\phi_{i}\in\mathcal{F}(M)$ such that
$\lim_{i\to\infty}\operatorname{Vol}(M,\phi_{i}^{*}\mathbf{g}_{\text{Hil}})=\operatorname{Vol}(M,\frac{(n-1)^{2}}{4n}g_{0}).$

In order to show that spherical Plateau solutions are unique up to intrinsic
isomorphism, we try to argue as follows. Consider a minimizing sequence of
maps $\phi_{i}\in\mathcal{F}(M)$, and denote by $C_{i}$ the integral currents
of $\mathcal{S}(\Gamma)/\Gamma$ induced by pushing forward the fundamental
class of $M$ by $\phi_{i}$. The barycenter map ${\mathrm{Bar}}$ enjoys the
Jacobian bound (2) which is almost achieved on a region
$\Omega_{i}\subset\operatorname{spt}(C_{i})$ that covers almost all of
$\operatorname{spt}(C_{i})$ as $i\to\infty$. This implies an important local
Lipschitz bound for ${\mathrm{Bar}}$, which holds on a whole neighborhood of
$\Omega_{i}$, and the differential of ${\mathrm{Bar}}$ at points of
$\Omega_{i}$ is close to a linear isometry. Supposing that $C_{i}$ converges
to $C_{\infty}=(X_{\infty},d_{\infty},S_{\infty})$, we then construct a limit
map

${\mathrm{Bar}}_{\infty}:X_{\infty}\to M$

sending the current structure $S_{\infty}$ to the natural current structure
$\llbracket 1_{M}\rrbracket$ supported on $M$. Heuristically, as $i$ goes to
infinity, the Jacobian bound (2) for
${\mathrm{Bar}}:\operatorname{spt}(C_{i})\to(M,\frac{(n-1)^{2}}{4n}g_{0})$
should be almost saturated almost everywhere, which means that the
differential of ${\mathrm{Bar}}$ should be close to a linear isometry almost
everywhere. Passing to the limit, we should deduce that
${\mathrm{Bar}}_{\infty}$ is an isometry for the intrinsic metrics, which
essentially would conclude the proof. This strategy of constructing a limit
barycenter map has been exploited in the rigidity theorems of [BCG95,
Proposition 7.1] and [BBCG12]. There, the authors can argue that since their
limit barycenter map is $1$-Lipschitz and preserves the volume, it has to be
an isometry.

However, for this argument to work, it is essential that the limit space is
smooth outside of a small singular set. The new challenge in our case is the
lack of regularity for spherical Plateau solutions: a priori they are just
integral currents in Banach spaces. An additional difficulty is that the
supports $\operatorname{spt}(C_{i})$ are in general far away from
$\operatorname{spt}(C_{\infty})$ in the Gromov-Hausdorff topology. To resolve
these issues, we show a curve lifting property stating that a “generic” curve
$\sigma$ in $(M,\frac{(n-1)^{2}}{4n}g_{0})$ lifts via ${\mathrm{Bar}}$ to a
curve $\sigma^{\prime}$ in $\operatorname{spt}(C_{i})$ of approximately the
same length as $\sigma$, so that ${\mathrm{Bar}}:\operatorname{spt}(C_{i})\to
M$ is approximately an isometry for length structures on a large subset of
$\operatorname{spt}(C_{i})$. This information passes to the limit
${\mathrm{Bar}}_{\infty}:X_{\infty}\to M$, and gives the desired conclusion.
The curve lifting property is in turn proved with an averaging argument
involving the coarea formula.

Theorem 0.1:

Given a Riemannian metric $g$ on $M$ with same volume as $g_{0}$ and with
entropy close to $n-1$, there is a Lipschitz map

$\Psi:(M,\frac{(n-1)^{2}}{4n}g)\to(\mathcal{S}(\Gamma)/\Gamma,\mathbf{g}_{\mathrm{Hil}})$

which is almost a Riemannian isometry to its image, as observed by Besson-
Courtois-Gallot [BCG91]: this is what relates the spherical Plateau problem to
the volume entropy. Applying the proof of Theorem 0.1 to
$\mathrm{Bar}\circ\Psi$, we deduce that $(M,\frac{(n-1)^{2}}{4n}g)$ is close
in the intrinsic flat topology to an integral current space
$C_{\infty}=(X_{\infty},d_{\infty},S_{\infty})$ which is intrinsically
isomorphic to $(M,\frac{(n-1)^{2}}{4n}g_{0})$.

This is not enough to conclude because two non-isometric metric spaces can be
intrinsically isometric, and it could be that $(M,\frac{(n-1)^{2}}{4n}g)$ and
$(M,\frac{(n-1)^{2}}{4n}g_{0})$ are in fact far away in the intrinsic flat
topology or Gromov-Prokhorov topology. Nevertheless, if that were the case, it
would mean that there is a pair of points $A,B\in(M,\frac{(n-1)^{2}}{4n}g)$
whose $g$-distance is strictly shorter than the $g_{0}$-distance between any
pair of points $A^{\prime},B^{\prime}\in X_{\infty}=M$ approximating $A,B$.
The equidistribution of geodesic spheres in the unit tangent bundles of closed
hyperbolic manifolds [EM93] forces the existence of a positive proportion of
closed geodesics in $(M,\frac{(n-1)^{2}}{4n}g_{0})$ under a length bound, each
of which contains many length minimizing segments from a neighborhood of
$A^{\prime}$ to a neighborhood of $B^{\prime}$. By our assumption, we could
lift those closed geodesics to curves in $(M,\frac{(n-1)^{2}}{4n}g)$ with
substantially smaller length. But since the volume entropy measures the growth
rate of lengths of minimizing curves in homotopy classes, this would imply
that the volume entropy of $(M,g)$ is greater than $n-1$ by a definite amount,
which is a contradiction leading to the intrinsic flat stability of the
entropy inequality.

We conclude the proof by applying a lemma of Portegies [Por15]: if a sequence
of integral current spaces converges to a limit in the intrinsic flat topology
without volume loss, then viewed as metric measure spaces the sequence
converges to the limit in the Gromov-Prokhorov topology.

### Organisation

Section 1 is devoted to some preliminary results. In Subsection 1.1, we review
Ambrosio-Kirchheim’s theory of metric currents, including the intrinsic flat
topology of Sormani-Wenger, and Wenger’s compactness theorem. We prove a
technical lemma on how to put together certain converging sequences of
integral current spaces. In Subsection 1.2, we illustrate how we use the
coarea formula with a toy example. Then in Subsection 1.3, we define the
barycenter map in our setting, and we state the main Jacobian bound due to
Besson-Courtois-Gallot. We also review in Subsection 1.4 an equidistribution
result for geodesic spheres in the unit tangent bundle of hyperbolic
manifolds.

In Section 2, we define the spherical Plateau problem for a closed oriented
hyperbolic manifold and prove the intrinsic uniqueness of spherical Plateau
solutions in Theorem 0.2.

In Section 3, we apply the equidistribution theorem for geodesic spheres and
the arguments of Theorem 0.2 to establish stability for the volume entropy
inequality in terms of the intrinsic flat topology. This is shown to imply the
desired result Theorem 0.1.

### Acknowledgements

I am grateful to Gérard Besson, Gilles Courtois, Juan Souto, John Lott, Ursula
Hamenstädt, Hyun Chul Jang and Ben Lowe for insightful and helpful discussions
during the writing of this article.

A.S. was partially supported by NSF grant DMS-2104254. This research was
conducted during the period A.S. served as a Clay Research Fellow.

##  1\. Preliminaries

###  1.1. Currents in metric spaces

The theory of currents in metric spaces begins with works of De Giorgi, and
Ambrosio-Kirchheim [AK00a]. It extends the theory of currents in finite
dimensional manifolds due to De Giorgi, Federer-Fleming. For the most part, in
this paper we will only stay in the standard framework of smooth maps and
smooth manifolds. Nevertheless, a key reason for caring about metric currents
is that this general theory enables to formulate powerful compactness results
like Theorem 1.1 below. Besides, there is a profusion of standard tools (weak
convergence, area/coarea formulae, slicing, push-forward…) for which the most
natural language is given by Geometric Measure Theory.

The main references we will need on the theory of metric currents are [AK00a,
AK00b, Wen11, SW11]. We reviewed in some details the main definitions and
results of the theory in Section 1 of [Son23]. Integral currents in complete
metric spaces are, roughly speaking, a countable union of Lipschitz push-
forwards of classical integral currents in Euclidean spaces. They give a
workable notion of “oriented submanifolds” in complete metric spaces like
Hilbert manifolds or Banach spaces. An $n$-dimensional integral current $S$
has a well-defined notion of boundary $\partial S$ which is an
$(n-1)$-dimensional integral current, a notion of volume measure denoted by
$\|S\|$ and a notion of total volume called mass $\mathbf{M}(S)$. Such a
current $S$ is concentrated on a so-called canonical set
$\operatorname{set}(S)$, itself included in the support
$\operatorname{spt}(S)$ of the measure $\|S\|$. The restriction of $S$ to a
Borel set $A$ is denoted by $S\llcorner A$, and its push-forward by a
Lipschitz map $\phi$ is called $\phi_{\sharp}S$. See [AK00a, Section 3], see
also [Son23, Subsections 1.1, 1.2] for more details.

The space of integral currents in a given complete metric space is endowed
with the weak topology and flat topology, and the latter is finer than the
former, see [Wen07, Subsection 1.1] [Son23, Subsection 1.3]. The mass is lower
semicontinuous with respect to convergence in those topologies [AK00a].

The area formula expresses the mass of an integral current by its image under
a Lipschitz map [AK00b, Section 8], [AK00a, Section 9], [Son23, Subsection
1.4]. The coarea formula, a kind of dual formula, expresses the mass of an
integral current in terms of a double integral involving level sets of a
Lipschitz map [AK00b, Section 9], [Son23, Subsection 1.4]. The slicing theorem
is a kind of generalization of Sard’s theorem and tells us that most level
sets of a Lipschitz map are integral currents themselves [AK00a, Theorems 5.6
and 5.7].

Following the notion of integral currents in complete metric spaces, one can
define a more intrinsic notion of integral currents. That was achieved by
Sormani-Wenger [SW11]. Basically an integral current space is a triple
$(X,d,S)$ where $(X,d)$ is a metric space and $S$ is an integral current in
the completion of $(X,d)$ (to make the choice of $X$ canonical, one requires
that $X$ is the “canonical set” of the current $S$) [Son23, Definition 1.3 and
Subsection 1.1]. A simple example of integral current space is given by a
closed oriented Riemannian $n$-manifold $(N,h)$: the metric space is $N$
endowed with the geodesic distance of $h$, and the integral current structure
is the usual current induced by the fundamental class $[N]\in
H_{n}(N;\mathbb{Z})$.

There is also an intrinsic notion of flat topology, called intrinsic flat
topology [SW11]. Similarly to the definition of Gromov-Hausdorff topology, two
integral current spaces are close in the intrinsic flat topology whenever they
can be isometrically embedded in a common Banach space in which they are close
in the usual flat topology [Son23, Definition 1.4] .

A key result is Wenger’s compactness theorem:

######  Theorem 1.1.

[Wen11][SW11, Theorem 4.19] Given a sequence of boundaryless integral current
spaces

$(X_{m},d_{m},S_{m})$

with uniformly bounded mass and diameter, there is a subsequence converging to
an integral current space in the intrinsic flat topology.

We need a technical lemma which embeds different converging sequences in a
common Banach space.

######  Lemma 1.2.

For $i\geq 0$ and $k\geq 0$, let $C_{i}$ be an $n$-dimensional integral
current in a complete metric space $(E_{i},d_{i})$ of uniformly bounded
diameter, and consider

$\text{a Borel set }U_{k,i}\subset E_{i},\quad\text{a compact set
}K_{k,i}\subset E_{i},$

such that $C_{i}\llcorner U_{k,i}$ is an integral current space. Suppose that

$\mathbf{M}(C_{i})+\mathbf{M}(\partial C_{i})\quad\text{ and
}\quad\mathbf{M}(C_{i}\llcorner U_{k,i})+\mathbf{M}(\partial(C_{i}\llcorner
U_{k,i}))$

are uniformly bounded as $i\to\infty$. Suppose that $\\{C_{i}\\}_{i\geq 0}$
and $\\{C_{i}\llcorner U_{k,i}\\}_{i\geq 0}$ converge in the intrinsic flat
topology respectively to $(X_{\infty},d_{\infty},S_{\infty})$ and
$(W_{k},d_{k},T_{k})$, and that $\\{K_{k,i}\\}_{i\geq 0}$ converges in the
Gromov-Hausdorff topology to $K_{k}$.

Then after extracting a subsequence, there exist a Banach space $\mathbf{Z}$
and isometric embeddings

$\operatorname{spt}(S_{\infty})\hookrightarrow\mathbf{Z},\quad\operatorname{spt}(T_{k})\hookrightarrow\mathbf{Z},\quad
K_{k}\hookrightarrow\mathbf{Z},\quad j_{i}:E_{i}\to\mathbf{Z},$

such that inside $\mathbf{Z}$:

  * •

$(j_{i})_{\sharp}(C_{i})$ converges in the flat topology to $S_{\infty}$,

  * •

for all $k$, $(j_{i})_{\sharp}(C_{i}\llcorner U_{k,i})$ converges in the flat
topology to $T_{k}$,

  * •

for all $k$, $j_{i}(K_{k,i})$ converges in the Hausdorff topology to $K_{k}$.

###### Proof.

This lemma is a corollary of the arguments in [Wen11]. Going through the proof
of [Wen11, Theorem 1.2], one checks that it implies the following
decomposition: for each $i\geq 0$,

$C_{i}=F_{i}^{1}+...+F_{i}^{i}+R_{i}^{i}$

where $F_{i}^{q}$ and $R_{i}^{i}$ are integral currents such that for each
fixed $q$, the sequence $\\{\operatorname{spt}(F_{i}^{q})\\}_{i}$ is equi-
compact with uniformly bounded diameter (see [AK00a, Definition 6.5]).
Moreover for each integer $L$,

(3)
$\limsup_{i\to\infty}\sum_{q=1}^{L}\mathbf{M}(F_{i}^{q})+\mathbf{M}(\partial
F_{i}^{q})<\infty,$

and for any $\eta>0$, there is $L_{\eta}$ so that for all $L>L_{\eta}$ and
$i>L$:

(4)
$\mathbf{d}_{\mathcal{F}}(F_{i}^{L+1}+...+F_{i}^{i}+R^{i}_{i},\mathbf{0})\leq\eta$

where $\mathbf{d}_{\mathcal{F}}$ is the intrinsic flat distance and
$\mathbf{0}$ is the zero integral current. Similarly, for each $k\geq 0$ and
$i\geq 0$, there are analogous decompositions

$C_{i}\llcorner U_{k,i}=F_{k,i}^{1}+...+F_{k,i}^{i}+R_{k,i}^{i}.$

For each $m\geq 0$, set

$B_{i}^{m}:=\big{(}\bigcup_{k=0}^{m}K_{k,i}\big{)}\cup\big{(}\bigcup_{q=0}^{\min\\{m,i\\}}\operatorname{spt}(F_{i}^{q})\big{)}\cup\big{(}\bigcup_{k=0}^{m}\bigcup_{q=0}^{\min\\{m,i\\}}\operatorname{spt}(F_{k,i}^{q})\big{)}.$

The sequence $\\{B_{i}^{m}\\}_{i\geq 0}$ is still equi-compact for each $m\geq
0$. We can now apply [Wen11, Proposition 5.2 and Section 5]: by extracting a
subsequence with a diagonal argument and renumbering, we conclude that there
is a metric space $\mathbf{Z}$ (which can be assumed to be a Banach space by
embedding into an $\ell^{\infty}$ space) and there are isometric embeddings

$j_{i}:(E_{i},d_{i})\hookrightarrow\mathbf{Z}$

such that the following is true.

  * •

$(j_{i})_{\sharp}(C_{i})$ converges in the flat topology to a limit integral
current of $\mathbf{Z}$; this follows from (3), (4), the compactness and
closure theorems of [AK00a], and [Wen07] as in the proof of [Wen11, Theorem
1.2]. We identify that limit with $S_{\infty}$ by uniqueness of intrinsic flat
limits [Wen11, Theorem 1.3].

  * •

Similarly for each $k>0$, $(j_{i})_{\sharp}(C_{i}\llcorner U_{k,i})$ converges
in the flat topology to a limit integral current of $\mathbf{Z}$ that we
identify with $T_{k}$.

  * •

For each $k>0$, $j_{i}(K_{k,i})$ converges in the Hausdorff topology to a
compact subset of $\mathbf{Z}$ which we identify with $K_{k}$ by uniqueness of
compact Gromov-Hausdorff limits.

∎

###  1.2. An application of the coarea formula

Below is an example illustrating how the standard coarea formula will be used
later.

Toy example: Let $\lambda>0$. Consider a sequence of immersed oriented compact
$n$-submanifolds with boundary $\\{\Sigma_{i}\\}_{i\geq 0}$ in a Riemannian
Hilbert manifold $(E,d)$. Suppose that there is a sequence of
$\lambda$-Lipschitz smooth maps

$\\{\pi^{\prime}_{i}:\operatorname{spt}(T_{i})\to\mathbb{R}^{n}\\}_{i\geq 0},$

and that for each $i$, consider there are subsets
$A_{i},B_{i}\subset\Sigma_{i}$ so that

$\lim_{i\to\infty}\mathcal{H}^{n-1}(\pi^{\prime}_{i}(\partial\Sigma_{i})\cap[0,3]\times[0,1]^{n-1})=0,$
$\pi^{\prime}_{i}(A_{i})\subset[0,1]\times[0,1]^{n-1},$
$\pi^{\prime}_{i}(B_{i})\subset[2,3]\times[0,1]^{n-1},$

and for some $\kappa>0$ independent of $i$, for all $i$:

$\quad\mathcal{H}^{n}(\pi^{\prime}_{i}(A_{i}))\geq\kappa,$
$\mathcal{H}^{n}(\pi^{\prime}_{i}(B_{i}))\geq\kappa.$

Let $\mathrm{proj}:\mathbb{R}^{n}\to\mathbb{R}^{n-1}$ be the projection on the
last $n-1$ coordinates. By Fubini’s theorem, for each $i$ we can find a vector
$\overrightarrow{t_{i}}\in\\{0\\}\times[-1,1]^{n-1}$ such that if we set

$\displaystyle\mathcal{W}_{i}:=\\{x\in[0,1]^{n-1};\quad$
$\displaystyle\mathcal{H}^{1}(\mathrm{proj}^{-1}(x)\cap\pi^{\prime}_{i}(A_{i}))>0\text{
and}$
$\displaystyle\mathcal{H}^{1}(\mathrm{proj}^{-1}(x)\cap(\pi^{\prime}_{i}(B_{i})+\overrightarrow{t_{i}}))>0\\}$

then we have

$\mathcal{H}^{n-1}(\mathcal{W}_{i})>\kappa^{\prime}$

for some $\kappa^{\prime}>0$ independent of $i$. By composing each
$\pi^{\prime}_{i}$ with a diffeomorphism with uniformly bounded Lipschitz
constant if necessary, we can assume that $\overrightarrow{t_{i}}=0$. Applying
the coarea formula and Sard’s theorem twice, first to the $\lambda$-Lipschitz
maps $\pi_{i}:=\mathrm{proj}\circ\pi^{\prime}_{i}$, then to the map
$(x_{1},...x_{n})\mapsto x_{1}$, we find for each $i$ some straight segment
$\sigma_{i}$ joining $a_{i}\in\pi^{\prime}_{i}(A_{i})$ to
$b_{i}\in\pi^{\prime}_{i}(B_{i})$, such that $\mathrm{proj}(\sigma_{i})$ is a
point in $[0,1]^{n-1}$ and such that

$\varkappa_{i}:=(\pi^{\prime}_{i})^{-1}(\sigma_{i})\subset\Sigma_{i}$

is a compact smooth curve which avoids $\partial\Sigma_{i}$ for all $i$ large
enough:

$\varkappa_{i}\cap\partial\Sigma_{i}=\varnothing.$

Moreover,

$\pi^{\prime}_{i}(\partial\varkappa_{i})\subset\pi^{\prime}_{i}(A_{i})\cup\pi^{\prime}_{i}(B_{i})$

This example generalizes when replacing $\mathbb{R}^{n}$ with a manifold.

###  1.3. The barycenter map and the Jacobian bound

The barycenter map played a crucial role in the work of Besson-Courtois-Gallot
on the volume entropy inequality [BCG95, BCG96] (see also [BCG99, Sam99, CF03,
Sou08] for a small sample of other uses of the barycenter map).

For the reader’s convenience, all the main properties of the barycenter map
are proved in our setting in [Son23, Section 2] and the main Jacobian bound is
recalled below. We choose to express the barycenter map using the
$\ell^{2}$-space on a group, instead of the $L^{2}$-space on a boundary as in
[BCG95]. The advantage is that only a minimal amount of knowledge is needed,
and that it extends directly to other more general situations (3-manifolds,
connected sums, Plateau Dehn fillings, see [Son23, Sections 4, 5, 6]). On the
other hand the fact that distance functions are non-smooth contrarily to
Busemann functions is a minor inconvenience in our setting.

Let $(M,g_{0})$ be a closed oriented hyperbolic manifold. Let
$(\tilde{M},g_{0})$ be its universal cover, namely the hyperbolic $n$-space.
Let $\Gamma:=\pi_{1}(M)$. The latter acts properly cocompactly and freely on
$(\tilde{M},g_{0})$. Recall that $\mathcal{S}(\Gamma)$ is the unit sphere in
the Hilbert space $\ell^{2}(\Gamma)$.

Fix a point $o\in\tilde{M}$. For $w\in\tilde{M}$, let
$\rho_{w}(.):=\operatorname{dist}_{g_{0}}(w,.)$. For
$f\in\mathcal{S}(\Gamma)$, consider the functional

(5) $\begin{split}\mathcal{B}_{f}&:\tilde{M}\to[0,\infty]\\\
\mathcal{B}_{f}(x)&:=\sum_{\gamma\in\Gamma}|f(\gamma)|^{2}\rho_{\gamma.o}(x).\end{split}$

######  Definition 1.3.

The set $\mathbb{S}^{+}$ of _admissible_ functions is defined as the set of
functions in $\mathcal{S}(\Gamma)$ satisfying the following conditions:

  1. (1)

the support of $f$ is finite,

  2. (2)

for any two points $a\neq b\in\tilde{M}$, there is $\gamma\in\Gamma$ such that
$f(\gamma)\neq 0$, and the points $a$, $b$ and $\gamma.o$ are not aligned,
namely they are not contained in a geodesic ray of $(\tilde{M},g_{0})$.

The _barycenter map_ is then defined as

$\mathrm{Bar}:\mathbb{S}^{+}\to\tilde{M}$ $\mathrm{Bar}(f):=\text{ the unique
point minimizing $\mathcal{B}_{f}$}.$

The barycenter map is well-defined [Son23, Section 2]. The subset
$\mathbb{S}^{+}\subset\mathcal{S}(\Gamma)$ is invariant by $\Gamma$, and
$\mathrm{Bar}$ is continuous and $\Gamma$-equivariant. The quotient map
$\mathbb{S}^{+}/\Gamma\to M$ is also denoted by $\mathrm{Bar}$.

Given any smooth embedding
$\varphi:\mathbb{R}^{k}\to\mathbb{S}^{+}\subset\mathcal{S}(\Gamma)$ where
$1\leq k\leq n$, The tangent $k$-plane $Q$ of such $\varphi(\mathbb{R}^{k})$
at $p:=\varphi(y)$ will be called a tangent $k$-plane of $\mathbb{S}^{+}$
based at $p$. The map

$\mathrm{Bar}:\varphi(\mathbb{R}^{k})\to\tilde{M}$

is smooth around $p$ whenever
$\mathrm{Bar}(p)\in\tilde{M}\setminus\\{\gamma.o\\}_{\gamma\in\Gamma}$. In
this case, its differential along $Q$ is denoted by
$d\mathrm{Bar}\big{|}_{Q}:Q\to T_{\mathrm{Bar}(p)}\tilde{M}$.

The main result in this Subsection is the following (see [Son23, Lemma 2.4]
for a proof):

######  Lemma 1.4.

[BCG95] Suppose that $n\geq 3$. Let $f\in\mathbb{S}^{+}$ be such that
$x:=\mathrm{Bar}(f)$ belongs to
$\tilde{M}\setminus\\{\gamma.o\\}_{\gamma\in\Gamma}$. Let $Q$ be a tangent
$n$-plane of $\mathbb{S}^{+}$ based at $f$. Then

(6)
$|\operatorname{Jac}\mathrm{Bar}\big{|}_{Q}|\leq\big{(}\frac{{4n}}{(n-1)^{2}}\big{)}^{n/2}.$

Moreover for any $\eta>0$ small enough and $d>0$, there exists $c_{\eta,d}>0$
with $\lim_{\eta\to 0}c_{\eta,d}=0$ for every $d$, such that the following
holds. If

$\operatorname{dist}_{g_{0}}(x,\\{\gamma.o\\}_{\gamma\in\Gamma})\geq d>0$

and

$|\operatorname{Jac}\mathrm{Bar}\big{|}_{Q}|\geq\big{(}\frac{{4n}}{(n-1)^{2}}\big{)}^{n/2}-\eta,$

then for all norm $1$ tangent vector $\vec{u}\subset Q$,

(7)
$|d\mathrm{Bar}\big{|}_{Q}(\vec{u})|\geq\big{(}\frac{4n}{(n-1)^{2}}\big{)}^{1/2}-c_{\eta,d}$

and for any connected curve $\alpha\subset\mathbb{S}^{+}$ of length less than
$\eta$ starting at $f$, we have

(8)
$\operatorname{length}_{g_{0}}(\mathrm{Bar}(\alpha))\leq(\big{(}\frac{4n}{(n-1)^{2}}\big{)}^{1/2}+c_{\eta,d})\operatorname{length}(\alpha).$

###  1.4. Equidistribution of geodesic spheres in hyperbolic manifolds

Consider $(M,g_{0})$ a closed hyperbolic manifold, with universal cover
$\tilde{M}$. Fix $x_{0}\in M$ and let $\tilde{x}_{0}$ be a lift of $x_{0}$ by
the natural projection $\tilde{M}\to M$. Let $T^{1}M$ denote the unit tangent
bundle of $M$. Let $\tilde{S}(\tilde{x}_{0},t)$ be the geodesic sphere of
radius $t$ centered at $\tilde{x}_{0}$ in $\tilde{M}$, and let
$\tilde{S}_{1}(\tilde{x}_{0},t)$ be its lift to the unit tangent bundle
$T^{1}\tilde{M}$ by considering the outward unit normal of
$\tilde{S}(\tilde{x}_{0},t)$. Let $S(x_{0},t)$ denote the projection of
$\tilde{S}(\tilde{x},t)$ in $M$, and let $S_{1}(x_{0},t)$ be the projection of
$\tilde{S}_{1}(\tilde{x}_{0},t)$ to the unit tangent bundle $T^{1}M$. A
measure on $T^{1}M$ (resp. ${S}_{1}({x}_{0},t)$) is called invariant if it is
induced by a measure on $T^{1}\tilde{M}$ invariant by isometries of
$\tilde{M}$ (resp. induced by a measure on $\tilde{S}_{1}(\tilde{x}_{0},t)$
invariant by rotations of center $\tilde{x}_{0}$ in $\tilde{M}$).

As a corollary of the mixing property for the geodesic flow on closed
hyperbolic manifolds, the lift of geodesic spheres equidistribute in the unit
tangent bundle. This is for instance explained in [EM93, Section 2] for
surfaces and generalized in [EM93, Theorem 1.2] 111I thank Ben Lowe for
pointing out this reference.. The statement is the following:

######  Theorem 1.5.

Let $x_{0}\in M$. For any continuous function $f:T^{1}M\to\mathbb{R}$,

$\lim_{t\to\infty}\int_{{S}_{1}({x}_{0},t)}f(y)d\mu_{t}(y)=\int_{T^{1}M}f(y)dv_{T^{1}M}(y),$

where $d\mu_{t}$ is the unique invariant probability measure on
${S}_{1}({x}_{0},t)$ and $dv_{T^{1}M}$ is the unique invariant probability
measure on $T^{1}M$.

Now let $A,B\in M$ and let $U_{A},U_{B}$ be two disjoint balls centered at $A$
and $B$. Fix $\tilde{x}_{0}\in\tilde{M}$ as before. For a Borel subset of
$\tilde{M}$, denote by $\operatorname{Vol}_{k}$ its $k$-dimensional
$g_{0}$-volume. Let

$\pi:\tilde{M}\to M$

be the natural projection.

######  Corollary 1.6.

There is $\theta\in(0,1)$ depending on $M,U_{A},U_{B}$ such that for all $t$
large enough, there is an open subset $W_{t}\subset\tilde{S}(\tilde{x}_{0},t)$
such that

$\operatorname{Vol}_{n-1}(W_{t})\geq\theta\operatorname{Vol}_{n-1}(\tilde{S}(\tilde{x}_{0},t))$

with the following property: for any $y\in W_{t}$, if
$l:[0,\operatorname{length}(l)]\to\tilde{M}$ denotes the length minimizing
geodesic from $\tilde{x}_{0}$ to $y$ in $(\tilde{M},g_{0})$ parametrized by
arclength, there are disjoint intervals

$[a_{1},b_{1}],...,[a_{m},b_{m}]\subset[0,\operatorname{length}(l)]$

such that

  * •

for $j\in\\{1,...,m\\}$, the restriction of $\pi\circ l$ to $[a_{j},b_{j}]$ is
length minimizing among all paths in $M$ joining $\pi(l(a_{j}))$ to
$\pi(l(b_{j}))$,

  * •

for $j\in\\{1,...,m\\}$, the endpoints satisfy $\pi(l(a_{j}))\in U_{A}$ and
$\pi(l(b_{j}))\in U_{B}$,

  * •

$\frac{1}{\operatorname{length}(l)}\sum_{j=1}^{m}|b_{j}-a_{j}|\geq\theta>0.$

###### Proof.

There is an open subset $O$ of the unit tangent bundle $T^{1}M$ such that for
any tangent vector $v$ in $O$, the base point $p$ of $v$ lies in $U_{A}$, and
for some $t_{0}>0$ depending only on $M,U_{A},U_{B}$, the geodesic starting at
$p$ with direction $v$ and length $t_{0}$ is length minimizing and ends at a
point $q\in U_{B}$. By Theorem 1.5, a uniformly positive proportion of
${S}_{1}({x}_{0},t)$ wit respect to $d\mu_{t}$ meets the open set $O$ for all
$t$ large. In the universal cover, it means that a uniform proportion of
$\tilde{S}_{1}(\tilde{x}_{0},t)$ meets lifts of the open set $O$ for all $t$
large. By Fubini’s theorem, we deduce that for all $t$ large, there is a
uniformly positive proportion of $\tilde{S}_{1}(\tilde{x}_{0},t)$ such that
geodesics from $\tilde{x}_{0}$ to that portion of
$\tilde{S}_{1}(\tilde{x}_{0},t)$ meet lifts of $O$ for a uniformly positive
proportion of their lengths. This explains the corollary. ∎

##  2\. The spherical Plateau problem for hyperbolic manifolds

###  2.1. The spherical Plateau problem

Let us define the spherical Plateau problem for closed oriented hyperbolic
manifolds. It is part of a more general framework [Son23]. Let $M$ be a closed
oriented hyperbolic manifold, whose fundamental group is denoted by $\Gamma$.
Let $\mathcal{S}$ be the unit sphere in $\ell^{2}(\Gamma)$. The
$\ell^{2}$-norm induces a Hilbert-Riemannian metric
$\mathbf{g}_{\mathrm{Hil}}$ on $\mathcal{S}(\Gamma)$. The group $\Gamma$ acts
isometrically on $\mathcal{S}(\Gamma)$ by the left regular representation: for
all $\gamma\in\Gamma$, $x\in\Gamma$, $f\in\mathcal{S}(\Gamma)$,

$\gamma.f(x):=f(\gamma^{-1}x).$

Since $\Gamma$ is infinite and torsion-free, $\Gamma$ acts properly and freely
on the infinite dimensional sphere $\mathcal{S}(\Gamma)$. The quotient space
$\mathcal{S}(\Gamma)/\Gamma$ is topologically a classifying space for
$\Gamma$. It is also a Hilbert manifold endowed with the induced Hilbert-
Riemannian metric $\mathbf{g}_{\mathrm{Hil}}$. The diameter of
$(\mathcal{S}(\Gamma)/\Gamma,\mathbf{g}_{\mathrm{Hil}})$ is bounded from above
by $\pi$.

Given base points $p_{0}\in M$, $q_{0}\in\mathcal{S}(\Gamma)/\Gamma$, there is
a smooth immersion $M\to\mathcal{S}(\Gamma)/\Gamma$ inducing the identity map
from $\pi_{1}(M,p_{0})$ to $\pi_{1}(\mathcal{S}(\Gamma)/\Gamma,q_{0})$, which
is unique up to homotopies sending $p_{0}$ to $q_{0}$. Other choices of
$p_{0},q_{0}$ yield homotopic maps, so that determines a unique homotopy class
of maps which we call “admissible”. Set

$\mathscr{F}(M):=\\{\phi:M\to\mathcal{S}(\Gamma)/\Gamma;\quad\text{ $\phi$ is
an admissible smooth immersion}\\}.$

Any map $\phi\in\mathscr{F}(M)$ defines the pull-back Riemannian metric
$\phi^{*}(\mathbf{g}_{\mathrm{Hil}})$ on $M$.

Besson-Courtois-Gallot introduced the spherical volume of $M$ in [BCG91]. It
can be equivalently be defined as follows.

######  Definition 2.1.

The _spherical volume_ of $M$ is defined as

$\operatorname{SphereVol}(M):=\inf\\{\operatorname{Vol}(M,\phi^{*}(\mathbf{g}_{\mathrm{Hil}}));\quad\phi\in\mathscr{F}(M)\\}.$

The spherical volume of closed oriented hyperbolic manifolds was computed by
Besson-Courtois-Gallot. See [Son23, Theorem 4.1] for a proof in our setting.

######  Theorem 2.2.

[BCG95, BCG96] Let $(M,g_{0})$ be a closed oriented hyperbolic manifold. Then

(9)
$\operatorname{SphereVol}(M)=\operatorname{Vol}(M,\frac{(n-1)^{2}}{{4n}}g_{0}).$

A sequence $\phi_{i}\in\mathscr{F}(M)$ is said to be _minimizing_ if

$\lim_{i\to\infty}\operatorname{Vol}(M,\phi_{i}^{*}(\mathbf{g}_{\mathrm{Hil}}))=\operatorname{SphereVol}(M).$

Denote by $\llbracket 1_{M}\rrbracket$ the integral current in $(M,g_{0})$
induced by $M$ and its orientation. For a Lipschitz map
$\phi:M\to\mathcal{S}(\Gamma)/\Gamma$, recall that $\phi_{\sharp}(\llbracket
1_{M}\rrbracket)$ denotes the push-forward integral current in
$\mathcal{S}(\Gamma)/\Gamma$. We can now define spherical Plateau solutions.

######  Definition 2.3.

We call spherical Plateau solution for $M$ any $n$-dimensional integral
current space $C_{\infty}$ which is the limit in the intrinsic flat topology
of a sequence $C_{i}:=(\phi_{i})_{\sharp}\llbracket 1_{M}\rrbracket$ where
$\phi_{i}\in\mathscr{F}(M)$ is a minimizing sequence.

For any sequence $\phi_{i}\in\mathscr{F}(M)$ such that

$\lim_{i\to\infty}\operatorname{Vol}(M,\phi_{i}^{*}(\mathbf{g}_{\mathrm{Hil}}))=\operatorname{SphereVol}(M),$

the mass and diameter of $(\phi_{i})_{\sharp}\llbracket 1_{M}\rrbracket$ are
uniformly bounded, so by Wenger’s compactness (Theorem 1.1) there is a
subsequence of $(\phi_{i})_{\sharp}\llbracket 1_{M}\rrbracket$ converging in
the intrinsic flat topology. The need for an abstract compactness result like
Theorem 1.1 is explained in [Son23, Remark 3.3].

######  Remark 2.4.

While for our present purpose, it is enough to consider the set
$\mathscr{F}(M)$ of admissible smooth immersions from $M$ to
$\mathcal{S}(\Gamma)/\Gamma$, we believe that it is better to formulate the
general spherical Plateau problem in terms of integral currents with compact
support in $\mathcal{S}(\Gamma)/\Gamma$ representing a homology class $h\in
H_{n}(\Gamma;\mathbb{Z})$. This is the point of view presented in [Son23,
Section 3]. In fact, by [Bru08] and a standard polyhedral approximation result
for integral currents in Hilbert manifolds, it is possible to prove that these
two setups lead to the same notions of spherical volume and spherical Plateau
solutions at least when the countable group $\Gamma$ is torsion-free.

###  2.2. Intrinsic uniqueness for hyperbolic manifolds

From a geometric point of view, a natural question is the uniqueness of
spherical Plateau solutions for closed hyperbolic manifolds. It seems quite
difficult to control the extrinsic metrics on spherical Plateau solutions. To
bypass such issues, it is helpful to consider integral current spaces up to
“intrinsic isomorphism” and discard the non-infinitesimal information.

Consider two integral current spaces $C_{1}:=(X_{1},d_{1},T_{1})$,
$C_{2}:=(X_{2},d_{2},T_{2})$. The intrinsic metric (also called length
structure) on $X_{j}$ induced by $d_{j}$ is denoted by $L_{d_{j}}$. Note that
if the identity map $(X_{j},d_{j})\to(X_{j},L_{d_{j}})$ is locally Lipschitz,
then the current $T_{j}$ determines an integral current in $(X_{j},L_{d_{j}})$
still denoted by $T_{j}$.

######  Definition 2.5.

The integral current spaces $C_{1}$ and $C_{2}$ are said to be _intrinsically
isomorphic_ if the identity maps $(X_{j},d_{j})\to(X_{j},L_{d_{j}})$ are
locally Lipschitz, and $(X_{1},L_{d_{1}})$ is isometric to $(X_{2},L_{d_{2}})$
via a map $\varphi$ such that $\varphi_{\sharp}(T_{1})=T_{2}$.

In case $(X_{2},d_{2},T_{2})=(M,\operatorname{dist}_{g},\llbracket
1_{M}\rrbracket)$ for some metric $g$ on $M$, we just say that $C_{1}$ is
intrinsically isomorphic to $(M,g)$. To avoid confusion, note that
“intrinsically isomorphic” does not imply nor is implied by “at intrinsic flat
distance $0$ from each other”.

Our main result in this section shows that in dimensions at least $3$, the
spherical Plateau solutions for closed hyperbolic manifolds are unique up to
intrinsic isomorphism, see Definition 2.5.

######  Theorem 2.6.

Let $(M,g_{0})$ be a closed oriented hyperbolic manifold of dimension at least
$3$. Then any spherical Plateau solution for $M$ is intrinsically isomorphic
to $(M,\frac{(n-1)^{2}}{{4n}}g_{0})$.

###### Proof.

For the reader’s convenience, the proof is summarized in [Son23, Section 4],
and the notations from the theory of metric currents are defined in [Son23,
Section 1]. Let $\phi_{i}\in\mathscr{F}(M)$ be a minimizing sequence, namely

(10)
$\lim_{i\to\infty}\operatorname{Vol}(M,\phi_{i}^{*}(\mathbf{g}_{\mathrm{Hil}}))=\operatorname{SphereVol}(M)=\operatorname{Vol}(M,\frac{(n-1)^{2}}{4n}g_{0}),$

where the second equality follows from Theorem 2.2. We suppose that the
integral currents

$C_{i}:=(\phi_{i})_{\sharp}\llbracket 1_{M}\rrbracket$

converge in the intrinsic flat topology to a spherical Plateau solution

$C_{\infty}=(X_{\infty},d_{\infty},S_{\infty}).$

Set $\Gamma:=\pi_{1}(M)$. By a perturbation argument, we can assume without
loss of generality that for all $i$, for all $y\in M$, any lift of
$\phi_{i}(y)\in\mathcal{S}(\Gamma)/\Gamma$ in
$\mathcal{S}(\Gamma)\subset\ell^{2}(\Gamma)$ has finite support. In
particular, we can assume that

$\operatorname{spt}(C_{i})\subset\mathbb{S}^{+}/\Gamma$

where $\mathbb{S}^{+}$ is defined in Section 1.3. By a further perturbation,
we can even assume that $\phi_{i}$ are smooth embeddings, namely
$\operatorname{spt}(C_{i})=\phi_{i}(M)$ is a closed embedded $n$-submanifold
of $\mathcal{S}(\Gamma)/\Gamma$.

From now on, we will use the notation

$g^{\prime}:=\frac{(n-1)^{2}}{4n}g_{0}.$

In the sequel, Jacobians, lengths and distances will be computed with respect
to the metric $g^{\prime}$ on $M$. Let $o\in\tilde{M}$ and call its projection
$\mathbf{p}\in M$. Let

$\mathrm{Bar}:\mathbb{S}^{+}/\Gamma\to M$

be the barycenter map, see Section 1.3. By $\Gamma$-equivariance, for any $i$,

(11) ${\mathrm{Bar}}_{\sharp}(C_{i})=\llbracket 1_{M}\rrbracket.$

Step 1 (Definitions of $\Omega_{i}$, $N_{d}$ and $D_{k}$):

By lower semicontinuity of the mass under intrinsic flat convergence [SW11]:

(12)
$\mathbf{M}(C_{\infty})\leq\operatorname{SphereVol}(M)=\operatorname{Vol}(M,g^{\prime})$

(the equality above is Theorem 2.2).

If a point $q\in\operatorname{spt}(C_{i})$ is sent by the map ${\mathrm{Bar}}$
to $M\setminus\\{\mathbf{p}\\}$, the $n$-dimensional Jacobian of
${\mathrm{Bar}}$ along the tangent $n$-plane of $\operatorname{spt}(C_{i})$ at
$q$ is well-defined and is bounded from above by $1$ by (6) in Lemma 1.4 with
respect to the metric $g^{\prime}$ on $M$. These remarks imply by the area
formula and (12) that

$\mathbf{M}(C_{i})\geq\operatorname{Vol}(M,g^{\prime})=\operatorname{SphereVol}(M).$

Since $C_{i}$ has mass converging to $\operatorname{SphereVol}(M)$, by the
area formula, the Jacobian of ${\mathrm{Bar}}$ has to be close to $1$ on a
larger and larger part of $\operatorname{spt}(C_{i})$ as $i\to\infty$, meaning
that there are open subsets

$\Omega_{i}\subset\operatorname{spt}(C_{i})$

such that at every point $q\in\Omega_{i}$, the map ${\mathrm{Bar}}$ is
differentiable along the tangent $n$-plane at $q$, and

(13)
$\begin{split}\lim_{i\to\infty}\mathbf{M}(C_{i}\llcorner\Omega_{i})=\lim_{i\to\infty}\mathbf{M}(C_{i})&=\operatorname{SphereVol}(M),
\\\
\lim_{i\to\infty}\|\operatorname{Jac}{{\mathrm{Bar}}}-1\|_{L^{\infty}(\Omega_{i})}&=0,\\\
\forall
y\in\Omega_{i},\quad{\mathrm{Bar}}^{-1}({\mathrm{Bar}}(y))\cap\operatorname{spt}(C_{i})&=\\{y\\},\end{split}$

where we recall that the Jacobian is computed with $g^{\prime}$ and
$\operatorname{Jac}{{\mathrm{Bar}}}$ denotes the Jacobian along the tangent
$n$-plane, see Section 1.1. For $r>0$, set

$\Omega_{i,r}:=\text{the $r$-neighborhood of $\Omega_{i}$ in
$\mathbb{S}^{+}/\Gamma\subset\mathcal{S}(\Gamma)/\Gamma$}.$

Let $d>0$ and set

$N_{d}:=\\{q\in M;\quad\operatorname{dist}_{g^{\prime}}(q,\mathbf{p})<d\\}.$

Inequality (8) of Lemma 1.4 ensures that a Lipschitz bound holds uniformly in
a neighborhood of $\Omega_{i}\setminus{{\mathrm{Bar}}}^{-1}(N_{d})$: for any
$\epsilon>0$, there is $r_{\epsilon}>0$, depending also on $d$, such that if
$i$ is large enough, then for
$f\in\Omega_{i}\setminus{{\mathrm{Bar}}}^{-1}(N_{d})$ and
$f^{\prime}\in\mathbb{S}^{+}/\Gamma$ joined to $f$ by a curve
$\alpha\subset\mathbb{S}^{+}/\Gamma$ of length at most $r_{\epsilon}>0$, we
have

(14)
$\operatorname{length}_{g^{\prime}}({\mathrm{Bar}}(\alpha))\leq(1+\epsilon)\operatorname{length}(\alpha).$

Given $f,f^{\prime}\in\mathbb{S}^{+}/\Gamma$ and a curve in
$\mathcal{S}(\Gamma)/\Gamma$ joining those two elements, after a small
perturbation that curve can be assumed to be inside $\mathbb{S}^{+}/\Gamma$.
As a consequence of (14), we get the following local Lipschitz bound: for any
$\tilde{r}\in(0,r_{1})$, the restriction of ${\mathrm{Bar}}$ to the subset
$\Omega_{i,\tilde{r}}\setminus{{\mathrm{Bar}}}^{-1}(N_{d})$ is
$\lambda$-Lipschitz for some $\lambda$ depending on $d$ but not on $i$.

By the local Lipschitz bound and (13), the coarea formula, Sard’s theorem and
Wenger’s compactness theorem (see Theorem 1.1 in Section 1.1), after taking a
subsequence if necessary, there is $\tilde{r}\in(0,r_{1})$ independent of $k$
and $i$, and a decreasing sequence of positive numbers $\\{d_{k}\\}_{k\geq 0}$
converging to $0$ such that for each $k$ and all $i$,

$D_{k,i}:=C_{i}\llcorner\big{(}\Omega_{i,\tilde{r}}\setminus{{\mathrm{Bar}}}^{-1}(N_{d_{k}})\big{)}$

is an integral current, with $\operatorname{spt}(D_{k,i})$ a compact
submanifold with boundary, satisfying the following: for each $k$,
$\mathbf{M}(D_{k,i})+\mathbf{M}(\partial D_{k,i})$ is uniformly bounded as
$i\to\infty$, $D_{k,i}$ converges to some limit integral current space $D_{k}$
in the intrinsic flat topology as $i\to\infty$, and if $\overline{N}_{d_{k}}$
denotes the closure of $N_{d_{k}}$ in $M$,

(15) $\lim_{i\to\infty}\|\partial
D_{k,i}\|\big{(}{{\mathrm{Bar}}}^{-1}(M\setminus\overline{N}_{d_{k}})\big{)}=0.$

In particular, there is a Banach space $\mathbf{Z}^{\prime}$ and isometric
embeddings

$j_{i}:\operatorname{spt}(D_{k,i})\hookrightarrow\mathbf{Z}^{\prime},\quad\operatorname{spt}(D_{k})\subset\mathbf{Z}^{\prime}$

(with a slight abuse of notations), and integral currents $U_{i},V_{i}$ of
$\mathbf{Z}^{\prime}$ such that

(16) $\begin{split}(j_{i})_{\sharp}(D_{k,i})-D_{k}&=U_{i}+\partial V_{i},\\\
\lim_{i\to\infty}\mathbf{M}(U_{i})&=\lim_{i\to\infty}\mathbf{M}(V_{i})=0.\end{split}$

Later in the proof, we will see that $\\{\operatorname{spt}(D_{k})\\}$ forms a
nested sequence whose union yields $X_{\infty}$ minus a point.

Step 2 (Construction of the limit map
${\mathrm{Bar}}_{k,\infty}:\operatorname{spt}(D_{k})\to M\setminus
N_{d_{k}}$):

Fix $k\geq 0$. We have seen that by (14), for $\lambda$ depending on $k$ but
not on $i$, the restriction

(17) ${\mathrm{Bar}}:\operatorname{spt}(D_{k,i})\to M\setminus
N_{d_{k}}\quad\text{is $\lambda$-Lipschitz.}$

By (13) and (11),

(18) ${\mathrm{Bar}}_{\sharp}(D_{k,i})\quad\text{converges in the flat
topology to $\llbracket 1_{M\setminus N_{d_{k}}}\rrbracket.$}$

Let $L^{\infty}(M\setminus N_{d_{k}})$ be the Banach space of bounded real
functions on $M\setminus N_{d_{k}}$. It is well-known that $(M\setminus
N_{d_{k}},g^{\prime})$ embeds isometrically inside $L(M\setminus N_{d_{k}})$
by a Kuratowski embedding, and that $L^{\infty}(M\setminus N_{d_{k}})$ is an
injective metric space in the sense that given any other metric space $Y$, a
subset $A\subset Y$, and a $\lambda$-Lipschitz map $\phi:A\to
L^{\infty}(M\setminus N_{d_{k}})$, there exists an extension $\phi:Y\to
L^{\infty}(M\setminus N_{d_{k}})$ which is $\lambda$-Lipschitz. Applying this
to the maps ${\mathrm{Bar}}\circ j_{i}^{-1}$ and using (16), (17), (18), we
get a sequence of $\lambda$-Lipschitz maps

$\theta_{i}:\operatorname{spt}((j_{i})_{\sharp}(D_{k,i}))\cup\operatorname{spt}(U_{i})\cup\operatorname{spt}(V_{i})\to
L^{\infty}(M\setminus N_{d_{k}}),$

so that
$\theta_{i}\big{|}_{\operatorname{spt}((j_{i})_{\sharp}(D_{k,i}))}={\mathrm{Bar}}\circ
j_{i}^{-1}$ and the sequence $(\theta_{i})_{\sharp}(D_{k})$ converges in the
flat topology to $\llbracket 1_{M\setminus N_{d_{k}}}\rrbracket$ inside
$L^{\infty}(M\setminus N_{d_{k}})$, in which $(M\setminus
N_{d_{k}},g^{\prime})$ is embedded isometrically as explained before.

Using $\theta_{i}$, we can construct a map
${\mathrm{Bar}}_{k,\infty}:\operatorname{spt}(D_{k})\to M\setminus N_{d_{k}}$
as follows. The set $\operatorname{spt}(D_{k})$ is separable, so there is a
dense sequence of points $\\{z_{l}\\}_{l\geq
0}\subset\operatorname{spt}(D_{k})$. For any $z_{l}$, the distance between
$\theta_{i}(z_{l})$ and $M\setminus N_{d_{k}}$ inside $L^{\infty}(M\setminus
N_{d_{k}})$ converges to $0$: indeed, $z_{l}$ is the limit of a sequence of
points $z_{l,i}\in\operatorname{spt}((j_{i})_{\sharp}(D_{k,i}))$ which are
sent by the $\lambda$-Lispchitz maps $\theta_{i}$ inside $M\setminus
N_{d_{k}}$. By compactness of $M\setminus N_{d_{k}}$ and a diagonal argument,
after extracting a subsequence, for each $z_{l}$, $\theta_{i}(z_{l})$
converges to a point $w_{l}\in M\setminus N_{d_{k}}$. Set

${\mathrm{Bar}}_{k,\infty}(z_{l}):=w_{l}.$

Since the maps $\theta_{i}$ are $\lambda$-Lipschitz,
${\mathrm{Bar}}_{k,\infty}$ extends uniquely to a $\lambda$-Lipschitz map

${\mathrm{Bar}}_{k,\infty}:\operatorname{spt}(D_{k})\to M\setminus N_{d_{k}}.$

Note that by (14) and its construction, ${\mathrm{Bar}}_{k,\infty}$ is locally
$1$-Lipschitz in the following sense: for any $\epsilon>0$, there is
$r_{\epsilon}$ depending on $\epsilon,d$, such that if
$a,b\in\operatorname{spt}(D_{k})$ are at distance at most $r_{\epsilon}$, then

(19)
$\operatorname{dist}_{g^{\prime}}({\mathrm{Bar}}_{k,\infty}(a),{\mathrm{Bar}}_{k,\infty}(b))\leq(1+\epsilon)\operatorname{dist}(a,b).$

Moreover, ${\mathrm{Bar}}_{k,\infty}$ preserves the current structures in the
sense that

(20) $({\mathrm{Bar}}_{k,\infty})_{\sharp}(D_{k})=\llbracket 1_{M\setminus
N_{d_{k}}}\rrbracket.$

To check that, in view of the fact that $(\theta_{i})_{\sharp}(D_{k})$
converges in the flat topology to $\llbracket 1_{M\setminus
N_{d_{k}}}\rrbracket$ inside $L^{\infty}(M\setminus N_{d_{k}})$ and by
uniqueness of weak limits, it is enough to show the following: consider
Lipschitz functions $f,\pi_{1},...,\pi_{n}:L^{\infty}(M\setminus
N_{d_{k}})\to\mathbb{R}$ such that $f$ is bounded, then

$({\mathrm{Bar}}_{k,\infty})_{\sharp}(D_{k})(fd\pi_{1}\wedge...\wedge
d\pi_{n})=\lim_{i\to\infty}(\theta_{i})_{\sharp}(D_{k})(fd\pi_{1}\wedge...\wedge
d\pi_{n}).$

But this clearly holds by [AK00a, Definition 3.1,(ii)], because the functions
$\pi_{j}\circ\theta_{i}:\operatorname{spt}(D_{k})\to\mathbb{R}$ (resp.
$f\circ\theta_{i}$) are uniformly Lipschitz and converge pointwise to
$\pi_{j}\circ{\mathrm{Bar}}_{k,\infty}:\operatorname{spt}(D_{k})\to\mathbb{R}$
(resp. $f\circ{\mathrm{Bar}}_{k,\infty}$) after taking the subsequence used to
define ${\mathrm{Bar}}_{k,\infty}$.

We record the fact that (19) and (20) imply with [SW11, Lemma 2.42] that

(21) $\mathbf{M}(D_{k})\geq\operatorname{Vol}({M\setminus
N_{d_{k}}},g^{\prime}).$

Step 3 (${\mathrm{Bar}}_{k,\infty}$ is an isometry for intrinsic metrics):

Let $L_{k}$ be the path metric on $\operatorname{spt}(D_{k})$ induced by its
metric (a priori $L_{k}$ can take $\infty$ as value). As usual $(M\setminus
N_{d_{k}},{g^{\prime}})$ denotes the manifold $M\setminus N_{d_{k}}$ with the
path metric induced by $g^{\prime}$. The latter will be denoted by
$\operatorname{dist}_{k,g^{\prime}}$. In this step, we show that

(22)
${\mathrm{Bar}}_{k,\infty}:(\operatorname{spt}(D_{k}),L_{k})\to(M\setminus
N_{d_{k}},{g^{\prime}})\quad\text{is a bijective isometry.}$

Before starting the proof of (22), we make the following observation. Recall
that the Banach space $\mathbf{Z}^{\prime}$ was defined at the end of Step 1.
For $\eta>0$, set

$O^{k}_{\eta}:=\text{$\eta$-neighborhood of $\operatorname{spt}(D_{k})$ inside
the Banach space $\mathbf{Z}^{\prime}$}.$

Then for every $\eta>0$,

(23)
$\lim_{i\to\infty}\mathbf{M}((j_{i})_{\sharp}(D_{k,i})\llcorner(\mathbf{Z}^{\prime}\setminus
O^{k}_{\eta}))=0,$

This equality follows from (21), from lower semicontinuity of the mass, and
the fact that
$\lim_{i\to\infty}\mathbf{M}((j_{i})_{\sharp}(D_{k,i}))=\operatorname{Vol}({M\setminus
N_{d_{k}}},g^{\prime})$ by (13).

It is convenient to show the following “curve lifting” property.

Curve lifting property: Let $\eta>0$. Let $x,y\in\operatorname{spt}(D_{k})$
and let

$l:=\operatorname{dist}_{k,g^{\prime}}({\mathrm{Bar}}_{k,\infty}(x),{\mathrm{Bar}}_{k,\infty}(y)).$

Then there exists a compact connected Lipschitz curve $\omega$ contained in
$O^{k}_{\eta}$, starting at $x$, ending at $y$, and moreover

$\operatorname{length}(\omega)\leq l+\eta.$

Let us prove the curve lifting property. Let $b_{x}$, $b_{y}$ be the metric
balls in the Banach space $\mathbf{Z}^{\prime}$, of radius
$\hat{r}\in(0,\frac{\eta}{100})$ centered at
$x,y\in\operatorname{spt}(D_{k})$. By construction of
${\mathrm{Bar}}_{k,\infty}$, if $\hat{r}$ is chosen small enough then for all
$i$ large and every $q\in(j_{i})^{-1}(b_{x})$ (resp.
$q\in(j_{i})^{-1}(b_{y})$), we have

$\operatorname{dist}_{k,g^{\prime}}({\mathrm{Bar}}(q),{\mathrm{Bar}}_{k,\infty}(x))\leq\eta/10\quad\text{(resp.
$\operatorname{dist}_{k,g^{\prime}}({\mathrm{Bar}}(q),{\mathrm{Bar}}_{k,\infty}(y))\leq\eta/10$).}$

By lower semicontinuity of the mass, for each $i$ large,

$\|(j_{i})_{\sharp}(D_{k,i})\|(b_{x})>2\kappa\quad\text{and}\quad\|(j_{i})_{\sharp}(D_{k,i})\|(b_{y})>2\kappa$

for some $\kappa>0$ depending on $\hat{r}$ but independent of $i$. For $i$
large, by (13) and the area formula,

(24)
$\begin{split}\mathcal{H}^{n}\big{(}{\mathrm{Bar}}(\Omega_{i}\cap\operatorname{spt}(D_{k,i})\cap
j_{i}^{-1}(b_{x}))\big{)}&\geq\kappa,\\\
\quad\mathcal{H}^{n}\big{(}{\mathrm{Bar}}(\Omega_{i}\cap\operatorname{spt}(D_{k,i})\cap
j_{i}^{-1}(b_{y}))\big{)}&\geq\kappa.\end{split}$

The map ${\mathrm{Bar}}:\operatorname{spt}(C_{i})\to M$ is surjective by (11).
Its restriction to $\operatorname{spt}(D_{k,i})$ is $\lambda$-Lipschitz, see
(17).

After applying the coarea formula, (13), (17), (15), (23) and (24) as in the
toy example of Subsection 1.2, we find, for each $i$ large enough, two points

$y_{1,i}\in\Omega_{i}\cap\operatorname{spt}(D_{k,i})\cap
j_{i}^{-1}(b_{x}),\quad
y_{2,i}\in\Omega_{i}\cap\operatorname{spt}(D_{k,i})\cap j_{i}^{-1}(b_{y})$

and a smooth curve

$\sigma_{i}\subset M\setminus{N}_{d_{k}}$

with $\operatorname{length}(\sigma_{i})\leq l+\frac{\eta}{3}$, joining
${\mathrm{Bar}}(y_{1,i})$ to ${\mathrm{Bar}}(y_{2,i})$ such that the
restricted preimage

$\varkappa_{i}:={\mathrm{Bar}}^{-1}(\sigma_{i})\cap\operatorname{spt}(D_{k,i})$

is a compact curve avoiding $\operatorname{set}(\partial D_{k,i})$, whose
boundary satisfies

${\mathrm{Bar}}(\partial\varkappa_{i})\subset{\mathrm{Bar}}(y_{1,i})\cup{\mathrm{Bar}}(y_{2,i}).$

Because $y_{1,i}$ and $y_{2,i}$ belong to $\Omega_{i}$, ${\mathrm{Bar}}$ is
injective at $y_{1,i}$ and $y_{2,i}$, thus in fact

(25) $\partial\varkappa_{i}=\\{y_{1,i},y_{2,i}\\}.$

By (13), the restriction of $D_{k,i}$ to the complement of $\Omega_{i}$ has
mass converging to $0$ as $i\to\infty$; similarly, by (23), the restriction of
$(j_{i})_{\sharp}D_{k,i}$ to the complement of $O^{k}_{\eta}$ has mass
converging to $0$. Thus the coarea formula implies that we could choose
$\varkappa_{i}$ satisfying additionally:

(26)
$\begin{split}\lim_{i\to\infty}\mathcal{H}^{1}(\varkappa_{i}\setminus\Omega_{i})&=0,\\\
\lim_{i\to\infty}\mathcal{H}^{1}(j_{i}(\varkappa_{i})\setminus
O^{k}_{\eta/2})&=0.\end{split}$

Together with (13), (7) in Lemma 1.4, the area formula and the slicing
theorem, these properties imply:

(27)
$\mathcal{H}^{1}(\varkappa_{i})\leq(1+\epsilon_{i})(l+\eta/2)+\epsilon_{i}$

where $\lim_{i\to\infty}\epsilon_{i}=0$. By considering the curve
$j_{i}(\varkappa_{i})$ in $\mathbf{Z}^{\prime}$, using (26) and the fact that
$j_{i}(y_{1,i})$ (resp. $j_{i}(y_{2,i})$) is in $b_{x}$ (resp. $b_{y}$), we
easily construct a new curve $\omega_{i}$ in $O_{\eta}^{k}$ joining $x$ to
$y$, with length at most $l+\eta$ for $i$ large. This proves the curve lifting
property.

The curve lifting property implies the following useful properties. Firstly,
$\operatorname{spt}(D_{k})$ is compact. Suppose towards a contraction that
$\operatorname{spt}(D_{k})$ is not compact, then for some
$r^{\prime}\in(0,r_{1})$, there is an infinite sequence of points
$\\{x_{m}\\}_{m\geq 0}\subset\operatorname{spt}(D_{k})$ such that those points
are pairwise at distance at least $r^{\prime}$ in $\mathbf{Z}^{\prime}$. By
compactness of $M\setminus N_{d_{k}}$, for any $\epsilon>0$ there are
$m_{1}\neq m_{2}$ such that

$\operatorname{dist}_{k,g^{\prime}}({\mathrm{Bar}}_{k,\infty}(x_{m_{1}}),{\mathrm{Bar}}_{k,\infty}(x_{m_{2}}))\leq\epsilon.$

Then the curve lifting property implies that the distance between $x_{m_{1}}$
and $x_{m_{2}}$ is at most $\epsilon$, a contradiction when
$\epsilon<r^{\prime}/2$.

Secondly ${\mathrm{Bar}}_{k,\infty}:\operatorname{spt}(D_{k})\to M\setminus
N_{d_{k}}$ is bijective. Indeed we verify that ${\mathrm{Bar}}_{k,\infty}$ is
injective by a direct application of the curve lifting property. Surjectivity
follows from (20) and the compactness of $\operatorname{spt}(D_{k})$.

We are ready to prove (22). By (19), clearly ${\mathrm{Bar}}_{k,\infty}$ is
distance non-increasing for the intrinsic metrics. Take two points $u,v\in
M\setminus N_{d_{k}}$ and let $\eta>0$. Let
$x:={\mathrm{Bar}}_{k,\infty}^{-1}(u)$,
$y:={\mathrm{Bar}}_{k,\infty}^{-1}(v)$. By applying the curve lifting property
repeatedly and making $\eta\to 0$, by compactness of
$\operatorname{spt}(D_{k})$ we get a limit Lipschitz curve in
$\operatorname{spt}(D_{k})$ joining $x$ to $y$, with length at most
$\operatorname{dist}_{k,g^{\prime}}(u,v)$. Thus ${\mathrm{Bar}}_{k,\infty}$ is
indeed an isometry for the intrinsic metrics as wanted.

Step 4 (Conclusion): At this point, we have proved for each $k$ that
$(\operatorname{spt}(D_{k}),L_{k})$ is isometric to $(M\setminus
N_{d_{k}},{g^{\prime}})$ via a map ${\mathrm{Bar}}_{k,\infty}$. It remains to
glue these sets and maps together. Recall that
$C_{\infty}=(X_{\infty},d_{\infty},S_{\infty})$ denotes the spherical Plateau
solution for $M$ we fixed at the beginning of the proof.

By Lemma 1.2 in Section 1.1, we can choose a common Banach space
$\mathbf{Z}^{\prime\prime}$ so that there exist isometric embeddings

$j^{\prime}_{i}:\operatorname{spt}(C_{i})\to\mathbf{Z}^{\prime\prime},\quad\operatorname{spt}(S_{\infty})\subset\mathbf{Z}^{\prime\prime},\quad\operatorname{spt}(D_{k})\subset\mathbf{Z}^{\prime\prime}$

such that $(j^{\prime}_{i})_{\sharp}(C_{i})$ converges to $S_{\infty}$ in the
flat topology inside $\mathbf{Z}^{\prime\prime}$, and for all $k\geq 0$,
$(j^{\prime}_{i})_{\sharp}(D_{k,i})$ converges to $D_{k}$ in the flat topology
inside $\mathbf{Z}^{\prime\prime}$. Next, by (23), for any $k<k^{\prime}$,
$\operatorname{spt}(D_{k})\subset\operatorname{spt}(D_{k^{\prime}})\subset\operatorname{spt}(S_{\infty})$,
and $D_{k}=S_{\infty}\llcorner\operatorname{spt}(D_{k})$. By a diagonal
argument and how we constructed the maps ${\mathrm{Bar}}_{k,\infty}$, we can
ensure that ${\mathrm{Bar}}_{k,\infty}$ is the restriction of
${\mathrm{Bar}}_{k^{\prime},\infty}$ to $\operatorname{spt}(D_{k})$ for any
$k<k^{\prime}$. Finally gluing these maps together we obtain a map

${\mathrm{Bar}}_{\infty}:\bigcup_{k\geq
0}\operatorname{spt}(D_{k})\to\bigcup_{k\geq 0}M\setminus
N_{d_{k}}=M\setminus\\{\mathbf{p}\\}$

which is bijective and a local isometry for the intrinsic metrics by Step 3.
Moreover, by (20),

$({\mathrm{Bar}}_{\infty})_{\sharp}(S_{\infty}\llcorner\bigcup_{k\geq
0}\operatorname{spt}(D_{k}))=\llbracket
1_{M\setminus\\{\mathbf{p}\\}}\rrbracket.$

Consequently, $\mathbf{M}(S_{\infty}\llcorner\bigcup_{k\geq
0}\operatorname{spt}(D_{k}))=\operatorname{Vol}(M,g^{\prime})=\operatorname{SphereVol}(M)$,
so the canonical set with its intrinsic metric $(X_{\infty},L_{d_{\infty}})$
has to be isometric to $(M,g^{\prime})$ via an extension of
${\mathrm{Bar}}_{\infty}$ sending $S_{\infty}$ to $\llbracket
1_{M}\rrbracket$. In other words,
$C_{\infty}=(X_{\infty},d_{\infty},S_{\infty})$ is intrinsically isomorphic to
$(M,g^{\prime})$. The proof is finished.

∎

######  Remark 2.7.

We observe that all the arguments in this section extend to the rank one
locally symmetric case by using [BCG96, Rua22], and so a version of Theorem
2.6 holds more generally for closed oriented manifolds which are locally
symmetric negatively curved [Son23, Theorem 4.3].

##  3\. Proof of the entropy stability theorem

###  3.1. Technical preparation

As before, $M$ is the closed oriented hyperbolic manifold, $\Gamma$ is its
fundamental group and $\mathcal{S}(\Gamma)$ is the unit sphere inside
$\ell^{2}(\Gamma)$, which is acted upon by $\Gamma$ via the regular
representation.

The barycenter map $\mathrm{Bar}:\mathbb{S}^{+}/\Gamma\to M$ was defined in
Subsection 1.3. The definition of the standard notion of $\epsilon$-isometry
can be found in [BBI22, Definition 7.3.27]. If $g$ is a Riemannian metric on
$M$, let $\operatorname{dist}_{g}$ be the geodesic distance on $M$ induced by
$g$. If $\Omega\subset M$, denote by
$(\Omega,\operatorname{dist}_{g}|_{\Omega})$ the metric space where
$\operatorname{dist}_{g}|_{\Omega}$ is the restriction of the geodesic
distance of $(M,g)$ to $\Omega$. We will need the following technical
extension of Theorem 2.6.

######  Theorem 3.1.

Let $(M,g_{0})$ be a closed oriented hyperbolic manifold of dimension at least
$3$. Let $g^{\prime}_{i}$ $(i\geq 1)$ be a sequence of Riemannian metrics on
$M$ and let $\phi_{i}\in\mathscr{F}(M)$ be a sequence satisfying the
following:

  * •

$\operatorname{Vol}(M,g^{\prime}_{i})=\operatorname{Vol}(M,\frac{(n-1)^{2}}{{4n}}g_{0})$,

  * •

the maps $\phi_{i}$ are $A$-Lipschitz for some $A>0$ independent of $i$, and
the maps $\phi_{i}$ send $(M,g^{\prime}_{i})$ to $\mathbb{S}^{+}/\Gamma$,

  * •

there are open subsets $R_{i}\subset M$ such that

$\lim_{i\to\infty}\operatorname{Vol}(R_{i},g^{\prime}_{i})=\operatorname{Vol}(M,\frac{(n-1)^{2}}{{4n}}g_{0}),$

$\phi_{i}$ is differentiable on $R_{i}$ and

$\lim_{i\to\infty}\big{\|}\sum_{u,v=1}^{n}|\mathbf{g}_{\mathrm{Hil}}(d\phi_{i}(e^{\prime}_{u}),d\phi_{i}(e^{\prime}_{v}))-\delta_{uv}|\big{\|}_{L^{\infty}(R_{i})}=0.$

Here $\\{e^{\prime}_{u}\\}_{u=1}^{n}$ denotes any choice of orthonormal basis
for the tangent spaces of $(M,g^{\prime}_{i})$ and $\mathbf{g}_{\mathrm{Hil}}$
is the canonical metric on $\mathcal{S}(\Gamma)/\Gamma$.

Then $(M,\operatorname{dist}_{g^{\prime}_{i}},\llbracket 1_{M}\rrbracket)$
subsequentially converges in the intrinsic flat topology to an integral
current space

$\hat{C}_{\infty}=(\hat{X}_{\infty},\hat{d}_{\infty},\hat{S}_{\infty})$

which is intrinsically isomorphic to $(M,\frac{h(g_{0})^{2}}{{4n}}g_{0})$ via
a map

$\mathrm{Bar}_{\infty}:\hat{X}_{\infty}\to M.$

Moreover for each $\epsilon_{0}>0$, there are finite subsets
$\tilde{F}_{i}\subset M$ such that for all $i$ large,

$\mathrm{Bar}_{\infty}^{-1}\circ\mathrm{Bar}\circ\phi_{i}:(\tilde{F}_{i},\operatorname{dist}_{g^{\prime}_{i}}|_{\tilde{F}_{i}})\to(\hat{X}_{\infty},\hat{d}_{\infty})$

is an $\epsilon_{0}$-isometry,

###### Proof.

In this proof set

$g^{\prime}:=\frac{(n-1)^{2}}{4n}g_{0}.$

Set

$\hat{d}_{i}:=\min\\{\operatorname{dist}_{g^{\prime}_{i}},6\operatorname{diam}(M,g^{\prime})\\}.$

This defines a metric on $M$ such that its diameter is at most
$6\operatorname{diam}(M,g^{\prime})$, and it is locally isometric to the
geodesic distance $\operatorname{dist}_{g^{\prime}_{i}}$. By Wenger’s
compactness theorem, the integral current spaces $(M,\hat{d}_{i},\llbracket
1_{M}\rrbracket)$ converge to an integral current space

$\hat{C}_{\infty}=(\hat{X}_{\infty},\hat{d}_{\infty},\hat{S}_{\infty})$

in the intrinsic flat topology, after picking a subsequence if necessary. Next
we show that $\hat{C}_{\infty}$ is intrinsically isomorphic to
$(M,g^{\prime})$.

Inspecting the proof of Theorem 2.6, we can replace

$C_{i},\quad\mathrm{Bar}:\operatorname{spt}(C_{i})\to M,\quad C_{\infty}$

by respectively

$(M,\hat{d}_{i},\llbracket
1_{M}\rrbracket),\quad\mathrm{Bar}\circ\phi_{i}:M\to M,\quad\hat{C}_{\infty}$

here. We will use accordingly the notations $\Omega_{i}$,
$\Omega_{i,\tilde{r}}$, $N_{d}$ $(d>0)$, $D_{k,i}$ and the steps from the
proof of Theorem 2.6. By repeating the arguments in the proof of Theorem 2.6
almost verbatim, as in Step 1, we first get open subsets $\Omega_{i}\subset M$
with $g^{\prime}_{i}$-volume converging to
$\operatorname{Vol}(M,\frac{(n-1)^{2}}{{4n}}g_{0})$, where the Jacobian of
$\mathrm{Bar}\circ\phi_{i}$ converges to $1$. Then we pick a sequence
$d_{k}\to 0$, and define the $\tilde{r}$-neighborhood $\Omega_{i,\tilde{r}}$
of $\Omega_{i}$ in $(M,g^{\prime}_{i})$ for $\tilde{r}>0$ small independent of
$i$. We set

$D_{k,i}:=\llbracket
1_{M}\rrbracket\llcorner\big{(}\Omega_{i,\tilde{r}}\setminus(\mathrm{Bar}\circ\phi_{i})^{-1}(N_{d_{k}})\big{)},$

which converges to an integral current space
$D_{k}=(Y_{k},d_{Y_{k}},T_{Y_{k}})$ in the intrinsic flat topology.

As in Step 2, we construct a limit barycenter map

$\mathrm{Bar}_{\infty}:(\operatorname{spt}(D_{k}),d_{Y_{k}})\to(M\setminus
N_{d_{k}},g^{\prime})$

by using the Lipschitz bound on
$\mathrm{Bar}\circ\phi_{i}:\operatorname{spt}(D_{k,i})\to M\setminus
N_{d_{k}}$.

We show as in Step 3 that $\mathrm{Bar}_{\infty}$ is an intrinsic isometry. As
in Step 3 and Step 4, we glue together the maps constructed above to get
$\mathrm{Bar}_{\infty}:\bigcup_{k}\operatorname{spt}(D_{k})\to
M\setminus\\{\mathbf{p}\\}$ and finally the desired intrinsic isomorphism

$\mathrm{Bar}_{\infty}:\hat{X}_{\infty}\to M$

(i.e. an intrinsic isometry from $(\hat{X}_{\infty},\hat{d}_{\infty})$ to
$(M,g^{\prime})$, sending the current structure $\hat{S}_{\infty}$ to the
current structure $\llbracket 1_{M}\rrbracket$).

We still need to show that $(M,\operatorname{dist}_{g^{\prime}_{i}},\llbracket
1_{M}\rrbracket)$, not just $(M,\hat{d}_{i},\llbracket 1_{M}\rrbracket)$,
subsequentially converges. From the above, we know that there are a Banach
space $B$, and isometric embeddings

$j_{i}:(M,\hat{d}_{i})\to B,\quad j_{\infty}:\hat{X}_{\infty}\to B,$

such that $(j_{i})_{\sharp}(M,\hat{d}_{i},\llbracket 1_{M}\rrbracket)$
converges to $(j_{\infty})_{\sharp}\hat{S}_{\infty}$ in the flat topology
inside $B$, and $(\hat{X}_{\infty},\hat{d}_{\infty})$ is intrinsically
isometric to $(M,g^{\prime}_{0})$ via an isometry sending $\hat{S}_{\infty}$
to $\llbracket 1_{M}\rrbracket$. Notice that for any $x\in M$ and
$R\in(0,3\operatorname{diam}(M,g^{\prime}_{0}))$, the metric balls
$B_{\operatorname{dist}_{g^{\prime}_{i}}}(x,R)\subset(M,\operatorname{dist}_{g^{\prime}_{i}})$
and $B_{\hat{d}_{i}}(x,R)\subset(M,{\hat{d}_{i}})$ are globally isometric. In
particular, if $O_{r}$ denotes the $r$-neighborhood of
$j_{\infty}(\hat{X}_{\infty})$ in $B$, we have: if
$r\in(0,\operatorname{diam}(M,g^{\prime}_{0}))$, for any $i$ and pair of
points $x,y\in M$ such that $j_{i}(x),j_{i}(y)\in O_{r}$, then

(28) $\operatorname{dist}_{g^{\prime}_{i}}(x,y)=\hat{d}_{i}(x,y).$

Moreover, for any fixed $r>0$, we have

(29) $\lim_{i\to\infty}\mathbf{M}((j_{i})_{\sharp}(M,\hat{d}_{i},\llbracket
1_{M}\rrbracket)\llcorner O_{r})=\operatorname{Vol}(M,g^{\prime}_{0}).$

That follows from lower semicontinuity of the mass under flat convergence, and
the first assumption in the statement of the theorem that the full mass of
$(M,\hat{d}_{i},\llbracket 1_{M}\rrbracket)$ is equal to
$\operatorname{Vol}(M,g^{\prime}_{0})$. By (28) and (29) and the slicing
theorem for metric currents [AK00a, Theorems 5.6, 5.7] [Son23, Subsection
1.5], we can choose for each $i$,
$r_{i}\in(\operatorname{diam}(M,g^{\prime}_{0})/2,\operatorname{diam}(M,g^{\prime}_{0}))$
such that if $\tilde{O}_{i}\subset M$ denotes the following open subset

$\tilde{O}_{i}:=(j_{i})^{-1}(O_{r_{i}}),$

then
$(j_{i})_{\sharp}(\tilde{O}_{i},\operatorname{dist}_{g^{\prime}_{i}}|_{\tilde{O}_{i}},\llbracket
1_{M}\rrbracket)$ are integral currents converging to
$(j_{\infty})_{\sharp}\hat{S}_{\infty}$ in $B$ in the flat topology. By
definition, this means that
$(\tilde{O}_{i},\operatorname{dist}_{g^{\prime}_{i}}|_{\tilde{O}_{i}},\llbracket
1_{M}\rrbracket)$ converges to $\hat{C}_{\infty}$ in the intrinsic flat
topology. Since by (29) the $g_{i}^{\prime}$-mass of the complement of
$\tilde{O}_{i}$ in $M$ becomes arbitrarily small as $i\to\infty$, we conclude
that $(M,\operatorname{dist}_{g^{\prime}_{i}},\llbracket 1_{M}\rrbracket)$
also converges to $\hat{C}_{\infty}$ in the intrinsic flat topology as
desired.

Finally, we need to check the last sentence in the statement of the theorem.
If $d_{k}$ is small enough compared to $\epsilon_{0}$,
$(\operatorname{spt}(D_{k}),d_{Y_{k}})$ is $\epsilon_{0}$-isometric to
$(\hat{X}_{\infty},\hat{d}_{\infty})$ (use that
$(\bigcup_{k}\operatorname{spt}(D_{k}),\hat{d}_{\infty})$ is isometric to
$(\hat{X}_{\infty},\hat{d}_{\infty})$ minus a point). Next since $D_{k,i}$
converges in the intrinsic flat topology to $D_{k}$, there is a Banach space
$\mathbf{Z}^{\prime}$ and isometric embeddings

$\operatorname{spt}(D_{k,i})\subset\mathbf{Z}^{\prime},\quad\operatorname{spt}(D_{k})\subset\mathbf{Z}^{\prime}$

such that $D_{k,i}$ converges in the flat topology to $D_{k}$ inside
$\mathbf{Z}^{\prime}$ (by abuse of notations we identify $D_{k,i}$, $D_{k}$
with their push-forwards inside $\mathbf{Z}^{\prime}$). Take a point
$x\in\operatorname{spt}(D_{k})\subset\mathbf{Z}^{\prime}$, then for $i$ large,
there is a point $y\in j_{i}(\operatorname{spt}(D_{k,i}))$ arbitrarily close
to $x$. By construction of $\mathrm{Bar}_{\infty}$ and the Lipschitz bound for
$\mathrm{Bar}$ on $\operatorname{spt}(D_{k,i})$, for $i$ large, the images
$\mathrm{Bar}\circ\phi_{i}(y)$ and $\mathrm{Bar}_{\infty}(x)$ are arbitrarily
close. Since $\mathrm{Bar}_{\infty}$ is an intrinsic isometry,
$\mathrm{Bar}_{\infty}^{-1}\circ\mathrm{Bar}\circ\phi_{i}(y)$ is arbitrarily
close to $x$ inside $\mathbf{Z}^{\prime}$ for $i$ large. Using that, it is
simple to find finite subsets $\tilde{F}_{i}\subset M$ such that for all $i$
large,

$\mathrm{Bar}_{\infty}^{-1}\circ\mathrm{Bar}\circ\phi_{i}:(\tilde{F}_{i},\operatorname{dist}_{g^{\prime}_{i}}|_{\tilde{F}_{i}})\to(\operatorname{spt}(D_{k}),d_{Y_{k}})\subset(\hat{X}_{\infty},\hat{d}_{\infty})$

is an $\epsilon_{0}$-isometry,

∎

###  3.2. From equidistribution of geodesic spheres to intrinsic flat
stability

Let us define maps $\Psi_{c}$ relating the volume entropy of a Riemannian
metric on $M$ and the spherical volume of $M$. Those were also introduced by
Besson-Courtois-Gallot [BCG91, Proof of Lemma 3.1]. Let $g$ be a Riemannian
metric on $M$. The universal cover of $M$ is $\tilde{M}$ and its fundamental
group is $\Gamma$. Let $h(g)$ be its volume entropy. Denote by $\mathcal{D}$ a
measurable fundamental domain in $\tilde{M}$ for the action of $\Gamma$ and
let $\gamma.\mathcal{D}$ be its image by an element $\gamma\in\Gamma$. Besson-
Courtois-Gallot considered for $c>h(g)$ maps similar to the following:

$\displaystyle\Psi_{c}$ $\displaystyle:\tilde{M}\to\mathcal{S}(\Gamma)$
$\displaystyle
x\mapsto\\{\gamma\mapsto\frac{1}{\|e^{-\frac{c}{2}\operatorname{dist}_{g}(x,.)}\|_{L^{2}(\tilde{M},g)}}\big{[}\int_{\gamma.\mathcal{D}}e^{-c\operatorname{dist}_{g}(x,u)}\operatorname{dvol}_{g}(u)\big{]}^{1/2}\\}.$

Those maps satisfy the following properties, which hold in any dimension
$n\geq 2$:

######  Lemma 3.2 ([BCG91]).

For a Riemannian metric $g$ on $M$, $\Psi_{c}$ is a $\Gamma$-equivariant
Lipschitz map, and for almost any $x\in\tilde{M}$, it satisfies

(30) $\sum_{j=1}^{n}|d_{x}{\Psi}_{c}(e_{j})|^{2}\leq\frac{c^{2}}{4},$

where $\\{e_{j}\\}$ is a $g$-orthonormal basis of $T_{x}\tilde{M}$.

###### Proof.

Let us outline the proof. Consider $S_{2}(\tilde{M},g)$ the unit sphere in
$L^{2}(\tilde{M},g)$. Set for $c>h(g)$:

$\overline{\Psi}_{c}:\tilde{M}\to S_{2}(\tilde{M},g)$
$\overline{\Psi}_{c}:x\mapsto\\{y\mapsto\frac{1}{\|e^{-\frac{c}{2}\operatorname{dist}_{g}(x,.)}\|_{L^{2}(\tilde{M},g)}}e^{-\frac{c}{2}\operatorname{dist}_{g}(x,y)}\\},$

and set

$\mathcal{A}:S_{2}(\tilde{M},g)\to\mathcal{S}(\Gamma)$
$\mathcal{A}:f\mapsto\\{\gamma\mapsto\big{[}\int_{\gamma.\mathcal{D}}f^{2}(u)\operatorname{dvol}_{g}(u)\big{]}^{1/2}\\}.$

These maps are manifestly $\Gamma$-equivariant, and note that
$\Psi_{c}=\mathcal{A}\circ\overline{\Psi}_{c}$. One easily checks that
$\mathcal{A}$ is $1$-Lipschitz. To prove the lemma, it remains to study
$\overline{\Psi}_{c}$. By the Pythagorean theorem,

$\displaystyle\|d_{x}\overline{\Psi}_{c}\|^{2}_{L^{2}(\tilde{M},g)}$
$\displaystyle\leq\frac{1}{\|e^{-\frac{c}{2}\operatorname{dist}_{g}(x,.)}\|^{2}_{L^{2}(\tilde{M},g)}}\int_{\tilde{M}}\|d_{x}e^{-\frac{c}{2}\operatorname{dist}_{g}(x,y)}\|^{2}\operatorname{dvol}_{g}(y)$
$\displaystyle\leq\frac{c^{2}/4}{\|e^{-\frac{c}{2}\operatorname{dist}_{g}(x,.)}\|^{2}_{L^{2}(\tilde{M},g)}}\int_{\tilde{M}}\|d_{x}\operatorname{dist}_{g}(.,y)\|^{2}e^{-c\operatorname{dist}_{g}(x,y)}\operatorname{dvol}_{g}(y).$

Taking the trace and using that the norm of the gradient of the distance
function is well-defined almost everywhere and equal to $1$, we get at almost
every $x\in M$, in a $g$-orthonormal basis $\\{e_{j}\\}$ of $T_{x}{M}$:

$\sum_{j=1}^{n}\|d_{x}\overline{\Psi}_{c}(e_{j})\|^{2}_{L^{2}(\tilde{M},g)}\leq\frac{c^{2}}{4}.$

This proves the lemma.

∎

######  Theorem 3.3.

Let $(M,g_{0})$ be a closed oriented hyperbolic manifold of dimension at least
$3$. Let $\\{g_{i}\\}_{i\geq 1}$ be a sequence of Riemannian metrics on $M$
with $\operatorname{Vol}(M,g_{i})=\operatorname{Vol}(M,g_{0})$. If

$\lim_{i\to\infty}h(g_{i})=h(g_{0})=n-1,$

then $(M,\operatorname{dist}_{g_{i}},\llbracket 1_{M}\rrbracket)$ converges to
$(M,\operatorname{dist}_{g_{0}},\llbracket 1_{M}\rrbracket)$ in the intrinsic
flat topology.

###### Proof.

Set

$g^{\prime}_{i}:=\frac{(n-1)^{2}}{4n}g_{i},\quad
g^{\prime}:=\frac{(n-1)^{2}}{4n}g_{0}$

The volume of $(M,g^{\prime}_{i})$ is by assumption equal to
$\operatorname{Vol}(M,g^{\prime})$. Note that after rescaling,

$h(g^{\prime})=2\sqrt{n}.$

Let $c^{\prime}_{i}>h(g^{\prime}_{i})$ be a sequence of positive numbers such
that

(31) $\lim_{i\to\infty}c^{\prime}_{i}=2\sqrt{n}.$

By Lemma 3.2, the maps

$\Psi_{c_{i}}:\tilde{M}\to\mathcal{S}(\Gamma)$

are $\Gamma$-equivariant. The quotient maps

$\Psi_{c_{i}}:(M,g^{\prime}_{i})\to\mathcal{S}(\Gamma)/\Gamma$

can be perturbed to be smooth immersions. Those new maps now belong to
$\mathscr{F}(M)$. After a further small perturbation, we obtain smooth
immersions

$\phi_{i}\in\mathscr{F}(M)$

sending $(M,g^{\prime}_{i})$ inside $\mathbb{S}^{+}/\Gamma$. Moreover, by (30)
and (31), we can assume that for all $x\in M$:

(32) $\sum_{j=1}^{n}|d_{x}\phi_{i}(e^{\prime}_{j})|^{2}\leq n+\nu_{i}$

for some positive $\nu_{i}\to 0$ (with respect to $g^{\prime}_{i}$), where
$\\{e^{\prime}_{j}\\}$ is an orthonormal basis for $g^{\prime}_{i}$. By (32)
and the inequality of arithmetic and geometric means,

(33) $|\operatorname{Jac}\phi_{i}|\leq(1+\frac{\nu_{i}}{n})^{n/2}$

where the Jacobian is computed with respect to $g^{\prime}_{i}$. By Theorem
2.2,

$\operatorname{SphereVol}(M)=\operatorname{Vol}(M,g^{\prime}),$

on the other hand, by assumption, we have
$\operatorname{Vol}(M,g^{\prime}_{i})=\operatorname{Vol}(M,g^{\prime})$.
Hence, by (33), $|\operatorname{Jac}\phi_{i}|_{g^{\prime}_{i}}$ converges to
$1$ on an open region $R_{i}\subset M$ with $g^{\prime}_{i}$-volume converging
to $\operatorname{Vol}(M,g^{\prime})$, which forces

$\lim_{i\to\infty}\big{\|}\sum_{u,v=1}^{n}|\mathbf{g}_{\mathrm{Hil}}(d\phi_{i}(e^{\prime}_{u}),d\phi_{i}(e^{\prime}_{v}))-\delta_{uv}|\big{\|}_{L^{\infty}(R_{i})}=0.$

We can thus apply Theorem 3.1: a subsequence of
$(M,\operatorname{dist}_{g^{\prime}_{i}},\llbracket 1_{M}\rrbracket)$
converges in the intrinsic flat topology to an integral current space
$\hat{C}_{\infty}=(\hat{X}_{\infty},\hat{d}_{\infty},\hat{S}_{\infty})$ which
is intrinsically isomorphic to $(M,\operatorname{dist}_{g^{\prime}},\llbracket
1_{M}\rrbracket)$. We will show that $\hat{C}_{\infty}$ is in fact equal to
$(M,\operatorname{dist}_{g^{\prime}},\llbracket 1_{M}\rrbracket)$ as an
integral current space, which will be enough to conclude since that will hold
for any converging subsequence of
$(M,\operatorname{dist}_{g^{\prime}_{i}},\llbracket 1_{M}\rrbracket)$.

From now on, we make the identifications $\hat{X}_{\infty}=M$ and
$\hat{S}_{\infty}=\llbracket 1_{M}\rrbracket$ to clarify the notations.
Suppose towards a contradiction that
$\hat{C}_{\infty}=(M,\hat{d}_{\infty},\llbracket 1_{M}\rrbracket)$ and
$(M,\operatorname{dist}_{g^{\prime}},\llbracket 1_{M}\rrbracket)$ are
intrinsically isomorphic but not isomorphic; this means that there are points
$A,B\in M$ such that

$\hat{d}_{\infty}(A,B)<\operatorname{dist}_{g^{\prime}}(A,B).$

We thus can find small open geodesic balls $U_{A},U_{B}$ centered at $A,B$
such that

(34) $\exists\lambda\in(0,1)\text{ such that }\forall x\in U_{A},\forall y\in
U_{B},\quad\hat{d}_{\infty}(x,y)<\lambda\operatorname{dist}_{g^{\prime}}(x,y).$

The map

$\Xi_{i}:=(\mathrm{Bar}_{\infty})^{-1}\circ\mathrm{Bar}\circ\phi_{i}:M\to M$

is a homotopy equivalence. By Theorem 3.1, for any $\epsilon_{0}>0$, there are
finite subsets $\tilde{F}_{i}\subset M$ such that the restrictions

(35)
$\Xi_{i}:(\tilde{F}_{i},\operatorname{dist}_{g^{\prime}_{i}})\to(M,\hat{d}_{\infty})$

are also $\epsilon_{0}$-isometries for all $i$ large.

By standard properties of $\epsilon$-isometries [BBI22, Exercise 7.5.11], we
have the following lifting property whenever $\epsilon_{0}$ is small enough
compared to the injectivity radius of $(M,g^{\prime})$: given a geodesic loop
$\sigma$ in $(M,g^{\prime})$, there is an “$\epsilon_{0}$-lift”

$\xi_{i}\subset M$

which is a piecewise smooth curve inside $(M,g^{\prime}_{i})$ such that for
all $i$ large

$\operatorname{length}_{g^{\prime}_{i}}(\xi_{i})\leq(1+\eta(\epsilon_{0}))(\operatorname{length}_{g^{\prime}}(\sigma))+\eta(\epsilon_{0})$

where

(36) $\lim_{\epsilon_{0}\to 0}\eta(\epsilon_{0})=0,$

and $\Xi_{i}(\xi_{i})$ is homotopic to $\sigma$. Recall that this can be
achieved as follows: we discretize $\sigma$ as a collection of points, which
we lift to $\tilde{F}_{i}$ via the $\epsilon_{0}$-isometry $\Xi_{i}$, and we
connect those lifted points into a curve. Note that for two non-homotopic
geodesic loops $\sigma_{1},\sigma_{2}$, the two $\epsilon_{0}$-lifts
$\xi_{1},\xi_{2}$ obtained by the above procedure are non-homotopic (use the
continuous homotopy equivalence $\Xi_{i}$). A similar lifting property holds
for geodesic segments, and also for geodesic segments in the universal cover
$\tilde{M}$. We will call the set of those properties the “standard lifting
procedure”. We call “$\epsilon_{0}$-lift” any object obtained by applying the
standard lifting procedure to $(\ref{epsilonisometry})$ (not to be confused
with “lift” by covering maps).

Let

$V^{i}_{A}:=\Xi_{i}^{-1}(U_{A})\cap\tilde{F}_{i},\quad
V^{i}_{B}:=\Xi_{i}^{-1}(U_{B})\cap\tilde{F}_{i}.$

These two sets are not empty whenever $\epsilon_{0}$ is chosen small enough
depending only on the size of $U_{A},U_{B}$. In the lifting procedure above,
we can always make sure that points in $U_{A}$ or $U_{B}$ lift to $V^{i}_{A}$
or $V^{i}_{B}$. Since $\Xi_{i}$ is an $\epsilon_{0}$-isometry between
$(\tilde{F}_{i},g^{\prime}_{i})$ and $(M,\hat{d}_{\infty})$, by (34), if
$x_{1}\in V^{i}_{A},y_{1}\in V^{i}_{B}$ are the $\epsilon_{0}$-lifts of
respectively $x\in U_{A},y\in U_{B}$, we have for some $\lambda_{1}\in(0,1)$
independent of $x,y,x_{1},y_{1}$ when $\epsilon_{0}$ is small enough:

(37)
$\operatorname{dist}_{g^{\prime}_{i}}(x_{1},y_{1})<\lambda_{1}\operatorname{dist}_{g^{\prime}}(x,y)$

Given a Riemannian metric $g$ on $M$ and a point $\mathbf{x}\in M$, let
$\mathcal{L}_{\leq L}(g,\mathbf{x})$ be the collection of homotopy classes of
loops based at $\mathbf{x}$ which contain at least one geodesic loop based at
$\mathbf{x}\in M$ of $g$-length at most $L$. It is well-known that the volume
entropy of $g$ is:

$h(g)=\lim_{L\to\infty}\frac{\log(\mathbf{card}\mathcal{L}_{\leq
L}(g,\mathbf{x}))}{L}$

where $\mathbf{card}$ denotes the cardinality of a set, and it does not depend
on the choice of base point $\mathbf{x}$.

Fix a base point $x_{0}\in(M,g^{\prime})$ and a lift
$\tilde{x}_{0}\in\tilde{M}$. By uniqueness of geodesic loops in homotopy
classes of loops inside hyperbolic manifolds, we identify $\mathcal{L}_{\leq
L}(g^{\prime},x_{0})$ with the set of geodesic loop based at $x_{0}$ with
length at most $L$.

The crucial argument is, as we will explain below, that the equidistribution
of lifts of geodesic spheres to the unit tangent bundle plus the distance
comparison inequality (37) force the volume entropy of $(M,g_{i})$ to be
strictly larger than $n-1$.

The volume entropy of the hyperbolic $n$-plane $(\tilde{M},g_{0})$ is $n-1$,
meaning that

(38) $\lim_{L\to\infty}\frac{\log(\mathbf{card}\mathcal{L}_{\leq
L}(g^{\prime},x_{0}))}{L}=2\sqrt{n},$

because $g^{\prime}=\frac{(n-1)^{2}}{4n}g_{0}$.

By Corollary 1.6, the inequality (37) and the standard lifting procedure, we
deduce that there is some $\theta_{1}\in(0,1)$ depending on $M,U_{A},U_{B}$
but independent of $i$ large satisfying the following. In the geodesic spheres
$\tilde{S}(\tilde{x}_{0},L)$ of universal cover $(\tilde{M},g^{\prime})$, for
$L$ large there is an open subset $W_{L}\subset\tilde{S}(\tilde{x}_{0},L)$
such that

$\operatorname{Vol}_{n-1,g^{\prime}}(W_{L})\geq\theta_{1}\operatorname{Vol}_{n-1,g^{\prime}}(\tilde{S}(\tilde{x}_{0},L)),$

and for any $y\in W_{L}$, the minimizing geodesic $l$ from $\tilde{x}_{0}$ to
$y$ $\epsilon_{0}$-lifts to a piecewise smooth curve $l^{\prime}$ in
$(\tilde{M},g^{\prime}_{i})$ which in turn is homotopic (with fixed endpoints)
to a curve of $g^{\prime}_{i}$-length at most $\theta_{1}L$.

For all $i$ large, fix an $\epsilon_{0}$-lift $\mathbf{x}_{0,i}$ of the base
point $x_{0}$ inside $(M,g^{\prime}_{i})$ via the $\epsilon_{0}$-isometry
$\Xi_{i}$, and a lift $\tilde{\mathbf{x}}_{0,i}\in(\tilde{M},g^{\prime}_{i})$
of $\mathbf{x}_{0,i}$ in the universal cover. By basic hyperbolic geometry
(volume of geodesic spheres and balls etc.) and the standard lifting
procedure, the previous paragraph implies that for $i$ large enough, for all
$L$ large enough there is $\theta_{2}\in(0,1)$ depending on $M,U_{A},U_{B}$
but independent of $i,L$ such that

  * •

there are distinct points $p_{1},...,p_{K}\in\tilde{M}$ which are lifts of
$x_{0}$ to the universal cover $\tilde{M}$,

  * •

their number satisfies

(39) $K\geq\theta_{2}\exp((1+\theta_{2})2\sqrt{n}L),$

  * •

there are distinct geodesic segments
$c_{1},...,c_{K}\subset(\tilde{M},g^{\prime})$ starting from $\tilde{x}_{0}$
and ending respectively at $p_{1},...,p_{K}$,

  * •

$c_{1},...,c_{K}$ $\epsilon_{0}$-lift to piecewise smooth curves in
$(\tilde{M},g^{\prime}_{i})$, which are respectively homotopic (with fixed
endpoints) to curves $c_{i,1},...,c_{i,K}\subset(\tilde{M},g^{\prime}_{i})$ of
$g^{\prime}_{i}$-lengths bounded from above by

$(1+\eta(\epsilon_{0}))L+\eta(\epsilon_{0}),$

  * •

the endpoints of $c_{i,1},...,c_{i,K}$ are $\tilde{\mathbf{x}}_{0,i}$ and
other lifts of $\mathbf{x}_{0,i}$ inside the universal cover
$(\tilde{M},g^{\prime}_{i})$.

Since by (36) $\eta(\epsilon_{0})$ can be made arbitrarily small, when
$\epsilon_{0}$ is chosen small enough, we conclude from (38) and (39) that for
any $i$ large enough, for all $L$ large:

$\log(\mathbf{card}\mathcal{L}_{\leq
L}(g^{\prime}_{i},{\mathbf{x}}_{0,i}))\geq\log(\mathbf{card}\mathcal{L}_{\leq(1+\frac{\theta_{2}}{2})L}(g^{\prime},x_{0})).$

In particular,

$\displaystyle h(g^{\prime}_{i})$
$\displaystyle\geq\liminf_{L\to\infty}\frac{\log(\mathbf{card}\mathcal{L}_{\leq(1+\frac{\theta_{2}}{2})L}(g^{\prime},x_{0}))}{L}$
$\displaystyle=(1+\frac{\theta_{2}}{2})h(g^{\prime})$
$\displaystyle>h(g^{\prime}).$

Since $\theta_{2}>0$ does not depend on $i$, this is a contradiction with our
assumption that $h(g_{i})$ converges to $n-1$ (namely $h(g^{\prime}_{i})$
converges to $h(g^{\prime})$) and the proof is finished.

∎

Recall that intrinsic flat convergence implies weak convergence (see Section
1.1). Given a Riemannian metric $g$ on $M$, the mass measure of the integral
current space $(M,\operatorname{dist}_{g},\llbracket 1_{M}\rrbracket)$ is
equal to the usual volume measure $\operatorname{dvol}_{g}$ on $M$. The proof
of Theorem 0.1 is then completed by combining Theorem 3.3 and the following
general lemma proved by Portegies:

######  Lemma 3.4.

[Por15, Lemma 2.1] Suppose $Z$ is a complete metric space, and $\\{T_{i}\\}$
is a sequence of integral currents in $Z$ converging weakly to an integral
current $T$. Moreover, assume that $\mathbf{M}(T_{i})$ converges to
$\mathbf{M}(T)$. Then the mass measure $\|T_{i}\|$ converges weakly to $\|T\|$
as measures on $Z$.

## References

  * [AK00a] Luigi Ambrosio and Bernd Kirchheim.  Currents in metric spaces.  Acta Math., 185(1):1–80, 2000.
  * [AK00b] Luigi Ambrosio and Bernd Kirchheim.  Rectifiable sets in metric and Banach spaces.  Math. Ann., 318(3):527–555, 2000.
  * [All21] Brian Allen.  Almost non-negative scalar curvature on Riemannian manifolds conformal to tori.  The Journal of Geometric Analysis, 31(11):11190–11213, 2021.
  * [Aub05] Erwann Aubry.  Pincement sur le spectre et le volume en courbure de Ricci positive.  In Annales scientifiques de l’Ecole normale supérieure, volume 38, pages 387–405, 2005.
  * [BBCG12] Laurent Bessières, Gérard Besson, Gilles Courtois, and Sylvestre Gallot.  Differentiable rigidity under Ricci curvature lower bound.  Duke Math. J., 161(1):29–67, 2012.
  * [BBI22] Dmitri Burago, Yuri Burago, and Sergei Ivanov.  A course in metric geometry, volume 33.  American Mathematical Society, 2022.
  * [BCG91] Gérard Besson, Gilles Courtois, and Sylvestre Gallot.  Volume et entropie minimale des espaces localement symétriques.  Invent. Math., 103(2):417–445, 1991.
  * [BCG95] Gérard Besson, Gilles Courtois, and Sylvestre Gallot.  Entropies et rigidités des espaces localement symétriques de courbure strictement négative.  Geom. Funct. Anal., 5(5):731–799, 1995.
  * [BCG96] Gérard Besson, Gilles Courtois, and Sylvestre Gallot.  Minimal entropy and Mostow’s rigidity theorems.  Ergodic Theory Dynam. Systems, 16(4):623–649, 1996.
  * [BCG99] Gérard Besson, Gilles Courtois, and Sylvestre Gallot.  Lemme de schwarz réel et applications géométriques.  Acta Mathematica, 183(2):145–169, 1999.
  * [Bru08] Michael Brunnbauer.  Homological invariance for asymptotic invariants and systolic inequalities.  Geom. Funct. Anal., 18(4):1087–1117, 2008.
  * [But22] Karen Butt.  Quantitative marked length spectrum rigidity.  arXiv preprint arXiv:2203.12128, 2022.
  * [CC97] Jeff Cheeger and Tobias H. Colding.  On the structure of spaces with Ricci curvature bounded below. I.  J. Differential Geom., 46(3):406–480, 1997.
  * [CF03] Christopher Connell and Benson Farb.  The degree theorem in higher rank.  Journal of Differential Geometry, 65(1):19–59, 2003.
  * [CL22] Jianchun Chu and Man-Chun Lee.  Conformal tori with almost non-negative scalar curvature.  Calculus of Variations and Partial Differential Equations, 61(3):114, 2022.
  * [Col96a] Tobias H. Colding.  Large manifolds with positive Ricci curvature.  Invent. Math., 124(1-3):193–214, 1996.
  * [Col96b] Tobias H. Colding.  Shape of manifolds with positive Ricci curvature.  Invent. Math., 124(1-3):175–191, 1996.
  * [Cou98] Gilles Courtois.  Des questions de stabilité.  Séminaire de théorie spectrale et géométrie, 17:159–162, 1998.
  * [CRX19] Lina Chen, Xiaochun Rong, and Shicheng Xu.  Quantitative volume space form rigidity under lower Ricci curvature bound I.  Journal of Differential Geometry, 113(2):227–272, 2019.
  * [DS23] Conghan Dong and Antoine Song.  Stability of Euclidean 3-space for the Positive Mass Theorem.  arxiv preprint, 2023.
  * [EM93] Alex Eskin and Curt McMullen.  Mixing, counting, and equidistribution in Lie groups.  Duke mathematical journal, 71(1):181–209, 1993.
  * [GKL22] Colin Guillarmou, Gerhard Knieper, and Thibault Lefeuvre.  Geodesic stretch, pressure metric and marked length spectrum rigidity.  Ergodic Theory and Dynamical Systems, 42(3):974–1022, 2022.
  * [GL19] Colin Guillarmou and Thibault Lefeuvre.  The marked length spectrum of Anosov manifolds.  Annals of Mathematics, 190(1):321–344, 2019.
  * [HLS17] Lan-Hsuan Huang, Dan A Lee, and Christina Sormani.  Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space.  Journal für die reine und angewandte Mathematik (Crelles Journal), 2017(727):269–299, 2017.
  * [KNPS21] Mikhail Karpukhin, Mickaël Nahon, Iosif Polterovich, and Daniel Stern.  Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces.  arXiv preprint arXiv:2106.15043, 2021.
  * [Lan11] Urs Lang.  Local currents in metric spaces.  J. Geom. Anal., 21(3):683–742, 2011.
  * [LNN20] Man-Chun Lee, Aaron Naber, and Robin Neumayer.  $d_{p}$ convergence and $\epsilon$-regularity theorems for entropy and scalar curvature lower bounds.  arXiv preprint arXiv:2010.15663, 2020.
  * [LS14] Dan A Lee and Christina Sormani.  Stability of the positive mass theorem for rotationally symmetric riemannian manifolds.  Journal für die reine und angewandte Mathematik (Crelles Journal), 2014(686):187–220, 2014.
  * [LW11] FRANÇOIS Ledrappier and XIAODONG Wang.  Pinching theorems for the volume entropy.  preprint, 2011.
  * [Pet99] Peter Petersen.  On eigenvalue pinching in positive Ricci curvature.  Inventiones mathematicae, 138(1):1–21, 1999.
  * [Por15] Jacobus W Portegies.  Semicontinuity of eigenvalues under intrinsic flat convergence.  Calculus of Variations and Partial Differential Equations, 54(2):1725–1766, 2015.
  * [Rua22] Yuping Ruan.  The Cayley hyperbolic space and volume entropy rigidity.  arXiv preprint arXiv:2203.14418, 2022.
  * [S+21] Christina Sormani et al.  Conjectures on convergence and scalar curvature.  arXiv preprint arXiv:2103.10093, 2021.
  * [Sam99] Andrea Sambusetti.  Minimal entropy and simplicial volume.  Manuscripta Math., 99(4):541–560, 1999.
  * [Son23] Antoine Song.  Metric currents, barycenter maps and the spherical Plateau problem.  arXiv preprint, 2023.
  * [Sou08] Juan Souto.  Two applications of the natural map.  Geometriae Dedicata, 133(1):51–57, 2008.
  * [SW11] Christina Sormani and Stefan Wenger.  The intrinsic flat distance between Riemannian manifolds and other integral current spaces.  J. Differential Geom., 87(1):117–199, 2011.
  * [Wen07] Stefan Wenger.  Flat convergence for integral currents in metric spaces.  Calc. Var. Partial Differential Equations, 28(2):139–160, 2007\.
  * [Wen11] Stefan Wenger.  Compactness for manifolds and integral currents with bounded diameter and volume.  Calc. Var. Partial Differential Equations, 40(3-4):423–448, 2011\.