# Minimal surface entropy and average area ratio

Ben Lowe and André Neves  Princeton University
Fine Hall
Princeton NJ 08544
USA<EMAIL_ADDRESS>University of Chicago
Department of Mathematics
Chicago IL 60637
USA<EMAIL_ADDRESS>
###### Abstract.

On any closed hyperbolizable $3$-manifold, we find a sharp relation between
the minimal surface entropy (introduced by Calegari-Marques-Neves) and the
average area ratio (introduced by Gromov), and we show that, among metrics $g$
with scalar curvature greater than or equal to $-6$, the former is maximized
by the hyperbolic metric. One corollary is to solve a conjecture of Gromov
regarding the average area ratio.

Our proofs use Ricci flow with surgery and laminar measures invariant under a
$\text{PSL}(2,\mathbb{R})$-action.

The second author is partly supported by NSF DMS-2005468 and a Simons
Investigator Grant.

##  1\. Introduction

The interplay between scalar curvature, area, and topology is a beautiful
chapter in mathematics. For an extended overview of the subject containing the
most recent developments, the reader can consult Gromov’s Four Lectures on
Scalar Curvature [15].

We are interested in studying the area functional on closed manifolds
admitting a hyperbolic metric. Variational methods, which were pioneered by
Schoen and Yau, are less effective in this setting because the restrictions
imposed by the second variation of area are not sharp in the hyperbolic case.
On the other hand, the study of length, curvature, and topology has a very
rich literature in the negative curvature case, partly because the fact that
the geodesic flow is Anosov brings an extra structure to the problem. In the
same vein, there is an extra structure for minimal surfaces coming from a
“natural” $\text{PSL}(2,\mathbb{R})$-action on the space of minimal immersions
(formalized very clearly by Labourie in [23]). The general principle we
follow, initiated by Calegari–Marques–Neves in [9], is to combine the rigidity
of that action (due to Ratner and Shah) with geometric methods to obtain sharp
relations between area, scalar curvature, and minimal surfaces. One
consequence is to answer a conjecture of Gromov regarding the least possible
value for the average area ratio.

Using spin methods, Min-Oo [27] proved a positive mass Theorem for
asymptotically hyperbolic manifolds and Ono [28], Davaux [11] proved sharp
spectral inequalities for hyperbolizable manifolds. Andersson, Cai, and
Galloway [2] proved a positive mass Theorem for asymptotically hyperbolic
manifolds using variational methods.

Let $(M,g)$ be a closed Riemannian $3$-manifold admitting a hyperbolic metric
$g_{0}$. We refer the reader to Section 2 for all the definitions.

###  1.1. Minimal Surface Entropy

Let $S_{\varepsilon}(M)$ denote the set of all homotopy class $\Pi$ of
essential surfaces whose limit set is a $(1+\varepsilon)$-quasicircle. We
define

$\text{area}_{g}(\Pi):=\inf\\{\text{area}_{g}(S):S\in\Pi\\}.$

Inspired by the following expression for the volume entropy $E_{vol}(g)$ on
negatively curved manifolds

$E_{vol}(g)=\lim_{L\to\infty}\frac{\ln\\#\\{{\rm length}_{g}(\gamma)\leq
L:\,\gamma\text{ closed geodesic in }(M,g)\\}}{L}$

Calegari, Marques, and Neves defined in [9] the minimal surface entropy111In
[9] the definition used $\liminf$ instead of $\limsup$ but all the results
proven there are also true with this definition.

(1) $E(g):=\lim_{\varepsilon\to
0}\limsup_{L\to\infty}\frac{\ln\\#\\{\text{area}_{g}(\Pi)\leq 4\pi(L-1):\Pi\in
S_{\varepsilon}(M)\\}}{L\ln L}.$

The authors showed in [9] that $E(g_{0})=2$ and that, among metrics $g$ with
sectional curvature less than or equal to $-1$, $E(g)\geq E(g_{0})=2$ and
equality implies $g=g_{0}$ (up to isometry). The inequality follows almost
directly from Gauss equation. On the other hand, to show that the hyperbolic
metric is the unique maximizer the authors had to use the rigidity of
$\text{PSL}(2,\mathbb{R})$-orbits in the frame bundle (due to Ratner [30] and
Shah [33]).

###  1.2. Average Area Ratio

Consider another closed hyperbolic $3$-manifold $N$ with an hyperbolic metric
$h_{0}$. The Grassmanian bundle over $N$ (or $M$) of unoriented tangent
$2$-planes to $N$ (or $M$) is denoted by $Gr_{2}(N)$ (or $Gr_{2}(M)$). The
metric $h_{0}$ induces a natural metric on $Gr_{2}(N)$.

Given $T:M\rightarrow N$ a smooth map, the following function is defined for
almost all $(x,P)\in Gr_{2}(N)$:

$(x,P)\in Gr_{2}(N)\,\mapsto\,|\wedge^{2}T^{-1}|(x,P):=\lim_{\delta\to
0}\frac{{\rm area}_{g}(T^{-1}(P_{\delta}))}{\delta},$

where $P_{\delta}\subset N$ is a disc transversal to $T$, tangent to $P$ at
$x\in N$, and area $\delta$.

Gromov [13, page 73] defined ${\rm Area}_{T}(g/h_{0})$ the average area ratio
of $T$ to be

${\rm Area}_{T}(g/h_{0}):=\frac{1}{{\rm
vol}_{h_{0}}(Gr_{2}(N))}\int_{Gr_{2}(N)}|\wedge^{2}T^{-1}|(x,P)dV_{h_{0}}(x,P).$

A more detailed definition is in (2).

In the same paper, Gromov used a second variation argument to show that ${\rm
Area}_{T}(g/h_{0})\geq{\rm degree}\,T/3$ if the scalar curvature of $g$
satisfies $R(g)\geq-6$. The reason he got the factor $1/3$ is because the
second variation of area is not sharp on hyperbolic manifolds. He conjectured
in Remark 2.4.C’ of [13] that assuming $R(g)\geq-6$

${\rm Area}_{T}(g/h_{0})\geq{\rm degree}\,T$ holds true and it is sharp.

This question was also addressed in Section 3.H of [14] and in the more recent
[15] at the end of Section 5.9.

###  1.3. Motivation

We follow the spirit of [13, 14] and explain, informally, one of the
motivations behind ${\rm Area}_{\rm Id}(g/g_{0})$. An important object in the
study of negatively curved manifolds is the $1$-dimensional foliation of the
unit tangent bundle where each leaf is an orbit for the geodesic flow.

Gromov proposed the following surface analogue: A $2$-dimensional foliation
$\mathcal{L}$ of $Gr_{2}(M)$ is a family
$\\{S_{\lambda}\\}_{\lambda\in\Lambda}$ of complete surfaces of $M$ so that
through every point in $(p,\tau)\in Gr_{2}(M)$ there is a unique $\lambda$ so
that $S_{\lambda}$ passes through $p$ with tangent plane $\tau$. Laminations
where each leaf is a stable minimal surface are analogous to the geodesic flow
foliation of the unit tangent bundle. There is a canonical foliation
$\mathcal{L}_{0}$ of $Gr_{2}(M)$ whose leafs are totally geodesic planes.

Given a metric $g$ on $M$ we obtain an area form on each leaf of
$\mathcal{L}$. Assuming $\mathcal{L}$ has a transversal measure we can
integrate the area of each leaf with respect to this measure and obtain
$vol_{g}(\mathcal{L})$.

The general question Gromov asks is how low can one make
$vol_{g}(\mathcal{L})$ subject to the constraint that $R(g)\geq-6$. There is a
“natural” transversal measure on $\mathcal{L}_{0}$ such that ${\rm Area}_{{\rm
Id}}(g/g_{0})$ coincides with
$vol_{g}(\mathcal{L}_{0})/vol_{g_{0}}(\mathcal{L}_{0})$. Therefore, Gromov’s
conjecture when $T={\rm Id}$ can be rephrased as

$vol_{g}(\mathcal{L}_{0})\geq vol_{g_{0}}(\mathcal{L}_{0})\quad\mbox{if}\quad
R(g)\geq-6.$

The following comparison is also interesting: if we consider a $3$-dimensional
foliation of $Gr_{3}(M)\simeq M$, there is only one leaf, the (unit)
transverse measure is trivial, and given a metric on $g$, the volume of this
foliation is $vol_{g}(M)$. Schoen conjectured that $vol_{g}(M)\geq
vol_{g_{0}}(M)$ among all metrics with $R(g)\geq-6$ and this was proven by
Perelman using Ricci flow (see [3] for the argument).

###  1.4. Main Theorems

The next theorem uses Ricci flow in the spirit of [8, 26] and
$\text{PSL}(2,\mathbb{R})$-invariant laminar measures.

######  Theorem 1.1.

Suppose $(M,g)$ has $R(g)\geq-6$. Then

$E(g)\leq 2$

and equality occurs if and only if $g$ is isometric to the hyperbolic metric.

This result was shown by the first author in [25] under the condition that the
(normalized) Ricci flow starting at $g$ converges to the hyperbolic metric
faster than $e^{-ct}$ for some $c>1$. As it is pointed out in [25], there are
closed manifolds $\mathbb{H}^{3}/\Gamma$ for which that will always happen and
closed manifolds $\mathbb{H}^{3}/\Gamma$ containing metrics for which that
will not happen. Manifolds containing closed embedded totally geodesic
surfaces fall into the second category. Later the second author shared his
notes containing an approach to the general case relying on
$\text{PSL}(2,\mathbb{R})$-invariant laminar measures. Both authors decided to
join their efforts and write a single paper containing that and other related
results.

We note that Agol, Storm, and Thurston [4] conjectured that $E_{vol}(g)$ is
maximized by the hyperbolic metric among all metrics with $R(g)\geq-6$.

The next theorem relates the average area ratio and the minimal surface
entropy in a sharp way.

######  Theorem 1.2.

For every Riemannian metric $(M,g)$ we have

${\rm Area}_{{\rm Id}}(g/g_{0})E(g)\geq 2$

and equality holds if and only if $g=cg_{0}$ for some constant $c>0$.

Appealing to the previous interpretation of ${\rm Area}_{{\rm Id}}(g/g_{0})$,
we can restate Theorem 1.2 as saying that for every metric $g$

$vol_{g}(\mathcal{L}_{0})E(g)\geq 2vol_{g_{0}}(\mathcal{L}_{0})$

with equality if and only if $g=cg_{0}$ for some $c>0$. This is reminiscent of
Besson, Courtois, and Gallot [7] which says that

$vol_{g}(M)^{1/3}E_{vol}(g)\geq 2vol_{g_{0}}(M)^{1/3}$

for every metric $g$ with equality if and only if $g$ is isometric to $g_{0}$.
It is a conjecture of Calegari-Marques-Neves [10] that

$vol_{g}(M)^{2/3}E(g)\geq 2vol_{g_{0}}(M)^{2/3}$

for every metric $g$ with equality occurring only for $g_{0}$ (up to
isometry).

We now deduce two corollaries. The first one follows from combining Theorem
1.1 and Theorem 1.2.

######  Corollary 1.3.

Consider $T:M\rightarrow N$ a local diffeomorphism with degree $d$. If $(M,g)$
has scalar curvature $R(g)\geq-6$ then

${\rm Area}_{T}(g/h_{0})\geq d$

and equality holds if and only if $T$ is a local isometry between $g$ and
$h_{0}$.

This result confirms Gromov’s conjecture for local diffeomorphisms. If we just
assume that $T$ has degree $d$ there may exist no local isometry between $M$
and $N$ and thus no obvious optimal map.

###### Proof of Corollary 1.3.

We have

${\rm Area}_{T}(g/h_{0})=d\,{\rm Area}_{{\rm Id}}(g/T^{*}(h_{0})).$

Hence, with $g_{0}:=T^{*}(h_{0})$, we have from Theorem 1.2 and Theorem 1.1
that

$2{\rm Area}_{T}(g/h_{0})\geq{E(g)}{\rm Area}_{T}(g/h_{0})=d{E(g)}{\rm
Area}_{{\rm Id}}(g/g_{0})\geq 2d$

and equality holds if and only if $g=g_{0}=T^{*}(h_{0})$. ∎

The second corollary is below.

######  Corollary 1.4.

Suppose $\Sigma\subset M$ is a totally geodesic closed immersion. If $(M,g)$
has scalar curvature $R(g)\geq-6$ then, with $\Pi$ denoting the homotopy class
of $\Sigma$,

${\rm area}_{g}(\Pi)\geq-2\pi\chi(\Sigma)$

with equality if and only if $g$ is an hyperbolic metric.

###### Proof.

This follows from the proof of Theorem 1.1, namely Theorem 5.1 applied with
$\Pi_{m}=\Pi$ for all $m\in\mathbb{N}$. ∎

Without the condition that $\Pi$ contains a totally geodesic immersion in the
hyperbolic metric the result fails already for $g=g_{0}$.

###  1.5. Sketch of proofs

We describe succinctly the main ideas behind the proofs of Theorem 1.1 and
Theorem 1.2.

Assume that $(M,g)$ has $R(g)\geq-6$. To prove Theorem 1.1 we need to show
that for every sequence of homotopy classes $\Pi_{m}\in S_{1/m}(M)$ we have

$\lim_{m\to\infty}{{\rm area}_{g}(\Pi_{m})}/{{\rm area}_{g_{0}}(\Pi_{m})}\geq
1.$

Let $(g_{t})_{t\geq 0}$ be a solution to normalized Ricci flow, which we first
assume exists for all time and converges to $g_{0}$. We show that if the
inequality above fails then for all $m$ sufficiently large we have for some
$\delta>0$ and all $t\geq 0$

${{\rm area}_{g_{t}}(\Pi_{m})}/{{\rm area}_{g_{0}}(\Pi_{m})}\leq 1-\delta
e^{-t}.$

Stability analysis shows that $g_{t}\simeq g_{0}+e^{-t}\bar{h}$ as
$t\to\infty$, where $\bar{h}$ is an eigentensor for the linearization of the
trace-free Ricci tensor at the hyperbolic metric. Letting $m\to\infty$ we
obtain measures $\mu_{t}$ on the frame bundle of $M$ so that

$\mu_{t}(1)=\lim_{m\to\infty}{{\rm area}_{g_{t}}(\Pi_{m})}/{{\rm
area}_{g_{0}}(\Pi_{m})}.$

Necessarily $\mu_{t}(1)\leq 1-\delta e^{-t}$. Using a form of Gauss equation
(see (12)) for these measures we have an identity of the type

$1=\mu_{t}(1)+\mu_{t}(\mbox{curvature terms}).$

Combining with the asymptotics $g_{t}\simeq g_{0}+e^{-t}\bar{h}$ we show

$1=\mu_{t}(1)+\mu_{t}(\mbox{curvature
terms})=\mu_{t}(1)+e^{-t}\mu_{+\infty}(\mbox{terms with }\bar{h})+o(e^{-t}).$

The measure $\mu_{+\infty}$ is ${\rm PSL}(2,\mathbb{R})$-invariant. Hence,
using Ratner’s classification theorem [30] and the fact that $\bar{h}$ is an
eigentensor, we show that

$\mu_{+\infty}(\mbox{terms with }\bar{h})=0.$

Thus $1=\mu_{t}(1)+o(e^{-t})$, which contradicts $\mu_{t}(1)\leq 1-\delta
e^{-t}$.

For the general case, we proceed as above but use Perelman’s Ricci flow with
surgery [29]. Because $M$ has an hyperbolic metric all surgeries correspond to
removing capped horns and replacing them by standard caps. The key point to
check is that essential surfaces minimizing area stay away from the capped
horns. We achieve this via area comparison.

We now sketch the argument behind Theorem 1.2. We start by improving a
construction of Labourie in [23] and find a sequence $\Sigma_{m}\subset M$ of
connected immersed minimal surfaces with respect to $g_{0}$ which becomes
equidistributed in the frame bundle. We show in Proposition 6.3 that

${\rm Area}_{{\rm Id}}(g/g_{0})=\lim_{m\to\infty}{\rm
area}_{g}(\Sigma_{m})/{\rm area}_{g_{0}}(\Sigma_{m}).$

A counting argument implies that

$E(g)\geq E(g_{0})\lim_{m\to\infty}{\rm area}_{g_{0}}(\Sigma_{m})/{\rm
area}_{g}(\Sigma_{m})$

and these two expressions give that $E(g){\rm Area}_{{\rm Id}}(g/g_{0})\geq
2$. If equality holds we show first that the metric $g$ is Zoll, i.e., every
totally geodesic disc with respect to $g_{0}$ is minimal with respect to $g$,
and then we show that $g$ must be a multiple of $g_{0}$. This proves Theorem
1.2.

###  1.6. Acknowledgments

The first author thanks his advisor Fernando Codá Marques for helpful
conversations related to this paper and for his support.

The second author thanks Danny Calegari.

##  2\. Preliminaries

Assume $(M,g)$ is a closed Riemannian $3$-manifold admitting a hyperbolic
metric $g_{0}$ and $N$ another closed hyperbolic $3$-manifold $N$ with an
hyperbolic metric $h_{0}$.

Suppose $T:M\rightarrow N$ is smooth map. Given $(x,P)\in Gr_{2}(M)$ we denote
by $|\wedge^{2}T|_{g}(x,P)$ the Jacobian of

$dT_{x}:P\rightarrow T_{T(x)}M,$

meaning that if $e_{1},e_{2}$ is an orthonormal basis of $P$ and
$u_{i}:=dT_{x}(e_{i}),$ $i=1,2,$

$|\wedge^{2}T|_{g}(x,P):=\sqrt{h_{0}(u_{1},u_{1})h_{0}(u_{2},u_{2})-h_{0}^{2}(u_{1},u_{2})}.$

Given $y\in N$ a regular value and $\tau=(y,V)\in Gr_{2}(N)$ set

$|\wedge^{2}T^{-1}|_{g}(y,V)=\sum_{x\in
T^{-1}(y)}\frac{1}{|\wedge^{2}T|_{g}(x,(dT_{x})^{-1}(V))}$

The function $\tau\mapsto|\wedge^{2}T^{-1}|_{g}(\tau)$ is defined almost
everywhere on $Gr_{2}(N)$.

For context, if $T$ is transverse to a closed surface $S\subset M$, then
$\Sigma:=T^{-1}(S)$ is a surface of $N$ with area

${\rm
area}_{g}(\Sigma)=\int_{{S}}|\wedge^{2}T^{-1}|_{g}(y,T_{y}S)dA_{h_{0}}(y).$

Gromov [13, page 73] defined the average area ratio of $T$ as

(2) ${\rm
Area}_{T}(g/h_{0}):=\int_{Gr_{2}(N)}|\wedge^{2}T^{-1}|_{g}(\tau)d\mu_{h_{0}}(\tau),$

where integration is with respect to the unit volume measure $\mu_{h_{0}}$ on
$Gr_{2}(N)$ induced by $h_{0}$. In particular, ${\rm Area}_{{\rm
Id}}(h_{0}/h_{0})=1$. In the definition above it is implicitly assumed that
$|\wedge^{2}T^{-1}|_{g}$ is in $L^{1}$. If that is not the case we define
${\rm Area}_{T}(g/h_{0})=\infty$.

We use $\pi_{1}(M)$ to denote as well its representation into
$\text{PSL}(2,\mathbb{C})$. A closed immersed surface $\Sigma\subset M$ is
essential if the immersion $\iota:\Sigma\rightarrow M$ is $\pi_{1}$-injective.
Essential surfaces lift to discs in $\mathbb{H}^{3}$. Using the representation
of $\pi_{1}(M)$ into ${\rm PSL}(2,\mathbb{C})$ we have
$\iota_{*}(\pi_{1}(\Sigma))<\text{PSL}(2,\mathbb{C})$ and so we can associate
its limit set $\Lambda(\Sigma)\subset\partial_{\infty}\mathbb{H}^{3}\simeq
S^{2}$ (for the definition see for instance [9, Section 2.2]).

Set $\mathcal{C}_{\varepsilon}$ to be the space of
$(1+\varepsilon)$-quasicircles in $\partial_{\infty}\mathbb{H}^{3}$ (see [9,
Definition 2.3] for precise definition). The group $\pi_{1}(M)$ acts on
$\mathcal{C}_{\varepsilon}$ and preserves $\mathcal{C}_{0}$ (which is the
space of all geodesic circles in
$S^{2}\simeq\partial_{\infty}\mathbb{H}^{3}$).

Let $S_{\varepsilon}(M)$ denote the set of all homotopy class $\Pi$ of
essential surfaces with limit set in $\mathcal{C}_{\varepsilon}$. Recall that
we defined

$\text{area}_{g}(\Pi):={\text{inf}}\\{\text{area}_{g}(S):S\in\Pi\\}.$

From Schoen-Yau [31] there is an immersed minimal surface (with respect to
$g$) ${\bf\Sigma}_{g}(\Pi)\in\Pi$ which realizes $\text{area}_{g}(\Pi)$.

From [32], assuming $\varepsilon$ is sufficiently small, for all
$\gamma\in\mathcal{C}_{\varepsilon}$ there is a unique embedded area-
minimizing disc $D(\gamma)\subset\mathbb{H}^{3}$ (with respect to $g_{0}$)
with $\partial_{\infty}D(\gamma)=\gamma\subset\partial_{\infty}\mathbb{H}^{3}$
and principal curvatures that can be made arbitrarily small by choosing
$\varepsilon$ small enough.

The same argument as in Theorem 4.3 of [24] (adapted to the case where the
minimal discs are not necessarily preserved by some surface group), shows that
we can find a neighborhood $\mathcal{U}$ of $g_{0}$ and $\bar{\varepsilon}$
small so that for each $\gamma\in\mathcal{C}_{\bar{\varepsilon}}$ there is a
unique non-degenerate area-minimizing disc $\Sigma_{g}(\gamma)$ with respect
to the metric $g$ so that $\partial_{\infty}\Sigma_{g}(\gamma)=\gamma$. The
discs $\Sigma_{g}(\gamma)$ and $D(\gamma)$ are at a bounded Hausdorff distance
from each other (independent of $g$) and if $g\to g_{0}$ then
$\Sigma_{g}(\gamma)$ converges to $D(\gamma)$ uniformly in $C^{2,\alpha}$.
Therefore we can decrease $\mathcal{U}$ and $\bar{\varepsilon}$ so that for
all $\gamma\in\mathcal{C}_{\bar{\varepsilon}}$ there is $f_{\gamma}\in
C^{\infty}(D(\gamma))$ (depending on $g$) such that its graph over $D(\gamma)$
is identical to $\Sigma_{g}(\gamma)$ and $|f_{\gamma}|_{2,\alpha}<1$. There is
an ambiguity on the sign of $f_{\gamma}$ but the quantities we consider, like
$|f_{\gamma}|$, will be sign independent.

If $\Pi\in S_{\bar{\varepsilon}}(M)$ and $g\in\mathcal{U}$ then, with
$\gamma:=\Lambda({\bf\Sigma}_{g}(\Pi))$, uniqueness implies that
$\Sigma_{g}(\gamma)$ covers a minimal surface ${\bf\Sigma}_{g}(\gamma)$ in
$\Pi$ which must coincide with ${\bf\Sigma}_{g}(\Pi)$ and thus satisfy

$\text{area}_{g}(\Pi)=\text{area}_{g}({\bf\Sigma}_{g}(\gamma)).$

Given $\gamma\in\mathcal{C}_{\varepsilon}$, let $n(\gamma)$ denote a
continuous unit normal vector field along $D(\gamma)$ with respect to $g_{0}$.
Consider the diffeomorphism (using the hyperboloid model)

(3) $F_{\gamma}:D(\gamma)\rightarrow\Sigma_{g}(\gamma),\quad
x\mapsto\cosh(f_{\gamma}(x))x+\sinh(f_{\gamma}(x))n(\gamma)(x).$

We omit the dependence of $g$ on $F_{\gamma}$ to avoid too much notation. When
$g=g_{0}$, $f_{\gamma}=0$ and thus $F_{\gamma}$ is the identity.

##  3\. Laminations and Laminar measures

We follow the presentation of Labourie in [23] and add some auxiliary results.

###  3.1. Laminations

Consider the space of stable minimal conformal immersions
$\mathcal{F}(\mathbb{H}^{3},\varepsilon)$ (with
$\varepsilon\leq\bar{\varepsilon}$) defined in Definition 5.1 of [23], i.e.,
the space of conformal minimal immersions

$\phi:\mathbb{H}^{2}\rightarrow(\mathbb{H}^{3},g_{0})\quad{\rm
with}\,\,\partial\phi:=\phi(\partial_{\infty}\mathbb{H}^{2})\in\mathcal{C}_{\varepsilon}.$

Because $\varepsilon\leq\bar{\varepsilon}$,
$D(\partial\phi)=\phi(\mathbb{H}^{2})$, $\phi$ is an embedding and stable for
the second variation of area. The topology we choose is the same as the one
considered in Definition 5.1 of [23] and it makes the map
$\phi\mapsto\partial\phi$ continuous (Theorem 5.2 of [23]). Thus

$\cap_{k\in\mathbb{N}}\mathcal{F}(\mathbb{H}^{3},1/k)=\mathcal{F}(\mathbb{H}^{3},0).$

Similar to [23] we also consider
$\mathcal{F}(M,\varepsilon):=\mathcal{F}(\mathbb{H}^{3},\varepsilon)/\pi_{1}(M)$
with the quotient topology. The group $\text{PSL}(2,\mathbb{R})$ acts on
$\mathbb{H}^{2}$ and thus it acts on $\mathcal{F}(M,\bar{\varepsilon})$ in the
following way:

(4) $\tau\in\text{PSL}(2,\mathbb{R}),\quad
R_{\tau}:\mathcal{F}(M,\varepsilon)\rightarrow\mathcal{F}(M,\varepsilon)\quad
R_{\tau}(\phi):=\phi\circ\tau^{-1}.$

The space $\mathcal{F}(M,\bar{\varepsilon})$ together with the
$\text{PSL}(2,\mathbb{R})$-action is called the conformal minimal laminations
of $M$.

Fix a fundamental domain $\Delta\subset\mathbb{H}^{3}$ of $M$. Given
$\phi\in\mathcal{F}(M,\varepsilon)$, there is a unique lift to an element of
$\mathcal{F}(\mathbb{H}^{3},\varepsilon)$, denoted by $\phi$ as well, that is
uniquely determined by the requirement that $\phi(i)\in\Delta$. Thus for each
$\phi\in\mathcal{F}(M,\varepsilon)$ we obtain
$\partial\phi\in\mathcal{C}_{\varepsilon}$ but this map is not necessarily
continuous. Theorem 5.2 (i) of [23] says that the evaluation map which sends
$\phi\in\mathcal{F}(\mathbb{H}^{3},\bar{\varepsilon})$ to
$\phi(i)\in\mathbb{H}^{3}$ is proper. As a result we deduce at once the lemma
below.

######  Lemma 3.1.

The space $\mathcal{F}(M,\bar{\varepsilon})$ is sequentially compact.

Given $\phi\in\mathcal{F}(M,\bar{\varepsilon})$, let $C(\phi)>0$ be the
conformal factor of $\phi^{*}(g_{0})$. Denote the Gaussian curvature of
$D(\partial\phi)$ by $K(\phi)$. From Gauss equation and [32] we have, after
making $\bar{\varepsilon}$ smaller if necessary, $-2\leq K(\phi)\leq-1$. The
maximum principle applied to the equation satisfied by $C(\phi)$ implies that

(5) $\frac{1}{2}\leq\frac{1}{\sup_{D(\partial\phi)}|K(\phi)|}\leq C(\phi)\leq
1.$

Let $F(M)$ denote the frame bundle of $M$, i.e.,
$F(M)=\text{PSL}(2,\mathbb{C})/\pi_{1}(M)$. Fix $\\{e_{1},e_{2}\\}$ an
orthonormal basis of $\mathbb{H}^{2}$ and, given
$\phi\in\mathcal{F}(M,\bar{\varepsilon})$, set

$e_{1}(\phi):=d\phi_{i}(e_{1})C(\phi)^{-1/2}\quad\text{and}\quad
e_{2}(\phi):=d\phi_{i}(e_{2})C(\phi)^{-1/2}.$

There is a unique $n(\phi)\in T_{\phi(i)}M$ so that
$\\{e_{1}(\phi),e_{2}(\phi),n(\phi)\\}$ is a positive frame. Consider the
continuous map

(6) $\Omega:\mathcal{F}(M,\bar{\varepsilon})\rightarrow
F(M),\quad\Omega(\phi)=(\phi(i),\\{e_{1}(\phi),e_{2}(\phi),n(\phi)\\}).$

Given $\phi\in\mathcal{F}(M,\bar{\varepsilon})$, set $\gamma:=\partial\phi$
and denote the Jacobian of (see (3))

$F_{\gamma}\circ\phi:\mathbb{H}^{2}\rightarrow(\Sigma_{g}(\gamma),g)$

by $|Jac_{g}(F_{\gamma}\circ\phi)|$. Consider the function

(7)
$\Lambda_{g}:\mathcal{F}(M,\bar{\varepsilon})\rightarrow\mathbb{R},\quad\Lambda_{g}(\phi):=|Jac_{g}(F_{\gamma}\circ\phi)|(i).$

If $dA_{g}$ is the area element of $\Sigma_{g}(\gamma)$, then
$(F_{\gamma}\circ\phi)_{i}^{*}(dA_{g})=\Lambda_{g}(\phi)dx\wedge dy$, in
isothermal coordinates. This function is continuous because it is independent
of the particular lift of $\phi$ that was chosen.

Let $\nu_{g}(\gamma)$ denote a continuous unit normal vector field along
$\Sigma_{g}(\gamma)$ with respect to $g$, where
$\gamma\in\mathcal{C}_{\bar{\varepsilon}}$. Consider the following functions

(8)
$|A|_{g}^{2}:\mathcal{F}(M,\varepsilon)\rightarrow\mathbb{R},\quad\phi\mapsto|A|_{g}^{2}(\Sigma_{g}(\partial\phi))(F_{\partial\phi}\circ\phi(i)),$
(9)
$Ric(g)(\nu,\nu):\mathcal{F}(M,\varepsilon)\rightarrow\mathbb{R},\quad\phi\mapsto
Ric(g)_{|F_{\partial\phi}\circ\phi(i)}(\nu_{g}(\partial\phi),\nu_{g}(\partial\phi)),$
(10) $R(g):\mathcal{F}(M,\varepsilon)\rightarrow\mathbb{R},\quad\phi\mapsto
R(g)(F_{\partial\phi}\circ\phi(i)).$

The definition of all these functions is independent of the particular lift of
$\phi\in\mathcal{F}(M,\varepsilon)$ that was chosen and thus they are
continuous.

###  3.2. Laminar measures

A laminar measure $\mu$ on $\mathcal{F}(M,\bar{\varepsilon})$ is a probability
measure that is invariant under the $\text{PSL}(2,\mathbb{R})$-action given by
(4).

A laminar measure $\mu$ and the map $\Omega$ defined in (6) induces a
probability measure $\Omega_{*}\mu$ on $F(M)$. That measure is invariant under
a $\text{PSL}(2,\mathbb{R})$-action which will not coincide in general with
the homogeneous action of $\text{PSL}(2,\mathbb{R})$ as a subgroup of
$\text{PSL}(2,\mathbb{C})$. Another issue that needs to be addressed is the
fact the space of laminar measures is not necessarily weakly compact (a
related problem is put as an open question in [22]).

######  Lemma 3.2.

Let $\mu_{k}$ be a sequence of laminar measures on $\mathcal{F}(M,1/k)$ so
that $\Omega_{*}\mu_{k}$ converges weakly to a probability measure $\bar{\mu}$
on $F(M)$. Then $\bar{\mu}$ is invariant under the homogeneous action of
$\text{PSL}(2,\mathbb{R})$.

###### Proof.

Let $\Omega_{0}:\mathcal{F}(M,0)\rightarrow F(M)$ be the restriction of
$\Omega$ to $\mathcal{F}(M,0)$. Every $\phi\in\mathcal{F}(M,0)$ has the
property that $D(\partial\phi)$ is a totally geodesic disc and so
$\phi:\mathbb{H}^{2}\rightarrow M$ is an isometric immersion. Thus $\phi$ is
uniquely determined by $\phi(i),e_{1}(\phi),$ and $e_{2}(\phi)$. Hence
$\Omega_{0}$ is bijective and a homeomorphism.

Recall the $\text{PSL}(2,\mathbb{R})$-action on
$\mathcal{F}(M,\bar{\varepsilon})$ defined in (4). $\Omega_{0}$ induces a
$\text{PSL}(2,\mathbb{R})$-action on $F(M)$ in the following way:

$\tau\in\text{PSL}(2,\mathbb{R}),\quad F_{\tau}:F(M)\rightarrow F(M),\quad
F_{\tau}(x)=\Omega_{0}(R_{\tau}\circ\Omega_{0}^{-1}(x)).$

This action corresponds to the homogeneous action of
$\text{PSL}(2,\mathbb{R})$. Thus, given $f\in C^{0}(F(M))$, we need to check
that

$\bar{\mu}(f\circ F_{\tau})=\bar{\mu}(f)\quad\mbox{for all
}\tau\in\text{PSL}(2,\mathbb{R}).$

Consider the “projection” of $\mathcal{F}(M,\varepsilon)$ onto
$\mathcal{F}(M,0)$ given by

$P:=\Omega_{0}^{-1}\circ\Omega:\mathcal{F}(M,\varepsilon)\rightarrow\mathcal{F}(M,0).$

Set $\eta:=f\circ\Omega\circ R_{\tau}$. We have $f\circ
F_{\tau}\circ\Omega=\eta\circ P$ and thus

${\bar{\mu}}(f\circ F_{\tau})=\lim_{k\to\infty}\Omega_{*}\mu_{k}(f\circ
F_{\tau})=\lim_{k\to\infty}\mu_{k}(\eta\circ P).$

We also have $\Omega_{*}\mu_{k}(f)=\mu_{k}(\eta)$ for all $k\in\mathbb{N}$ by
$\text{PSL}(2,\mathbb{R})$-invariance and thus

${\bar{\mu}}(f)=\lim_{k\to\infty}\Omega_{*}\mu_{k}(f)=\lim_{k\to\infty}\mu_{k}(\eta).$

In light of these last two identities it suffices to check that

$\lim_{k\to\infty}|\mu_{k}(\eta\circ P)-\mu_{k}(\eta)|=0$

and this follows at once if we show that

$\lim_{k\to\infty}\sup_{\phi\in\mathcal{F}(M,1/k)}|\eta\circ
P(\phi)-\eta(\phi)|=0.$

If this identity does not hold we find $\delta>0$ and
$\phi_{k}\in\mathcal{F}(M,1/k)$ so that $|\eta\circ
P(\phi_{k})-\eta(\phi_{k})|\geq\delta$ for all $k\in\mathbb{N}$. From Lemma
3.1 we know that, after passing to a subsequence, $\phi_{k}$ converges to some
$\phi\in\mathcal{F}(M,0)$ which must satisfy $|\eta\circ
P(\phi)-\eta(\phi)|\geq\delta$. This is impossible because $P(\phi)=\phi$. ∎

Let $\Gamma$ be a Fuchsian subgroup of $\text{PSL}(2,\mathbb{R})$ so that
$\mathbb{H}^{2}/\Gamma$ is a closed hyperbolic surface with genus $l$. All
Fuchsian groups we consider will have this property with no need for further
mentioning.

$\text{PSL}(2,\mathbb{R})/\Gamma$ is the frame bundle of
$\mathbb{H}^{2}/\Gamma$. With respect to the invariant metric on
$\text{PSL}(2,\mathbb{R})/\Gamma$ we have
$vol(\text{PSL}(2,\mathbb{R})/\Gamma)=\alpha_{0}4\pi(l-1)$ for some universal
constant $\alpha_{0}$.

Suppose $\phi\in\mathcal{F}(M,\bar{\varepsilon})$ is equivariant with respect
to a representation of $\Gamma<\text{PSL}(2,\mathbb{R})$ in
$\pi_{1}(M)<\text{PSL}(2,\mathbb{C})$. Consider
$U\subset\text{PSL}(2,\mathbb{R})$ a fundamental domain of
$\text{PSL}(2,\mathbb{R})/\Gamma$. Following Proposition 5.5 of [23] we define
$\delta_{\phi}$ a laminar measure on $\mathcal{F}(M,\bar{\varepsilon})$

$\delta_{\phi}(f):=\frac{1}{vol(U)}\int_{U}f(\phi\circ\tau)d\bar{\nu}(\tau),\quad
f\in C^{0}(\mathcal{F}(M,\bar{\varepsilon}))$

where $\bar{\nu}$ is the bi-invariant measure on $\text{PSL}(2,\mathbb{R})$.

Equivariance implies that $D(\partial\phi)$ project to closed surface ${\bf
D(\partial\phi)}$ on $M$ and that $\phi$ descends to an immersion from
$\mathbb{H}^{2}/\Gamma$ to ${\bf D(\partial\phi)}$ that we still denote by
${\phi}$. The uniqueness property implies that $\Sigma_{g}(\partial\phi)$
projects to closed surface ${\bf\Sigma}_{g}(\partial\phi)$ on $M$ that is
homotopic to ${\bf D(\partial\phi)}$. The map $F_{\partial\phi}\circ\phi$ is
also equivariant and so descends to a map from $\mathbb{H}^{2}/\Gamma$ to
${\bf\Sigma}(\partial\phi)$ that we denote by
$\bf{F_{\partial\phi}}\circ{\phi}$. The unit normal vector field
$\nu_{g}(\gamma)$ induces a unit normal vector field along
${\bf\Sigma}_{g}(\gamma)$ that we also denote by $\nu_{g}(\gamma)$.

For context, suppose $f$ is a continuous function in $Gr_{2}(M)$ and set

$\hat{f}:\mathcal{F}(M,\bar{\varepsilon})\rightarrow\mathbb{R},\quad\phi\mapsto
f({\pi\circ F_{\partial\phi}}\circ\phi(i),d({\pi\circ
F_{\partial\phi}})_{|\phi(i)}(d\phi_{|i}({T_{i}}\mathbb{H}^{2}))).$

The function $\hat{f}$ is continuous. With $dA_{hyp}$ denoting the hyperbolic
volume form on $\mathbb{H}^{2}$ we have

(11)
$\int_{{\bf\Sigma}_{g}(\partial\phi)}f(x,T_{x}{\bf\Sigma}_{g}(\partial\phi))dA_{g}(x)\\\
=\int_{\mathbb{H}^{2}/\Gamma}f({\bf F_{\partial\phi}}\circ\phi(z),d{\bf
F_{\partial\phi}}_{|\phi(z)}(d_{z}\phi(\mathbb{H}^{2}))|Jac_{g}({\bf
F_{\partial\phi}\circ\phi})|(z)dA_{hyp}(z)\\\
=\frac{1}{\alpha_{0}}\int_{U}\hat{f}(\phi\circ\tau)\Lambda_{g}(\phi\circ\tau)d\bar{\nu}(\tau)=4\pi(l-1)\delta_{\phi}(\hat{f}\Lambda_{g}).$

### Gauss identity for laminar measures

From Gauss equation we have

$4\pi(l-1)=\text{area}_{g}({\bf\Sigma}_{g}(\partial\phi))\\\
+\int_{{\bf\Sigma}_{g}(\partial\phi)}Ric(g)({\nu}_{g}({\partial\phi}),{\nu}_{g}({\partial\phi}))-\frac{1}{3}R(g)+\frac{|A|^{2}}{2}-\frac{R(g)+6}{6}dA_{g}.$

When interpreted in terms of laminar measures this identity becomes

(12)
$1=\frac{\text{area}_{g}({\bf\Sigma}_{g}(\partial\phi))}{4\pi(l-1)}+\delta_{\phi}\left(\left[Ric(g)(\nu,\nu)-{R(g)/3}+{|A|^{2}_{g}}/{2}\right]\Lambda_{g}\right)\\\
-\delta_{\phi}\left(\frac{R(g)+6}{6}\Lambda_{g}\right),$

where the functions $|A|^{2}_{g}$, $Ric(g)(\nu,\nu)$, and $R(g)$ are as
defined in (8), (9), and (10).

##  4\. Proof of Theorem 1.1: Part I

Recall that $M$ is a closed manifold with an hyperbolic metric $g_{0}$.
Throughout this paper we refer to normalized Ricci flow as a one-parameter
family of metrics $(\bar{g}_{t})_{t\in I}$ which solve

(13) $\frac{d\bar{g}_{t}}{dt}=-2Ric(\bar{g}_{t})-4\bar{g}_{t}.$

######  Theorem 4.1.

There is a neighborhood $\mathcal{V}$ of $\bar{g}$ so that for all
$g\in\mathcal{V}$ with $R(g)\geq-6$ the following holds: For any sequence
$\Pi_{m}\in S_{1/m}(M)$ we have

$\liminf_{m\to\infty}\frac{{\rm area}_{g}(\Pi_{m})}{4\pi(l_{m}-1)}\geq 1,$

where $l_{m}$ is the genus of an essential surface in $\Pi_{m}$.

If equality holds then $g$ is isometric to $g_{0}$.

###### Proof.

Consider the neighborhood $\mathcal{U}$ of $g_{0}$ described in Section 2.
From [21, Appendix A] we see that we can find a neighborhood of $g_{0}$ in the
$C^{2,\alpha}$-topology so that for every initial condition in that
neighborhood, the normalized Ricci flow exists for all time and converges
exponentially fast in the $C^{2,\alpha}$-topology to an hyperbolic metric in
$\mathcal{U}$. Reasoning like in [16, Section 17] we can upgrade the
convergence and find a smaller open neighborhood
$\mathcal{V}\subset{\mathcal{U}}$ of $g_{0}$ so that for every
$g\in\mathcal{V}$ the normalized Ricci flow $(\bar{g}_{t})_{t\geq 0}$ starting
at $g$ exists for all time, does not leave $\mathcal{U}$, and converges
exponentially fast to an Einstein metric in $\mathcal{U}$, which must be
isometric to $g_{0}$ from Mostow rigidity. Furthermore, in [21] it is also
constructed a family of diffeomorphisms $\\{\Phi_{t}\\}_{t\geq 0}$ converging
strongly to some diffeomorphism so that $g_{t}:=\Phi_{t}^{*}\bar{g}_{t}$
solves the DeTurck-modified Ricci flow (which is strictly parabolic) and
$g_{t}$ converges to $g_{0}$ as $t\to\infty$.

The maximum principle implies that the condition $R(g)\geq-6$ is preserved by
the normalized Ricci flow because

$\frac{d}{dt}R(\bar{g}_{t})\geq\Delta_{g_{t}}R(\bar{g}_{t})+\frac{2}{3}R(\bar{g}_{t})(R(\bar{g}_{t})+6).$

###  4.1. Proof of inequality

Suppose for contradiction that the inequality fails for some metric
$g\in\mathcal{V}$. Thus we can find $\Pi_{m}\in S_{1/m}(M)\cap
S_{\bar{\varepsilon}}(M)$ and $\delta>0$ so that for all $m\in\mathbb{N}$

$\frac{\text{area}_{g}(\Pi_{m})}{4\pi(l_{m}-1)}\leq 1-\delta.$

Because $\bar{g}_{t}\in\mathcal{U}$ for all $t$, there is
$\phi_{m}\in\mathcal{F}(M,1/m)$ equivariant with respect to a representation
of a Fuchsian subgroup of ${\rm PSL}(2,\mathbb{R})$ in $\pi_{1}(M)$ so that
${\bf\Sigma}_{g_{t}}(\partial\phi_{m})\in\Pi_{m}$,
${\bf\Sigma}_{g_{t}}(\partial\phi_{m})$ depends smoothly on $t$, and
$\text{area}_{g_{t}}(\Pi_{m})=\text{area}_{g_{t}}({\bf\Sigma}_{g_{t}}(\partial\phi_{m}))$
for all $t\geq 0$. Fix $m\in\mathbb{N}$ and denote by $\nu_{t}$ the normal
vector to ${\bf\Sigma}_{g_{t}}(\partial\phi_{m})$ with respect to $g_{t}$.
Using $R(g_{t})\geq-6$ and Gauss equation

(14)
$\frac{d}{dt}\text{area}_{g_{t}}(\Pi_{m})=-\int_{{\bf\Sigma}_{g_{t}}(\partial\phi_{m})}R(g_{t})-Ric(g_{t})(\nu_{t},\nu_{t})+4\,dA_{g_{t}}\\\
=4\pi(l_{m}-1)-\text{area}_{g_{t}}(\Pi_{m})-\int_{{\bf\Sigma}_{g_{t}}(\partial\phi_{m})}\frac{|A|^{2}}{2}+\frac{R(g_{t})+6}{2}dA_{g_{t}}\\\
\leq 4\pi(l_{m}-1)-\text{area}_{g_{t}}(\Pi_{m}).$

Solving the ODE obtained by replacing the inequality sign above by an equality
sign we get that for all $t\geq 0$

(15) $\frac{\text{area}_{g_{t}}(\Pi_{m})}{4\pi(l_{m}-1)}\leq
1-e^{-t}\left(1-\frac{\text{area}_{g}(\Pi_{m})}{4\pi(l_{m}-1)}\right)\leq
1-\delta e^{-t}.$

Combining (12) with (15) and using the fact that $R(g_{t})+6\geq 0$ we get

(16) $\delta
e^{-t}\leq\delta_{\phi_{m}}\left(\left[Ric(g_{t})(\nu,\nu)-{R(g_{t})/3}+{|A|^{2}_{g_{t}}}/{2}\right]\Lambda_{g_{t}}\right).$

After passing to a subsequence, ${\Omega_{*}}\delta_{\phi_{m}}$ converges
weakly to a unit measure $\bar{\mu}$ on $F(M)$. From Lemma 3.2 we have that
$\bar{\mu}$ is invariant under the homogeneous action of
$\text{PSL}(2,\mathbb{R})$.

Let $L$ denote the linearization of the traceless Ricci tensor at $g_{0}$. For
every $2$-tensor $h$ on $M$ set $\theta(h):F(M)\rightarrow\mathbb{R}$ to be
the continuous function given by

(17) $\theta(h)(x,\\{u_{1},u_{2},n\\}):=L(h)_{x}(n,n).$

Set $h_{t}:=g_{t}-g_{0}$. We use $O(e^{at})$ to denote a term bounded by
$Ce^{at}$, where $C$ does not depend on $m$ or $t$.

######  Lemma 4.2.

$\delta\leq\bar{\mu}(e^{t}\theta(h_{t}))+O(e^{-1/3t})$.

###### Proof.

Write

  * •

$E_{1}:=\delta_{\phi_{m}}({|A|^{2}_{g_{t}}}/{2}\,\Lambda_{g_{t}})$;

  * •

$E_{2}:=\delta_{\phi_{m}}([Ric(g_{t})(\nu,\nu)-{R(g_{t})/3}](\Lambda_{g_{t}}-1))$;

  * •

$E_{3}:=\delta_{\phi_{m}}([Ric(g_{t})(\nu,\nu)-{R(g_{t})/3}]-\theta(h_{t})\circ\Omega)$.

We now estimate these terms. We know from Proposition A.2 that

$|h_{t}|_{C^{4}}=O(e^{-2/3t}).$

Fix $m\in\mathbb{N}$ and set $\gamma_{m}:=\partial\phi_{m}$. The surface
$\Sigma_{g_{t}}(\gamma_{m})$ has zero mean curvature with respect to $g_{t}$.
Thus the mean curvature $H_{g_{0}}(\Sigma_{g_{t}}(\gamma_{m}))$ of
$\Sigma_{g_{t}}(\gamma_{m})$ with respect to the hyperbolic metric has
$C^{0,\alpha}$-norm bounded by $O(e^{-2/3t})$. (This follows from the fact
that $h_{t}$ and $\nabla h_{t}$ have both that order).

Using the mean convex foliation of $\mathbb{H}^{3}$ coming from the discs
equidistant to $D(\gamma_{m})$ that was described by Uhlenbeck in [35, Theorem
3.3], the maximum principle implies that for some constant $c_{0}$

$\sup_{m\in\mathbb{N}}|f_{\gamma_{m}}|_{L^{\infty}}\leq
c_{0}\sup_{m\in\mathbb{N}}|H_{g_{0}}(\Sigma_{g_{t}}(\gamma_{m}))|_{L^{\infty}}=O(e^{-2/3t}).$

Elliptic regularity implies the existence of a constant $c_{1}>0$ such that

$\sup_{m\in\mathbb{N}}|f_{\gamma_{m}}|_{C^{2,\alpha}}\leq
c_{1}\sup_{m\in\mathbb{N}}(|f_{\gamma_{m}}|_{L^{\infty}}+|H_{g_{0}}(\Sigma_{g_{t}}(\gamma_{m}))|_{C^{0,\alpha}})=O(e^{-2/3t}).$

Thus $|A|^{2}_{g_{0}}(\Sigma_{g_{t}}(\gamma_{m}))=O(e^{-4/3t})$. If
$\lambda_{j}(g_{t})$ and $\lambda_{j}(g_{0})$, $j=1,2$ denote, respectively,
the principal curvatures of $\Sigma(\gamma_{m})$ with respect to $g_{t}$ and
$g_{0}$, we have

$\lambda_{j}(g_{t})=\lambda_{j}(g_{0})+O(|h_{t}|_{C^{1}}),\quad j=1,2.$

Hence $|A|^{2}_{g_{t}}(\Sigma_{g_{t}}(\gamma_{m}))=O(e^{-4/3t})$ and thus

(18)
$2E_{1}=\frac{1}{4\pi(l_{k}-1)}\int_{{\bf\Sigma}_{g_{t}}(\gamma_{m})}|A|^{2}dA_{g_{t}}=O(e^{-4/3t}).$

From Gauss equation, (5), and
$|A|^{2}_{g_{t}}(\Sigma_{g_{t}}(\gamma_{m}))=O(e^{-4/3t})$ we obtain that

$|C(\phi_{m})-1|=O(e^{-4/3t}).$

Moreover, from Proposition A.1 we obtain

$|Jac_{g_{t}}(F_{\gamma_{m}})|=1+O(|h|_{C^{1}})+O(|f_{\gamma_{m}}|_{C^{1}})=1+O(e^{-2/3t}).$

Thus
$|Jac_{g_{t}}(F_{\gamma_{m}}\circ\phi_{m})|=|Jac_{g_{t}}(F_{\gamma_{m}})|C(\phi_{m})=1+O(e^{-2/3t})$
and this means (see (7)) that

(19) $\Lambda_{g_{t}}(\phi_{m})=1+O(e^{-2/3t}).$

Recall the definitions in (6) and (17). From Proposition A.1 and the fact that
$|h_{t}|_{C^{4}}$ and $\sup_{m\in\mathbb{N}}|f_{\gamma_{m}}|_{C^{2,\alpha}}$
have order $O(e^{-2/3t})$, we see that

(20) $[Ric(g_{t})(\nu,\nu)-R(g_{t})/3](\phi_{m})\\\
=L(h_{t})_{\phi_{m}(i)}(n(\gamma_{m}),n(\gamma_{m}))+O(e^{{}^{-4/3t}})\\\
=\theta(h_{t})\circ\Omega(\phi_{m})+O(e^{{}^{-4/3t}}).$

Note that $\theta(h_{t})=O(|h_{t}|_{C^{2}})=O(e^{-2/3t})$ and so we obtain
from (20) and (19) that both

$E_{2}=\delta_{\phi_{m}}([\theta(h_{t})\circ\Omega+O(e^{{}^{-4/3t}})](\Lambda_{g_{t}}-1))=O(e^{{}^{-4/3t}})$

and $E_{3}=O(e^{{}^{-4/3t}})$. As a result,

$\delta_{\phi_{m}}\left(\left[Ric(g_{t})(\nu,\nu)-{R(g_{t})/3}+{|A|^{2}_{g_{t}}}/{2}\right]\Lambda_{g_{t}}\right)\\\
=\delta_{\phi_{m}}(\theta(h_{t})\circ\Omega)+E_{1}+E_{2}+E_{3}=\Omega^{*}\delta_{\phi_{m}}(\theta(h_{t}))+O(e^{{}^{-4/3t}})$

and so we have from (16)

$\delta
e^{-t}\leq\Omega^{*}\delta_{\phi_{m}}(\theta(h_{t}))+O(e^{{}^{-4/3t}}).$

Making $m\to\infty$ the result follows. ∎

Consider the operator on symmetric $2$-tensors

(21) $\mathcal{A}(h)=\Delta_{g_{0}}h-2(tr_{g_{0}}h)g_{0}+2h.$

We know from Proposition A.2 that $e^{t}h_{t}$ converges in $W^{2,2}$ to
$\bar{h}$, where $\bar{h}$ satisfies $\mathcal{A}(\bar{h})=-\bar{h}$. Thus
$e^{t}\theta(h_{t})\to\theta(\bar{h})$ in $L^{2}$ and the previous lemma
implies

$\delta\leq\bar{\mu}(\theta(\bar{h})).$

We now show that $\bar{\mu}(\theta(\bar{h}))=0$, which gives us a
contradiction and thus implies that for any sequence $\Pi_{m}\in S_{1/m}(M)$
we have

$\liminf_{m\to\infty}\frac{{\rm area}_{g}(\Pi_{m})}{{\rm
area}_{g_{0}}(\Pi_{m})}\geq 1.$

######  Proposition 4.3.

Let $\eta$ be an ergodic $\text{PSL}(2,\mathbb{R})$-invariant probability
measure on $F(M)$. Then $\eta(\theta(\bar{h}))=0$.

###### Proof.

Because $\bar{h}$ is the lowest eigenfunction for $\mathcal{A}$, it follows
from Koiso Bochner formula (see [21]) that $\bar{h}$ is trace free and
divergence free. Hence from the formula for the linearization of the traceless
Ricci tensor [6, Theorem 1.174] we have

$L(\bar{h})=-\frac{1}{2}\mathcal{A}(\bar{h})=\frac{{\bar{h}}}{2}.$

Thus for all $(x,\\{u_{1},u_{2},n\\})\in F(M)$ we have

(22) $\theta(\bar{h})((x,\\{u_{1},u_{2},n\\}))=\frac{1}{2}\bar{h}_{x}(n,n).$

From Ratner’s classification theorem [30] we have that $\eta$ is either the
homogeneous Lebesgue measure $\mu_{Leb}$ on $F(M)$ or there is a totally
geodesic surface $S\subset M$ so that $\eta$ is the homogeneous Lebesgue
measure on $F(S)$, the frame bundle of $S$.

Suppose the first case occurs. We have $tr_{g_{0}}\bar{h}=0$ and thus

$\eta(\theta(\bar{h}))=\mu_{Leb}(\theta(\bar{h}))=\frac{1}{\text{vol}(M)}\int_{M}\frac{1}{6}tr_{g_{0}}\bar{h}\,dV_{g_{0}}=0.$

Suppose the second case occurs and choose $n$ a normal vector field along $S$.
Consider the vector field $X$ tangent to $S$ such that

$g_{0}(X,Y)=\nabla_{n}\bar{h}(Y,n)\quad\mbox{for every vector field }Y\mbox{
tangent to }S.$

Set $f:S\rightarrow\mathbb{R}$, where $f(x):=\bar{h}_{x}(n(x),n(x))$.

######  Lemma 4.4.

With $\Delta_{S}$ and $div_{S}$ denoting, respectively, the Laplacian and
divergence on $S$, we have $\Delta_{S}f+6f=div_{S}X.$

###### Proof.

Fix $x\in S$ and an orthonormal basis $\\{e_{1},e_{2}\\}$ of $T_{x}S$.

By commuting derivatives and using the fact that $tr_{g_{0}}\bar{h}=0$ we
deduce

$\displaystyle\sum_{i=1}^{2}(\nabla_{n}\nabla_{e_{i}}\bar{h})(e_{i},n)=$
$\displaystyle\sum_{i=1}^{2}(\nabla_{e_{i}}\nabla_{n}\bar{h})(e_{i},n)+\sum_{i=1}^{2}\bar{h}(R(e_{i},n)e_{i},n)$
$\displaystyle+\sum_{i=1}^{2}\bar{h}(e_{i},R(e_{i},n)n)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}(\nabla_{e_{i}}\nabla_{n}\bar{h})(e_{i},n)-2\bar{h}(n,n)+\sum_{i=1}^{2}\bar{h}(e_{i},e_{i})$
$\displaystyle=$ $\displaystyle div_{S}X(x)-3f(x).$

Differentiating ${\rm div}\,\bar{h}=0$ in the direction of $n$ we obtain that
at $x\in S$

$(\nabla_{n}\nabla_{e_{1}}\bar{h})(e_{1},n)+(\nabla_{n}\nabla_{e_{2}}\bar{h})(e_{2},n)+(\nabla_{n}\nabla_{n}\bar{h})(n,n)=0.$

From (21) and $\mathcal{A}(\bar{h})=-\bar{h}$ we have that
$\Delta_{g_{0}}\bar{h}=-3\bar{h}$. Thus the last two identities we derived and
the fact that $S$ is totally geodesic imply

$\displaystyle\Delta_{S}f(x)$
$\displaystyle=(\Delta_{g_{0}}\bar{h})(n,n)-(\nabla_{n}\nabla_{n}\bar{h})(n,n)=-3f(x)-(\nabla_{n}\nabla_{n}\bar{h})(n,n)$
$\displaystyle=-3f(x)+\sum_{i=1}^{2}(\nabla_{n}\nabla_{e_{i}}\bar{h})(e_{i},n)=-6f(x)+div_{S}X(x).$

∎

From (22) we see that $\theta(\bar{h})=f/2$ when restricted to $F(S)\subset
F(M)$. Because $\eta$ is the unit Lebesgue measure on $F(S)$ we have from
Lemma 4.4 that

${\eta(\theta(\bar{h}))=\frac{1}{\text{area}_{g_{0}}(S)}\int_{S}\frac{f}{2}dA_{g_{0}}=\frac{1}{12\,\text{area}_{g_{0}}(S)}\int_{S}\Delta_{S}f-div_{S}XdA_{g_{0}}=0.}$

∎

The ergodic decomposition theorem for $\text{PSL}(2,\mathbb{R})$-actions
(Theorem 4.2.6 in [19]) gives the existence of a space $B$ with a probability
measure $\nu$ on $B$ so that for all $b\in B$ there is an ergodic
$\text{PSL}(2,\mathbb{R})$-invariant probability measure $\mu_{b}$ on $F(M)$
so that $\bar{\mu}(f)=\int_{B}\mu_{b}(f)d\nu(b)$ for all $f\in C^{0}(F(M))$.
Proposition 4.3 implies that $\bar{\mu}(\theta(\bar{h}))=0$.

###  4.2. Proof of rigidity

Suppose that for some metric $g\in\mathcal{V}$ with $R(g)\geq-6$ and some
sequence $\Pi_{m}\in S_{1/m}(M)$ we have

$\liminf_{m\to\infty}\frac{{\rm area}_{g}(\Pi_{m})}{4\pi(l_{m}-1)}=1,$

where $l_{m}$ is the genus of an essential surface in $\Pi_{m}$. Run
normalized Ricci flow $(g_{t})_{0\leq t\leq\bar{t}}$ starting at $g$ for a
short time interval and set

$a(t):=\liminf_{m\to\infty}\frac{{\rm area}_{g_{t}}(\Pi_{m})}{4\pi(l_{m}-1)}.$

From (15) we see we see that $a(0)=1$ implies that $a(t)\leq 1$ for all $0\leq
t\leq\bar{t}$ and thus $a(t)=1$ for all $0\leq t\leq\bar{t}$.

Suppose that $g$ is not Einstein. From the strong maximum principle applied to
the evolution equation of $R(g_{t})$ we obtain the existence of $\delta$ so
that $R(g_{t})\geq-6+2\delta$ for all $\bar{t}/2\leq t\leq\bar{t}$. Thus we
see from (14) that for all $\bar{t}/2\leq t\leq\bar{t}$ and all
$m\in\mathbb{N}$

$\frac{d}{dt}\text{area}_{g_{t}}(\Pi_{m})\leq
4\pi(l_{m}-1)-(1+\delta)\text{area}_{g_{t}}(\Pi_{m}).$

ODE comparison gives us a contradiction because

$a(\bar{t})\leq
a(\bar{t}/2)e^{-(1+\delta)\bar{t}/2}+\frac{1-e^{-(1+\delta)\bar{t}/2}}{1+\delta}=e^{-(1+\delta)\bar{t}/2}+\frac{1-e^{-(1+\delta)\bar{t}/2}}{1+\delta}<1.$

∎

##  5\. Proof of Theorem 1.1: Part II

We use Perelman’s Ricci flow with surgery [29] to remove the local condition
on Theorem 4.1.

######  Theorem 5.1.

Assume $g$ is a metric on $M$ such that $R(g)\geq-6$. For any sequence
$\Pi_{m}\in S_{1/m}(M)$ we have

$\liminf_{m\to\infty}\frac{{\rm area}_{g}(\Pi_{m})}{4\pi(l_{m}-1)}\geq 1,$

where $l_{m}$ is the genus of an essential surface in $\Pi_{m}$.

If equality holds then $g$ is isometric to $g_{0}$.

###### Proof.

We will use the notation and results of [29].

Note that Ricci flow and normalized Ricci flow (13) differ only by scaling.
With $\varepsilon_{0}$ small, Perelman finds in [29, Section 4] a sequence of
manifolds $M_{k}$ with $M_{0}=M$, a discrete set of times
$\\{t_{k}\\}_{k\in\mathbb{N}_{0}}$ with $t_{0}=0$, and a sequence of smooth
solutions to normalized Ricci flow $(\bar{g}_{t})_{t_{k}\leq t<t_{k+1}}$ on
$M_{k}$ with $\bar{g}_{0}=g$ so that the following holds: each $M_{k+1}$ is
obtained from $M_{k}$ by surgery as described in [29, Section 4.4] but,
because $M$ is irreducible and contains no embedded projective planes, the
only possible surgeries remove $\varepsilon_{0}$-caps and glue in (perturbed)
standard caps. Hence $M_{k}=M$ for all $k\in\mathbb{N}_{0}$ (after discarding
$3$-spheres) and, still following [29, Section 4], there is a sequence of
compact sets $\Lambda_{k}\subset M$ such that $M\setminus\Lambda_{k}$ is
diffeomorphic to a union of open balls, $\bar{g}_{t}$ converges smoothly to
$\bar{g}_{t_{k+1}}$ on $\Lambda_{k}$ as $t\to t_{k+1}$, and for all
$t<t_{k+1}$ close to $t_{k+1}$, the metric $\bar{g}_{t}$ is such that every
$p$ in the boundary of $M\setminus\Lambda_{k}$ is the center slice of some
$\varepsilon_{0}$-neck.

In Section 7.1 of [29] Perelman argues that $R(\bar{g}_{t})\geq-6$ for all
$t\geq 0$ (for the normalized Ricci flow with surgery). In Section 7.3 and
Section 7.4 of [29], Perelman shows that $(M,\bar{g}_{t})$ admits, for all $t$
sufficiently large, a thick-thin decomposition, where the thick part meets the
thin part along incompressible tori. Because $M$ is closed and admits an
hyperbolic metric, it has no incompressible tori and thus, for all $t$
sufficiently large, $(M,\bar{g}_{t})$ coincides with the thick part.
Therefore, we obtain from Lemma 7.2 of [29] that
$|Ric(\bar{g}_{t})+2\bar{g}_{t}|_{C^{0}}$ is small for all $t$ sufficiently
large. Thus we see from [36] that for all $t$ sufficiently large
$(\bar{g}_{t})_{t\geq 0}$ is a smooth solution to normalized Ricci flow and
converges to an hyperbolic metric on $M$. In particular there are only
finitely many surgery times $t_{1}<t_{2}<\ldots<t_{K}$. Using Mostow rigidity
we can assume that, after applying a diffeomorphism to ${g_{0}}$,
$(\bar{g}_{t})_{t\geq 0}$ converges smoothly to $g_{0}$ as $t\to\infty$.

Given a homotopy class $\Pi$ of essential surfaces, there is
$\Sigma_{t}\in\Pi$, a minimal surface with respect to $\bar{g}_{t}$, such that
${\rm area}_{\bar{g}_{t}}(\Pi)={\rm area}_{\bar{g}_{t}}(\Sigma_{t})$.

######  Lemma 5.2.

We can find $\varepsilon_{0}$ small so that for every homotopy class $\Pi$ of
essential surfaces and every $t<t_{k+1}$ close to $t_{k+1}$,
$\Sigma_{t}\subset\Lambda_{k}$.

###### Proof.

For all $t<t_{k+1}$ close to $t_{k+1}$, the metric $\bar{g}_{t}$ is such that
every $p$ in the boundary of $M\setminus\Lambda_{k}$ is the center slice of
some $\varepsilon_{0}$-neck. This means that there exists a neighborhood
$N\subset M$ and a diffeomorphism (depending on $t$ and $p$)

$\Phi:S^{2}\times[-{\varepsilon_{0}}^{-1},{\varepsilon_{0}}^{-1}]\rightarrow
N$

such that $p\in\Phi(S^{2}\times\\{0\\})$ and for some $\lambda>0$ (depending
on $x$ and $t$)

(23)
$|\lambda^{-2}\Phi^{*}{\bar{g}_{t}}-g_{S^{2}\times[-{\varepsilon_{0}}^{-1},{\varepsilon_{0}}^{-1}]}|_{C^{\lceil
1/{\varepsilon_{0}}\rceil}}<{\varepsilon_{0}}.$

Moreover, $\Phi(S^{2}\times\\{0\\})$ is homotopic to a boundary component of
$\Lambda_{k}$ because $M\setminus\Lambda_{k}$ consists of
$\varepsilon_{0}$-caps.

Assume for contradiction that $\Sigma_{t}$ passes through $p$. The region
$M\setminus\Lambda_{k}$ is diffeomorphic to a disjoint union of balls and so
there is a ball $B_{k}$ containing $N$ so that $\partial
B_{k}=\Phi(S^{2}\times\\{-1/{\varepsilon_{0}}\\})$. It cannot be the case that
$\Sigma_{t}$ is contained in $B_{k}$ and thus $\Sigma_{t}$ must intersect
$\partial B_{k}$. We can perturb $B_{k}$ slightly so that $\partial B_{k}$
intersects $\Sigma_{t}$ transversely in a union of circles. Let $D_{p}$ be the
connected region in $\Sigma_{t}\cap B_{k}$ that passes through $p$.

On the one hand, $\partial{D_{p}}$ can be filled in by a region in $\partial
B_{k}$ with area at most roughly $4\pi\lambda^{2}$. On the other hand, $D_{p}$
must intersect every cross-section $\Phi(S^{2}\times\\{y\\})$ for
$-1/{\varepsilon_{0}}<y<0$. By the monotonicity formula, there is a universal
constant $c$ such that for $y\in(-1/{\varepsilon_{0}}+1/2,-1/2)$,

$\text{area}_{\bar{g}_{t}}(D_{p}\cap\Phi(S^{2}\times(y-1/2,y+1/2))>c\lambda^{2}.$

It follows by choosing disjoint unit intervals in
$(-{1}/{\varepsilon_{0}}+1/2,-1/2)$ that if we chose ${\varepsilon_{0}}$ such
that $1/{\varepsilon_{0}}$ is greater than $5\pi/c$, then the area of $D_{p}$
will be larger than $(4\pi+1)\lambda^{2}$. By cutting out $D_{p}$ and gluing
in a region of $\partial B_{k}$ we could then produce a surface homotopic to
$\Sigma_{t}$ but with smaller area, which is a contradiction. ∎

This lemma implies that we can use [8, Lemma 9] and conclude that
$\bar{A}(t):={\rm area}_{\bar{g}_{t}}(\Pi)$ is a Lipschitz function and thus
differentiable almost everywhere. Let $\bar{t}$ be a point of
differentiability of $\bar{A}$ and consider the function $a(t):={\rm
area}_{\bar{g}_{t}}(\Sigma_{\bar{t}})$. We have $\bar{A}(t)\leq a(t)$ for all
$t$ and $\bar{A}(\bar{t})=a(\bar{t})$. Hence, arguing like in (14) we deduce

$\bar{A}^{\prime}(t)\leq
a^{\prime}(\bar{t})\leq-2\pi\chi(\Sigma_{\bar{t}})-\bar{A}(t).$

From this ODE we can argue like in the proof of Theorem 4.1 and conclude that
if for some sequence $\Pi_{m}\in S_{1/m}(M)$ we have

$\frac{{\rm area}_{g}(\Pi_{m})}{4\pi(l_{m}-1)}\leq 1-\delta,$

where $l_{m}$ is the genus of an essential surface in $\Pi_{m}$, then for all
$t\geq 0$

$\frac{{\rm area}_{\bar{g}_{t}}(\Pi_{m})}{4\pi(l_{m}-1)}\leq 1-\delta e^{-t}.$

We have that $\bar{g}_{t}$ converges smoothly to $g_{0}$ and so this
contradicts Theorem 4.1. Thus the inequality in Theorem 5.1 must hold.

If equality holds in Theorem 5.1, the very same argument used as in Theorem
4.1 shows that $g_{0}$ must be Einstein. ∎

We can now derive Theorem 1.1 from Theorem 5.1.

######  Theorem 5.3.

Suppose $(M,g)$ has $R(g)\geq-6$. Then $E(g)\leq 2$ and equality occurs if and
only if $g$ is hyperbolic.

###### Proof.

In what follows we will use the fact (from [32]) that for any $\Pi_{m}\in
S_{1/m}(M)$, if $l_{m}$ is the genus of an essential surface in $\Pi_{m}$,
then we have ${\rm area}_{g_{0}}(\Pi_{m})/{4\pi(l_{m}-1)}\to 1$ as
$m\to\infty$.

Choose $\delta>0$. Theorem 5.1 implies the the existence of $\varepsilon_{0}$
so that for all $\Pi\in S_{\varepsilon_{0}}(M)$ we have

${\rm area}_{g_{0}}(\Pi)\leq(1+\delta){\rm area}_{g}(\Pi).$

Hence for all $\varepsilon\leq\varepsilon_{0}$ we have

$\\#\\{\text{area}_{g}(\Pi)\leq 4\pi(L-1):\Pi\in S_{\varepsilon}(M)\\}\\\
\leq\\#\\{\text{area}_{g_{0}}(\Pi)\leq(1+\delta)4\pi(L-1):\Pi\in
S_{\varepsilon}(M)\\}.$

This inequality and the expression (1) for the minimal surface entropy imply

$E(g)\leq E(g_{0})(1+\delta)=2(1+\delta).$

The inequality follows from making $\delta\to 0$.

Suppose now that $E(g)=2$ for some metric with $R(g)\geq-6$. Reasoning as
above we see that if we could find $\delta>0$ and $\varepsilon_{0}$ so that
for all $\Pi\in S_{\varepsilon_{0}}(M)$

${\rm area}_{g_{0}}(\Pi)\leq(1-\delta){\rm area}_{g}(\Pi),$

then $E(g)\leq 2(1-\delta)$. Hence there is $\Pi_{m}\in S_{1/m}(M)$ so that

$\liminf_{m\to\infty}\frac{{\rm area}_{g}(\Pi_{m})}{{\rm
area}_{g_{0}}(\Pi_{m})}\leq 1$

and Theorem 5.1 implies that $g$ is hyperbolic. ∎

##  6\. Proof of Theorem 1.2

For each $m\in\mathbb{N}$, Labourie [23, Theorem 5.7] found
$N_{m}\in\mathbb{N}$, $\\{\phi^{i}_{m}\\}_{i=1}^{N_{m}}$ in
$\mathcal{F}(M,1/m)$, and $0<a_{m}^{1},\ldots,a_{m}^{N_{m}}<1$ with
$a_{m}^{1}+\ldots+a_{m}^{N_{m}}=1$, so that: $\phi^{i}_{m}$ is equivariant
with respect to a representation of a Fuchsian subgroup of ${\rm
PSL}(2,\mathbb{R})$ in $\pi_{1}(M)<\text{PSL}(2,\mathbb{C})$ and the laminar
measure

(24) $\delta_{m}:=\sum_{i=1}^{N_{m}}a^{i}_{m}\delta_{\phi^{i}_{m}}$

is such that $\Omega_{*}\delta_{m}$ converges to $\mu_{Leb}$, the unit
Lebesgue homogeneous measure on $F(M)$.

######  Proposition 6.1.

There is $\phi_{m}$ in $\mathcal{F}(M,1/m)$ equivariant with respect to a
representation of a Fuchsian group $\Gamma_{m}$ of ${\rm PSL}(2,\mathbb{R})$
in $\pi_{1}(M)<\text{PSL}(2,\mathbb{C})$ such that
$\Omega_{*}\delta_{\phi_{m}}$ converges to $\mu_{Leb}$ as $m\to\infty$.

###### Proof.

The space of all closed totally geodesic immersions in $(M,g_{0})$ is
countable and so

$\mathcal{T}:=\\{F(S)\subset F(M):\mbox{S is a closed totally geodesic
immersion in }(M,g_{0})\\}$

is also countable, where $F(S)$ denotes the frame bundle of $S$ which injects
naturally in $F(M)$.

Considering tubular neighborhoods, we can find a decreasing sequence of open
sets $\\{B_{k}\\}_{k\in\mathbb{N}}\subset F(M)$ so that for all
$k\in\mathbb{N}$

$\cup_{T\in\mathcal{T}}T\subset B_{k},\quad\mu_{Leb}(\partial
B_{k})=0,\quad\mbox{and}\quad\mu_{Leb}(B_{k})\leq 2^{-2k-3}.$

######  Lemma 6.2.

For each $j\in\mathbb{N}$, there is $j\leq m_{j}\in\mathbb{N}$ and
$\phi_{j}\in\\{\phi^{1}_{m_{j}},\ldots,\phi^{N_{m_{j}}}_{m_{j}}\\}$ so that

$\Omega_{*}\delta_{\phi_{j}}(B_{k})\leq 2^{-(k+1)}\quad\mbox{for all }k\leq
j.$

###### Proof.

For all $k\in\mathbb{N}$, $\Omega_{*}\delta_{m}(B_{k})\to\mu_{Leb}(B_{k})$ as
$m\to\infty$. Thus we can find a strictly increasing sequence of integers
$\\{m_{j}\\}_{j\in\mathbb{N}}$ so that

(25) $\Omega_{*}\delta_{m_{j}}(B_{k})\leq 2\mu_{Leb}(B_{k})\leq
2^{-2(k+1)}\quad\mbox{for all }k\leq j.$

Relabel $m_{j}$ to be $j$ so that $\delta_{m_{j}},\phi^{i}_{m_{j}},N_{m_{j}},$
and $a^{i}_{m_{j}}$ become $\delta_{j},\phi^{i}_{j}$, $N_{j}$, and
$a^{i}_{j}$, respectively.

Consider $\mu_{j}$ a unit measure on $\\{1,\ldots,N_{j}\\}$ so that
$\mu_{j}(i)=a^{i}_{j}$ and set

$J_{j,k}:=\\{i\in\\{1,\ldots,N_{j}\\}:\Omega_{*}\delta_{\phi^{i}_{j}}(B_{k})\geq
2^{-(k+1)}\\}.$

From (25) and the definition of $J_{j,k}$ we have that for all $j\geq k$

$\mu_{j}(J_{j,k})=\sum_{i\in J_{j,k}}a^{i}_{j}\leq 2^{k+1}\sum_{i\in
J_{j,k}}a^{i}_{j}\Omega_{*}\delta_{\phi^{i}_{j}}(B_{k})\leq
2^{k+1}\Omega_{*}\delta_{j}(B_{k})\leq 2^{-(k+1)}.$

Thus, for all $j\in\mathbb{N}$ and $k\leq j$,
$\mu_{j}(\cup_{k=1}^{j}J_{j,k})\leq\sum_{k=1}^{j}2^{-(k+1)}\leq 1/2.$ Hence

$A_{j}:=\\{1,\ldots,N_{j}\\}\setminus\cup_{k=1}^{j}J_{j,k}\neq\emptyset$

and we pick $l_{j}\in A_{j}$. The maps $\phi_{j}:=\phi^{l_{j}}_{m_{j}}$
satisfy the desired conditions. ∎

After passing to a subsequence, $\Omega_{*}\delta_{\phi_{j}}$ converges weakly
to a unit measure $\nu$ on $F(M)$. From Lemma 3.2 we have that $\nu$ is
invariant under the homogeneous action of $\text{PSL}(2,\mathbb{R})$. The
proof will be completed if we show that $\nu=\mu_{Leb}$.

Ratner’s classification theorem [30] implies that every ergodic probability
measures on $F(M)$ invariant under the homogeneous action of
$\text{PSL}(2,\mathbb{R})$ is either $\mu_{Leb}$ or is supported in some
$T\in\mathcal{T}$. Thus, from the ergodic decomposition theorem for
$\text{PSL}(2,\mathbb{R})$-actions (Theorem 4.2.6 in [19]), there is
$0\leq\theta\leq 1$ so that the measure $\nu$ decomposes as
$\nu=\theta\mu_{Leb}+(1-\theta)\mu_{\mathcal{T}}$, where $\mu_{\mathcal{T}}$
is some probability measure on $F(M)$ with support in
$\cup_{T\in\mathcal{T}}T$. Moreover we have from Lemma 6.2 that
$\nu(B_{k})\leq 2^{-(k+1)}$ for all $k\in\mathbb{N}$.

Recall that $\cup_{T\in\mathcal{T}}T\subset B_{k}$ for all $k\in\mathbb{N}$
and so

$1-\theta=(1-\theta)\mu_{\mathcal{T}}(B_{k})\leq\nu(B_{k})\leq 2^{-(k+1)}.$

Hence $\theta=1$, which means that $\nu=\mu_{Leb}$. ∎

######  Proposition 6.3.

${\rm Area}_{\rm Id}(g/g_{0})=\lim_{m\to\infty}\frac{{\rm area}_{g}({\bf
D}(\partial\phi_{m}))}{2\pi|\chi({\bf D}(\partial\phi_{m}))|}.$

###### Proof.

We have for all $(x,P)\in Gr_{2}(M)$

$|\wedge^{2}{\rm Id}^{-1}|_{g}(x,P)=|\wedge^{2}{\rm Id}|^{-1}_{g}(x,P).$

We abuse notation and also denote by $|\wedge^{2}{\rm Id}|_{g}$ the following
smooth positive function on the frame bundle $F(M)$

$|\wedge^{2}{\rm
Id}|_{g}:F(M)\rightarrow\mathbb{R},\quad(y,\\{e_{1},e_{2},n\\})\mapsto|\wedge^{2}{\rm
Id}|_{g}(y,{\rm span}\\{e_{1},e_{2}\\}).$

We have

${\rm Area}_{\rm Id}(g/g_{0})=\mu_{Leb}(|\wedge^{2}{\rm Id}|^{-1}_{g})$

and thus the fact that $\Omega_{*}\delta_{\phi_{m}}\to\mu_{Leb}$ implies

(26) ${\rm Area}_{\rm Id}(g/g_{0})=\mu_{Leb}(|\wedge^{2}{\rm
Id}|^{-1}_{g})=\lim_{m\to\infty}\Omega_{*}\delta_{\phi_{m}}(|\wedge^{2}{\rm
Id}|^{-1}_{g}).$

Consider $U_{m}\subset\text{PSL}(2,\mathbb{R})$ a fundamental domain of
$\text{PSL}(2,\mathbb{R})/\Gamma_{m}$. Then

$\Omega_{*}\delta_{\phi_{m}}(|\wedge^{2}{\rm
Id}|^{-1}_{g})=\frac{1}{vol(U_{m})}\int_{U_{m}}|\wedge^{2}{\rm
Id}|^{-1}_{g}\circ\Omega(\phi_{m}\circ\tau)d\bar{\nu}(\tau).$

Note that for all $\tau\in\text{PSL}(2,\mathbb{R})$ we have, with $z=\tau(i)$,

$|\wedge^{2}{\rm Id}|_{g}\circ\Omega(\phi_{m}\circ\tau)=|\wedge^{2}{\rm
Id}|_{g}(\phi_{m}(z),(d\phi_{m})_{z}(T_{z}\mathbb{H}^{2})).$

Thus, denoting the hyperbolic volume form on $\mathbb{H}^{2}$ by $dA_{hyp}$,

(27) $\Omega_{*}\delta_{\phi_{m}}(|\wedge^{2}{\rm Id}|^{-1}_{g})\\\
=\frac{1}{2\pi|\chi({\bf
D}(\partial\phi_{m}))|}\int_{\mathbb{H}^{2}/\Gamma_{m}}|\wedge^{2}{\rm
Id}|^{-1}_{g}(\phi_{m}(z),(d\phi_{m})_{z}(T_{z}\mathbb{H}^{2}))dA_{hyp}(z)\\\
=\frac{1}{2\pi|\chi({\bf D}(\partial\phi_{m}))|}\int_{{\bf
D}(\partial\phi_{m})}\frac{|\wedge^{2}{\rm Id}|^{-1}_{g}(y,T_{y}{\bf
D}(\partial\phi_{m}))}{C(\phi_{m})\circ\phi_{m}^{-1}(y)}dA_{g_{0}}(y).$

Set $A_{m}:=||A||_{L^{\infty}({\bf D}(\partial\phi_{m}))}$. From Gauss
equation and (5) we have

(28) $1\leq\frac{1}{C(\phi_{m})}\leq 1+A_{m}^{2}.$

The co-area formula says that

$\int_{{\bf D}(\partial\phi_{m})}{|\wedge^{2}{\rm Id}|^{-1}_{g}(y,T_{y}{\bf
D}(\partial\psi_{m}))}{dA_{g_{0}}}={\rm area}_{g}({\bf D}(\partial\phi_{m}))$

and thus, combining with (27) and (28), we obtain

$\frac{{\rm area}_{g}({\bf D}(\partial\phi_{m}))}{2\pi|\chi({\bf
D}(\partial\phi_{m}))|}\leq\Omega_{*}\delta_{\phi_{m}}(|\wedge^{2}{\rm
Id}|^{-1}_{g})\leq(1+A_{m}^{2})\frac{{\rm area}_{g}({\bf
D}(\partial\phi_{m}))}{2\pi|\chi({\bf D}(\partial\phi_{m}))|}.$

We have that $A_{m}\to 0$ as $m\to\infty$ and hence we deduce from this
inequality and (26) the desired result. ∎

###  6.1. Proof of inequality

The closed surfaces ${\bf D}(\partial\phi_{m})$ define a homotopy class
$\Pi_{m}\in S_{1/m}(M)$ for all $m\in\mathbb{N}.$ We denote the genus of ${\bf
D}(\partial\phi_{m})$ by $l_{m}$.

######  Proposition 6.4.

${\rm Area}_{{\rm Id}}(g/g_{0})E(g)\geq 2$.

###### Proof.

After passing to a subsequence, set

$\alpha:=\lim_{m\to\infty}\frac{{\rm area}_{g}(\Pi_{m})}{4\pi(l_{m}-1)}.$

Given $\delta$ we have that for all $m$ sufficiently large

${\rm area}_{g}(\Pi_{m})\leq(\alpha+\delta)4\pi(l_{m}-1).$

Let ${\bf D}_{m}^{k}$, $l_{m}^{k}$, and $\Pi_{m}^{k}\in S_{1/m}(M)$ denote,
respectively, a $k$-cover of ${\bf D}(\partial\phi_{m})$, its genus, and its
homotopy class. From the inequality above we have for all $k\in\mathbb{N}$ and
all $m\in\mathbb{N}$ sufficiently large

(29) $\frac{{\rm area}_{g}(\Pi_{m}^{k})}{4\pi(l_{m}^{k}-1)}\leq\frac{k\,{\rm
area}_{g}(\Pi_{m})}{4\pi(l_{m}^{k}-1)}=\frac{{\rm
area}_{g}(\Pi_{m})}{4\pi(l_{m}-1)}\leq\alpha+\delta.$

From the Müller-Puchta’s formula (see [9, Section 4]) there is $c(m)>0$ so
that ${\bf D}(\partial\phi_{m})$ has at least $(c(m)l_{m}^{k})^{2l_{m}^{k}}$
distinct covers of degree less than or equal to $k$. Thus, if we choose
$L_{m}^{k}$ so that $4\pi(L_{m}^{k}-1)=(\alpha+\delta)4\pi(l^{k}_{m}-1)$, we
have from (29) that for all $k\in\mathbb{N}$ and all $m\in\mathbb{N}$
sufficiently large

$\\#\\{\text{area}_{g}(\Pi)\leq 4\pi(L_{m}^{k}-1):\Pi\in
S_{1/m}(M)\\}\geq(c(m)l_{m}^{k})^{2l_{m}^{k}}.$

Hence for all $m$ sufficiently large

$\lim_{k\to\infty}\frac{\ln\\#\\{\text{area}_{g}(\Pi)\leq
4\pi(L_{m}^{k}-1):\Pi\in S_{1/m}(M)\\}}{L_{m}^{k}\ln L_{m}^{k}}\\\
\geq\lim_{k\to\infty}\frac{2l_{m}^{k}\ln(c(m)l_{m}^{k})}{L_{m}^{k}\ln
L_{m}^{k}}=\frac{2}{\alpha+\delta}.$

From the expression for $E(g)$ in (1) we obtain that $(\alpha+\delta)E(g)\geq
2$. Making $\delta\to 0$ we deduce $\alpha E(g)\geq 2$. From Proposition 6.3
we have

${\rm Area}_{{\rm Id}}(g/g_{0})=\lim_{m\to\infty}\frac{{\rm area}_{g}({\bf
D}(\partial\phi_{m}))}{2\pi|\chi({\bf
D}(\partial\phi_{m}))|}\geq\lim_{m\to\infty}\frac{{\rm
area}_{g}(\Pi_{m})}{4\pi(l_{m}-1)}=\alpha$

and thus ${\rm Area}_{{\rm Id}}(g/g_{0})E(g)\geq 2$. ∎

###  6.2. Proof of rigidity

Suppose that ${\rm Area}_{{\rm Id}}(g/g_{0})E(g)=2$. In this case, the proof
of the preceding proposition implies that

(30) $\lim_{m\to\infty}\frac{{\rm area}_{g}({\bf D}(\partial\phi_{m}))}{{\rm
area}_{g}(\Pi_{m})}=1.$

The mean curvature of a surface in $(M,g)$ is denoted by $H_{g}$.

######  Lemma 6.5.

$\lim_{m\to\infty}\frac{1}{{\rm area}_{g}({\bf
D}(\partial\phi_{m}))}\int_{{\bf D}(\partial\phi_{m})}|H_{g}|^{2}dA_{g}=0.$

###### Proof.

Suppose that, after passing to a subsequence, there is $\delta>0$ so that

(31) $\int_{{\bf D}(\partial\phi_{m})}|H_{g}|^{2}dA_{g}\geq 2\delta\,{\rm
area}_{g}({\bf D}(\partial\phi_{m}))$

for all $m\in\mathbb{N}$.

The second fundamental form of ${\bf D}(\partial\phi_{m})$ (with respect to
$g$) and any of its derivatives are uniformly bounded independently of $m$.
Arguing like in [12, Corollary 4.4] we find $t_{0}>0$, $C_{0}>0$, and $\\{{\bf
D}_{m}(t)\\}_{0\leq t\leq t_{0}}$ a solution to mean curvature flow with
initial condition ${\bf D}_{m}(0):={\bf D}(\partial\phi_{m})$ so that both the
mean curvature of ${\bf D}_{m}(t)$ and its derivative is bounded uniformly by
$C_{0}$.

With $H_{g}(t)$, $A_{g}(t)$, and $\nu_{t}$ denoting, respectively, the mean
curvature, second fundamental form, and normal vector of ${\bf D}_{m}(t)$, we
have the evolution equation

$\frac{d}{dt}H_{g}(t)=\Delta
H_{g}(t)+(|A_{g}(t)|^{2}+Rc(g)(\nu_{t},\nu_{t}))H_{g}(t).$

Thus there is a constant $C_{1}>0$ (depending only on $C_{0}$ and $g$) so that
for all $m\in\mathbb{N}$ and $0\leq t\leq t_{0}$

$\frac{d}{dt}\int_{{\bf D}_{m}(t)}|H_{g}(t)|^{2}dA_{g}\geq-C_{1}{\rm
area}_{g}({\bf D}_{m}(t))\geq-C_{1}{\rm area}_{g}({\bf D}(\partial\phi_{m})).$

Choose $t_{1}<\min\\{{\delta/C_{1}},t_{0}\\}$. From (31) and the inequality
above we have that for all $m\in\mathbb{N}$ and $0\leq t\leq t_{1}$

$\int_{{\bf D}_{m}(t)}|H_{g}|^{2}dA_{g}\geq\delta\,{\rm area}_{g}({\bf
D}(\partial\phi_{m}))\geq\delta\,{\rm area}_{g}({\bf D}_{m}(t)).$

Therefore

$\frac{d}{dt}{\rm area}_{g}({\bf D}_{m}(t))=-\int_{{\bf
D}_{m}(t)}|H_{g}|^{2}dA_{g}\leq-\delta\,{\rm area}_{g}({\bf D}_{m}(t))$

and thus for all $m\in\mathbb{N}$ we have

${\rm area}_{g}(\Pi_{m})\leq{\rm area}_{g}({\bf D}_{m}(t_{1}))\leq e^{-\delta
t_{1}}{\rm area}_{g}({\bf D}(\partial\phi_{m})).$

This contradicts (30). ∎

Recall that $\Omega^{*}\delta_{\phi_{m}}$ converges to $\mu_{Leb}$ as
$m\to\infty$. The previous lemma allows us to apply Theorem A.3 to conclude
the following:

For every $\gamma\in\mathcal{C}_{0}$, there is a lift
$D_{m}\subset\mathbb{H}^{3}$ of ${\bf D}(\partial\phi_{m})\subset M$ so that,
after passing to a subsequence, $D_{m}$ converges on compact sets to the
totally geodesic disc $D(\gamma)\subset\mathbb{H}^{3}$ and

$\lim_{m\to\infty}\int_{D_{m}\cap
B_{R}(0)}|H_{g}|^{2}dA_{g_{0}}=0\quad\mbox{for all }R>0.$

Thus we obtain that $D(\gamma)$ is a minimal disc for the metric $g$. The
arbitrariness of $\gamma$ implies that every totally geodesic disc of
$\mathbb{H}^{3}$ is minimal with respect to $g$.

The next theorem follows from adapting some of the arguments in [1].

######  Theorem 6.6.

For some $c>0$, $g=cg_{0}$.

###### Proof.

We denote the space of Killing symmetric $2$-tensors on a Riemannian manifold
$(X,h)$ by $K_{2}(X,h)$. They are characterized by the property (see [34,
Section 1]) that a symmetric $2$-tensor $k$ lies in $K_{2}(X,h)$ if and only
if for every geodesic $\gamma\subset X$ the function below is constant

$t\mapsto k(\gamma^{\prime}(t),\gamma^{\prime}(t)).$

Given a symmetric $2$-tensor $k$ on $X$, we denote the absolute value of its
determinant with respect to $h$ by $|k|_{h}$.

Let $\delta$ denote the Euclidean metric in $\mathbb{R}^{3}$ and
$B\subset\mathbb{H}^{3}$ denote the open unit ball. Consider the Beltrami-
Klein model $(B,g_{\rm hyp})$ for $(\mathbb{H}^{3},g_{0})$. The important
property of this model is that the image of geodesics and totally geodesic
discs in $(B,g_{\rm hyp})$ is the same as affine lines and affine planes in
$(B,\delta).$

In [1], among other results, the authors classify all metric $h$ on $S^{n}$
for which every totally geodesic hypersphere is minimal with respect to $h$.
They are those for which

$|h|_{g_{\rm round}}^{-2/(n+1)}h\in K_{2}(S^{n},g_{\rm round}).$

The same computation as the one performed in [1], which we repeat for the sake
of completeness, gives

######  Proposition 6.7.

A positive definite symmetric $2$-tensor $h$ on $B$ has the property that
every affine plane is minimal with respect to $h$ if and only if

$|h|_{g_{\rm hyp}}^{-1/2}h\in K_{2}(B,g_{\rm hyp}).$

###### Proof.

We will use the following theorem of Hangan [17] : A metric $h$ on $B$ is such
that all affine planes have zero mean curvature with respect to $h$ if and
only if

$|h|_{\delta}^{-1/2}h\in K_{2}(B,\delta).$

######  Lemma 6.8.

With $k$ a $2$-tensor on $B$ then

$k\in K_{2}(B,g_{\rm hyp})\iff|g_{\rm hyp}|_{\delta}^{-1/2}k\in
K_{2}(B,\delta)$

###### Proof.

From Hangan’s theorem we have that $|g_{\rm hyp}|_{\delta}^{-1/2}g_{\rm
hyp}\in K_{2}(B,\delta)$ and hence, for every Euclidean geodesic
$\gamma\subset B$, we have some constant $c_{0}=c_{0}(\gamma)$ such that for
all $t$

$g_{\rm hyp}(\gamma^{\prime}(t),\gamma^{\prime}(t))=|g_{\rm
hyp}|_{\delta}^{1/2}(\gamma(t))c_{0}.$

Note that geodesics with respect to $g_{\rm hyp}$ are also straight lines.
Hence if we perform a change of variable according to

$\frac{dt}{ds}(s)=|g_{\rm hyp}|_{\delta}^{-1/4}(\gamma(t(s)))$

we have that $s\mapsto\sigma(s):=\gamma(t(s))$ is a geodesic for $g_{\rm
hyp}$. Therefore

$\frac{d}{ds}k(\sigma^{\prime}(s),\sigma^{\prime}(s))=\frac{dt}{ds}(s)\frac{d}{dt}_{t=t(s)}\left(\frac{k(\gamma^{\prime}(t),\gamma^{\prime}(t))}{|g_{\rm
hyp}|_{\delta}^{1/2}(\gamma(t))}\right)$

and so

$\frac{d}{ds}k(\sigma^{\prime}(s),\sigma^{\prime}(s))=0\iff\frac{d}{dt}_{t=t(s)}\left(\frac{k(\gamma^{\prime}(t),\gamma^{\prime}(t))}{|g_{\rm
hyp}|_{\delta}^{1/2}(\gamma(t))}\right)=0.$

Thus $k$ is constant along hyperbolic geodesics if and only if $|g_{\rm
hyp}|_{\delta}^{-1/2}k$ is constant along Euclidean geodesics, which implies
the result. ∎

Using this lemma and the identity $|h|_{\delta}=|h|_{g_{\rm hyp}}|g_{\rm
hyp}|_{\delta}$ we have

$|h|_{\delta}^{-1/2}h\in K_{2}(B,\delta)\iff|h|_{g_{\rm hyp}}^{-1/2}h\in
K_{2}(B,g_{\rm hyp}).$

Hangan’s theorem [17] implies the desired result. ∎

Denote the lift of the metric $g$ to $\mathbb{H}^{3}$ by $g$ as well. We are
assuming that every totally geodesic disc in $\mathbb{H}^{3}$ is minimal with
respect to $g$ and so the previous proposition implies that
$G:=|g|_{g_{0}}^{-1/2}g\in K_{2}(M,g_{0})$.

The geodesic flow in $(M,g_{0})$ is ergodic and so we can choose a geodesic
$\sigma\subset M$ which is dense in the unit tangent bundle. From the fact
that $t\mapsto G(\sigma^{\prime}(t),\sigma^{\prime}(t))$ is constant we deduce
the existence of a constant $\alpha$ so that $G(Y,Y)=\alpha g_{0}(Y,Y)$ for
every vector field $Y$. This implies that $G=\alpha g_{0}$. Using the fact
that $G=|g|_{g_{0}}^{-1/2}g$ we deduce that $g$ is a positive multiple of
$g_{0}$.

∎

##  Appendix A Auxiliary Results

###  A.1. Asymptotic Expansion

With $g\in\mathcal{U}$, set $h:=g-g_{0}$. Without loss of generality we assume
that $|h|_{C^{4}}\leq 1$.

Given $\gamma\in\mathcal{C}_{\bar{\varepsilon}}$, Recall that
$\nu_{g}(\gamma)$ denotes a continuous unit normal vector field along
$\Sigma_{g}(\gamma)$ with respect to $g$ and $n(\gamma)$ denotes a continuous
unit normal vector field along $D(\gamma)$ with respect to $g_{0}$.

Recall the diffeomorphism (using the hyperboloid model)

$F_{\gamma}:D(\gamma)\rightarrow\Sigma_{g}(\gamma),\quad
x\mapsto\cosh(f_{\gamma}(x))x+\sinh(f_{\gamma}(x))n(\gamma)(x).$

Given $p\in M$ and $k\in\mathbb{N}_{0}$, $l\in\mathbb{N}$, we denote by
$O_{p}^{k}(|h|^{l})$ any quantity for which there is a constant $\alpha_{k,l}$
(independent of $p$ and $g\in\mathcal{U}$) so that its absolute value at $p$
is bounded by $\alpha_{k,l}\sum_{j=0}^{k}|\nabla^{k}h|^{l}(p)$. Likewise,
given $\gamma\in\mathcal{C}_{\bar{\varepsilon}}$ and $x\in D(\gamma)$, we
denote by $O_{x}^{k}(|f_{\gamma}|^{l})$ any quantify for which there is a
constant $\beta_{k,l}$ (independent of $x$, $\gamma$, and $g\in\mathcal{U}$)
so that its absolute value at $x$ is bounded by
$\alpha_{k,l}\sum_{j=0}^{k}|\nabla^{k}f_{\gamma}|^{l}(x)$.

Let $L$ denote the linearization at $g_{0}$ of

$h\mapsto\mathring{Ric}(g_{0}+h):=Ric(g_{0}+h)-\frac{R(g_{0}+h)}{3}(g_{0}+h).$

######  Proposition A.1.

With $\gamma\in\mathcal{C}_{\varepsilon}$, $x\in D(\gamma)$ and
$y=F_{\gamma}(x)$ we have

$Ric(g)_{y}(\nu_{g}(\gamma),\nu_{g}(\gamma))-\frac{1}{3}R(g)(y)=L(h)_{x}(n(\gamma),n(\gamma))\\\
+O^{3}_{x}(|h|^{2})+O^{1}_{x}(|f_{\gamma}|^{2}).$

With $|Jac_{g}F_{\gamma}|$ the Jacobian of
$F_{\gamma}:(D(\gamma),g_{0})\rightarrow(\Sigma_{g}(\gamma),g)$ we have

$|Jac_{g}F_{\gamma}|(x)=1+O_{x}^{1}(|h|)+O_{x}^{1}(|f_{\gamma}|).$

###### Proof.

Set

$X:=\\{(\gamma,f):\gamma\in\mathcal{C}_{\bar{\varepsilon}},f\in
C^{2}(D(\gamma)),|f|_{C^{2}}\leq 1\\}.$

Given $(\gamma,f)\in X$ set

$F(\gamma,f):D(\gamma)\rightarrow\Sigma_{g}(\gamma),\quad
x\mapsto\cosh(f(x))x+\sinh(f(x))n(\gamma)(x)$

and $\Sigma(\gamma,f):=F(\gamma,f)(D(\gamma))$. Let $n(\gamma,f)$ be the unit
normal vector field along $\Sigma(\gamma,f)$, defined so that $n(\gamma,f)$
depends smoothly on $(\gamma,f)$ and $n(\gamma,0)=n(\gamma)$. We have for all
$0\leq t\leq 1$

$\left|\frac{d^{k}}{(dt)^{k}}(n(\gamma,tf)(F(\gamma,tf)(x))\right|=O_{x}^{1}(|f|^{k}),\quad
k=1,2.$

Hence, if $G$ is a $2$-tensor on $M$ with $|G|_{C^{2}}\leq 1$, then

$\alpha(t):=G(n(\gamma,tf)(F(\gamma,tf)(x)),n(\gamma,tf)(F(\gamma,tf)(x)))$

satisfies

$|\alpha^{\prime}(0)|\leq(|\nabla
G|(x)+|G|(x))O_{x}^{1}(|f|)\quad\mbox{and}\quad\sup_{0\leq t\leq
1}|\alpha^{\prime\prime}(t)|=O_{x}^{1}(|f|^{2}).$

Therefore, with $y=F(\gamma,f)(x)$, we obtain from Taylor’s expansion

(32) $|G(n(\gamma,f)(y),n(\gamma,f)(y))-G(n(\gamma)(x),n(\gamma)(x))|\\\
\leq|\nabla G|^{2}(x)+|G(x)|^{2}+O_{x}^{1}(|f|^{2}).$

Setting $G=|\nabla^{k}h|^{2}g_{0}$ in this inequality we deduce that for
$k=0,1,2.$

(33) $|\nabla^{k}h|(y)=O^{k+1}_{x}(|h|)+O_{x}^{1}(|f|).$

Setting $G=L(h)$ in (32) and using the fact that $L$ is a second order
differential operator we deduce

(34)
$|L(h)_{x}(n(\gamma)(x),n(\gamma)(x))-L(h)_{y}(n(\gamma,f)(y),n(\gamma,f)(y))|\\\
=O_{x}^{3}(|h|^{2})+O_{x}^{1}(|f|^{2}).$

Let $\nu_{g}(\gamma,f)$ denote the unit normal vector field along
$\Sigma(\gamma,f)$ with respect to $g=g_{0}+h$ so that
$\nu_{g_{0}}(\gamma,f)=n(\gamma,f)$ and $\nu_{g}$ depends smoothly on its
parameters. Then

$|\nu_{g}(\gamma,f)(y)-n(\gamma,f)(y)|=O_{y}^{0}(|h|)=O_{x}^{1}(|h|)+O_{x}^{1}(|f|),$

where in the last identity we used (33). Using this identity and (33) we have

$L(h)_{y}(\nu_{g}(\gamma,f)(y),\nu_{g}(\gamma,f)(y))\\\
=L(h)_{y}(n(\gamma,f)(y),n(\gamma,f)(y))+O_{x}^{3}(|h|^{2})+O_{x}^{1}(|f|^{2}).$

Combining with (34) we deduce

$L(h)_{y}(\nu_{g}(\gamma,f)(y),\nu_{g}(\gamma,f)(y))\\\
=L(h)_{y}(n(\gamma)(x),n(\gamma)(x))+O_{x}^{3}(|h|^{2})+O_{x}^{1}(|f|^{2}).$

Using Taylor’s expansion we have that for every $y\in M$, every unit vector
field $Y\in T_{y}M$ and $g_{0}+h\in\mathcal{U}$

$|\mathring{Ric}(g_{0}+h)_{y}(Y,Y)-L(h)_{y}(Y,Y)|=O_{y}^{2}(|h|^{2})=O_{x}^{3}(|h|^{2})+O_{x}^{1}(|f|^{2}).$

Therefore

$\mathring{Ric}(g_{0}+h)_{y}(\nu_{g}(\gamma,f)(y),\nu_{g}(\gamma,f)(y))\\\
=L(h)_{y}(n(\gamma)(x),n(\gamma)(x))+O_{x}^{3}(|h|^{2})+O_{x}^{1}(|f|^{2}).$

The first statement in the proposition follows from choosing $f=f_{\gamma}$ in
the identity above.

We now prove the statement regarding $|Jac_{g}F_{\gamma}|$. With
$(\gamma,f)\in X$, denote by $|Jac_{g}F(\gamma,f)|(g)$ the Jacobian of
$F(\gamma,f):(D(\gamma),g_{0})\rightarrow(\Sigma(\gamma,f),g)$. With
$g=g_{0}+h\in\mathcal{U}$ and $y=F(\gamma,f)(x)$ we have

$\displaystyle|Jac_{g}F(\gamma,f)|(x)$
$\displaystyle=|Jac_{g_{0}}F(\gamma,f)|(x)+O^{0}_{y}(|h|)$
$\displaystyle=|Jac_{g_{0}}F(\gamma,f)|(x)+O^{1}_{x}(|h|)+O^{1}_{x}(|f|)$
$\displaystyle=|Jac_{g_{0}}F(\gamma,0)|(x)+O^{1}_{x}(|h|)+O^{1}_{x}(|f|)$
$\displaystyle=1+O^{1}_{x}(|h|)+O^{1}_{x}(|f|).$

∎

###  A.2. Stability of Ricci flow

We assume we have a solution to normalized Ricci flow $(\bar{g}_{t})_{t\geq
0}$ (13) and a smooth family of diffeomorphisms $\\{\Phi_{t}\\}_{t\geq 0}$
converging strongly to some diffeomorphism so that
$g_{t}:=\Phi_{t}^{*}\bar{g}_{t}$ solves the DeTurck-modified Ricci flow (which
is strictly parabolic) and converges to $g_{0}$ as $t\to\infty$.

Recall the operator $\mathcal{A}$ defined in (21). It has discrete spectrum
$1\leq\lambda_{1}\leq\lambda_{2}\leq\ldots$

######  Proposition A.2.

Set $h_{t}:=\Phi^{*}_{t}\bar{g}_{t}-g_{0}$, $t\geq 0$. Then

  * •

$|h_{t}|_{C^{4}}\leq O(e^{-2t/3})$;

  * •

$e^{t}h_{t}$ converges in $W^{2,2}$ as $t\to\infty$ to $\bar{h}$, where
$\bar{h}$ is a smooth $2$-tensor with $\mathcal{A}(\bar{h})=-\bar{h}$.

###### Proof.

In [21] it is shown that $|h_{t}|_{C^{2}}\leq O(e^{-2t/3})$. Standard
estimates (similar to [18, Lemma 5.3]) show that for all $k\in\mathbb{N}$,
$|Ric(\bar{g}_{t})+2\bar{g}_{t}|_{C^{k}}\leq O(e^{-2t/3})$ and thus
$|\bar{g}_{t}-\bar{g}_{+\infty}|_{C^{k}}\leq O(e^{-2t/3})$ as well. The
diffeomorphisms $\\{\Phi_{t}\\}_{t\geq 0}$ depend only on $g_{0}$ and
$\\{\bar{g}_{t}\\}_{t\geq 0}$, and one can check that for all
$k\in\mathbb{N}$, $|\Phi_{t}-\Phi_{+\infty}|_{C^{k}}\leq O(e^{-2t/3})$. In
particular, $|h_{t}|_{C^{4}}\leq O(e^{-2t/3})$.

In [21] it is also show that the tensors $h_{t}$ satisfy an equation of the
form

$\frac{dh_{t}}{dt}=\mathcal{A}(h_{t})+Q_{t},$

where $Q_{t}$ is a non-linear term depending on $g_{0},h_{t},\nabla
h_{t},\nabla^{2}h_{t}$. The important property we need is that
$|Q_{t}|_{C^{k}}\leq O(|h_{t}|_{C^{k+2}}^{2})$. From here on we consider
$Q_{t}$ as being a fixed non-homogeneous term where $|Q_{t}|_{C^{2}}\leq
O(e^{-4t/3})$.

Consider an $L^{2}$-orthonormal basis $\\{u_{k}\\}_{k\in\mathbb{N}}$ for the
space of symmetric $2$-tensors on $M$ made of eigentensors for $\mathcal{A}$.
Necessarily

$h_{t}=\sum_{k=0}^{\infty}\left(e^{-\lambda_{k}t}\langle
h_{0},u_{k}\rangle+\int_{0}^{t}e^{\lambda_{k}(\tau-t)}\langle
Q_{\tau},u_{k}\rangle d\tau\right)u_{k}.$

Set $\eta_{t}:=h_{t}-f(t),$ where

$f(t):=\sum_{\lambda_{k}=1}\left(e^{-t}\langle
h_{0},u_{k}\rangle+\int_{0}^{t}e^{(\tau-t)}\langle Q_{\tau},u_{k}\rangle
d\tau\right)u_{k}.$

Because $\eta_{t}$ is orthogonal to the $1$-eigentensors there is some
$\delta$ so that

$\int_{M}\mathcal{A}(\eta_{t})\eta_{t}dV_{g_{0}}\leq-(1+2\delta)\int_{M}\eta_{t}^{2}dV_{g_{0}}.$

Moreover we have $\partial_{t}\eta_{t}=\mathcal{A}(\eta_{t})+O(e^{-4t/3}),$
which when combined with the inequality above implies that

$\frac{d}{dt}\int_{M}\eta_{t}^{2}dV_{g_{0}}\leq-2(1+2\delta)\int_{M}\eta_{t}^{2}dV_{g_{0}}+O(e^{-4t/3})\int_{M}\eta_{t}dV_{g_{0}}\\\
\leq-2(1+2\delta)\int_{M}\eta_{t}^{2}dV_{g_{0}}+O(e^{-4t/3})\left(\int_{M}\eta_{t}^{2}dV_{g_{0}}\right)^{1/2}\\\
\leq-2(1+\delta)\int_{M}\eta_{t}^{2}dV_{g_{0}}+O(e^{-8t/3}).$

Hence $\int_{M}(e^{t}\eta_{t})^{2}dV_{g_{0}}\to 0$ as $t\to\infty$. Likewise,
$\eta^{1}_{t}:=\mathcal{A}(\eta_{t})$ is also orthogonal to all
$1$-eigentensors,
$\partial_{t}\eta^{1}_{t}=\mathcal{A}(\eta^{1}_{t})+O(e^{-4t/3})$, and the
same argument gives $\int_{M}(e^{t}\eta^{1}_{t})^{2}dV_{g_{0}}\to 0$. This
implies that $e^{t}\eta_{t}$ tends to zero in $W^{2,2}$ as $t\to\infty$. The
result follows because the $1$-eigentensor $e^{t}f(t)$ converges in $W^{2,2}$
as $t\to\infty$ to some smooth tensor $\bar{h}$. ∎

###  A.3. A theorem of Calegari-Marques-Neves

Consider a sequence $\phi_{i}\in\mathcal{F}(M,1/i)$ equivariant with respect
to a representation of a Fuchsian subgroup of $\text{PSL}(2,\mathbb{R})$ in
$\pi_{1}(M)<\text{PSL}(2,\mathbb{C})$. Let $G_{i}<\pi_{1}(M)$ be the image of
that representation. The group $G_{i}$ preserves
$D(\partial\phi_{i})\subset\mathbb{H}^{3}$ and recall that ${\bf
D}(\partial\phi_{i})={D}(\partial\phi_{i})/G_{i}.$ We assume

  * •

$\Omega_{*}\phi_{i}$ converges weakly to a measure $\nu$ on $F(M)$ as
$i\to\infty$, where the measure $\nu$ is such that so that $\nu(O)>0$ for
every open set $O$;

  * •

there is a sequence of immersed surfaces $\Sigma_{i}\subset M$ homotopic to
${\bf D}(\partial\phi_{i})$ and $f_{i}\in C^{0}(\Sigma_{i})$ so that

$\lim_{i\to\infty}\frac{1}{{\rm
area}_{g_{0}}(\Sigma_{i})}\int_{\Sigma_{i}}|f_{i}|dA_{g_{0}}=0.$

The Hausdorff distance between two sets in $\mathbb{H}^{3}$ is denoted by
$d_{H}$.

The following theorem corresponds to Theorem 6.1 of [9], where it is assumed
that $\Sigma_{i}$ is area-minimizing with respect to some metric. An
inspection of the proof shows that one only needs the areas of $\Sigma_{i}$
and ${\bf D}(\partial\phi_{i})$ to be comparable and their universal covers to
be at a uniform Hausdorff distance from each other.

######  Theorem A.3.

Assume the existence of $C>0$ and, for all $i\in\mathbb{N}$, a covering
$\Omega_{i}\subset\mathbb{H}^{3}$ of $\Sigma_{i}$ so that

${\rm area}_{g_{0}}(\Sigma_{i})\leq C{\rm area}_{g_{0}}({\bf
D}(\partial\phi_{i}))\quad\mbox{and}\quad C^{-1}\leq
d_{H}(\Omega_{i},D(\partial\phi_{i}))\leq C.$

For every $\gamma\in\mathcal{C}_{0}$ there is
$\eta_{i}\in\pi_{1}(M)<\text{Isom}\,(\mathbb{H}^{3})$ such that, after passing
to a subsequence, $\eta_{i}(D(\partial\phi_{i}))$ converges on compact sets to
$D(\gamma)$ and

$\lim_{i\to\infty}\int_{\eta_{i}(\Omega_{i})\cap
B_{R}(0)}|f_{i}\circ\eta_{i}^{-1}|dA_{g_{0}}=0\quad\mbox{for all }R>0.$

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