nameyeardelim, 50mm 73e année, 2020–2021 1179

# Asymptotic counting of minimal surfaces in hyperbolic $3$-manifolds

according to Calegari, Marques and Neves

François Labourie  LJAD, Université Côte d’Azur  francois.labourie@univ-
cotedazur.fr

(Mai 2021)

To Srishti Dhar Chatterji,
who attended séminaire Bourbaki for most of half a century.

###### Contents

  1. 1 Counting geodesics and equidistribution
    1. 1.1 Entropy and asymptotic counting of geodesics
      1. 1.1.1 Rigidity of the entropy
    2. 1.2 Equidistribution
  2. 2 Totally geodesic submanifolds of higher dimension
    1. 2.1 Closed totally geodesic submanifolds are rare
    2. 2.2 The set of pointed totally geodesic spaces and Shah’s Theorem
      1. 2.2.1 Totally geodesic hyperbolic planes, the frame bundle and the $\mathsf{PSL}_{2}(\mathbb{R})$-action
      2. 2.2.2 Shah and Ratner’s Theorem
    3. 2.3 In variable curvature
      1. 2.3.1 The boundary at infinity and circles
      2. 2.3.2 A characterization of hyperbolic metrics
  3. 3 Surface subgroups in hyperbolic $3$-manifolds
    1. 3.1 Quasi-Fuchsian groups and quasicircles
    2. 3.2 Surface subgroups in fundamental groups of closed hyperbolic $3$-manifolds
    3. 3.3 Counting surface subgroups
  4. 4 Minimal surfaces in $3$-manifolds
    1. 4.1 Minimal immersions
      1. 4.1.1 The first variation formula
      2. 4.1.2 The second variation formula
    2. 4.2 Minimal surfaces in $3$-manifolds
      1. 4.2.1 Back to quasi-Fuchsian manifolds
    3. 4.3 Compactness results
    4. 4.4 Almost Fuchsian minimal surfaces, Uhlenbeck’s result and the asymptotic Plateau problem
      1. 4.4.1 Almost Fuchsian minimal surfaces
      2. 4.4.2 Asymptotic Plateau problem
    5. 4.5 In variable curvature
      1. 4.5.1 Minimal surfaces and quasi-isometries
    6. 4.6 The case of fibered manifolds
  5. 5 Equidistribution in the phase space of minimal surfaces
    1. 5.1 A phase space for stable minimal surfaces
    2. 5.2 Laminar measures and conformal currents
    3. 5.3 Equidistribution
  6. 6 Calegari–Marques–Neves asymptotic counting
    1. 6.1 Counting surfaces: from genus to area
    2. 6.2 The upper bound in variable curvature
    3. 6.3 The upper bound in constant curvature
    4. 6.4 The lower bound
    5. 6.5 The equality case
      1. 6.5.1 Finding surfaces which are more and more hyperbolic
      2. 6.5.2 From more and more hyperbolic to more and more totally geodesic
      3. 6.5.3 Conclusion

## Introduction

The study of the geodesic flow in closed negatively curved manifolds is a
beautiful mix of topology, Riemannian geometry, geometric group theory and
ergodic theory. We know in this situation that closed geodesics are in one-to-
one correspondence with conjugacy classes of elements of the fundamental
group, or equivalently, with the set of homotopy classes of maps of circles in
the manifold. Even though closed geodesics are infinite in number, we have a
good grasp —thanks to the notion of topological entropy— of how the number of
these geodesics grows with respect to the length. We also have a computation
of this topological entropy in hyperbolic spaces by [Bowen:1972] and
[Margulis:1969ve] and rigidity results for this entropy by [Besson:1995um] and
[Hamenstadt:1990tx].

While the statements of this first series of results seem to deal only with
closed geodesics, the foliation of the unit tangent bundle by orbits of the
geodesic flow plays a fundamental role. The study of invariant measures by the
geodesic flow is a crucial tool, and the equidistribution of closed geodesics
by [Bowen:1972] and [Margulis:1969ve] for hyperbolic manifolds a central
result. We refer to section 1 for more precise definitions, results and
references.

For many reasons —as we discuss in section 2— closed totally geodesic
submanifolds of dimension at least $2$ are quite rare. However, in constant
curvature, the foliation of the Grassmannian of $k$-planes coming from totally
geodesic planes is a natural generalization of the geodesic flow and several
crucial results of [Ratner:1991wl, Ratner:1991tu, Shah:1991wr] as well as
[McMullen:2017ub] describe closed invariant sets and invariant measures. This
foliation stops to make sense in variable curvature, at least far away from
the constant curvature situation, although for metrics close to hyperbolic
ones, a result by [Gromov:1991uv] —see also [Lowe:2020vu]— shows that the
foliation of the Grassmann bundle persists when one replaces totally geodesic
submanifolds by minimal ones.

If we move in the topological direction, going from circles to surfaces,
Kahn–Marković Surface Subgroup Theorem ([Kahn:2009wh]) provides the existence
of many surface subgroups in the fundamental group of a hyperbolizable
$3$-manifold $M$. A subsequent result of [Kahn:2010uo] gives an asymptotic of
the number of these surface groups with respect to the genus —see Theorem 3.3.

However this asymptotic counting does not involve the underlying Riemannian
geometry as opposed to the topological entropy that we discussed in the first
paragraph. The next step is to use fundamental results of [Schoen:1979] and
[Sacks:1982], which tells us that every such surface group can be realized by
a minimal surface —although non necessarily uniquely.

In [Calegari:2020uo], the authors propose a novel idea: count asymptotically
with respect to the area these minimal surfaces, but when the boundary at
infinity of those minimal surfaces becomes more and more circular, or more
precisely are $K$-quasicircles, with $K$ approaching $1$. The precise
definition of this counting requires the description of quasi-Fuchsian groups
and their boundary at infinity, done in section 3, and their main result
(Theorem 6) is presented in section 6. These results define an entropy-like
constant $E(M,h)$ for minimal surfaces in a Riemannian manifold $(M,h)$ of
curvature less than $-1$. The main result of [Calegari:2020uo] is to compute
it for hyperbolic manifolds, gives bounds in the general case and most notably
proves a rigidity result: $E(M,h)=2$ if and only if $h$ is hyperbolic.
Altogether, these results mirror those for closed geodesics.

When one moves to studying solution of elliptic partial differential
equations, for instance minimal surfaces or pseudo-holomophic curves, the
situation is different from the chaotic behavior of the geodesic flow. While
there is a huge literature about moduli spaces of solutions when one imposes
constraints such as homology classes, we do not have that many results
describing a moduli space of all solutions: possibly immersed with dense
images, in other words to continue the process for minimal surfaces described
in the introduction of [Gromov:1991uv] for geodesics: if one wishes to
understand closed geodesics not as individuals but as members of a community
one has to look at all (not only closed) geodesics in X which form an
$1$-dimensional foliation of the projectivized tangent bundle.

The presentation of these notes shifts around the ideas used in
[Calegari:2020uo] and follows more directly the philosophy introduced in
[Gromov:1991uv]. We focus on the construction of such a moduli space —that we
call the phase space of stable minimal surfaces— and its topological
properties —see section 5.1 and Theorem 5.1. These properties are a rephrasing
of Theorem 4.5.1 about quasi-isometric properties of stable minimal surfaces,
relying on results of [Seppi:2016ut] and a “Morse type Lemma” argument by
[Calegari:2020uo, Theorem 3.1]. This space is the analogue, in our situation,
of the geodesic flow and the $\mathbb{R}$-action is replaced by an
$\mathsf{SL}_{2}(\mathbb{R})$-action.

Then we move to studying $\mathsf{SL}_{2}(\mathbb{R})$-invariant measures on
this phase space and show they are related to what we call laminar currents
which are the analogues in our situation of geodesic currents —see
[Bonahon:1997tl]. The main result is now an equidistribution result in this
situation: Theorem 1.2. This theorem follows from the techniques of the proof
of Surface Subgroup Theorem using the presentation given in [Kahn:2018wx].

This Equidistribution Theorem and the construction of the phase space allows,
by comparing the counting with respect to area and the genus —as in
[Kahn:2010uo]— to proceed quickly to the proof of the results of
[Calegari:2020uo] when, for the rigidity result, we assume that $h$ is close
enough to a hyperbolic metric.

The whole article of [Calegari:2020uo] mixes beautiful ideas from many
subjects, adding to the mix of topology, Riemannian geometry, geometric group
theory and ergodic theory used in the study of the geodesic flow, a pinch of
geometric analysis. The approach given in these notes is not just to present
the proof but also to take the opportunity to tour some of the fundamental
results in these various mathematics111The introduction of [Calegari:2020uo]
also addresses minimal hypersurfaces in higher dimension that we do not
discuss here. We take some leisurely approach and explain some of the main
results and take the time to give a few simple proofs and elementary
discussions: the clever proof of Thurston showing that there are only finitely
many surface groups of a given genus in the fundamental group of a hyperbolic
manifold, the discussion of stable minimal surfaces, the geometric analysis
trick that derives from a rigidity result (here the characterization of the
plane as the unique stable minimal surface in $\mathbb{R}^{3}$) some
compactness results (proposition 9).

During the preparation of these notes, I benefited from the help of many
colleagues, as well as the insight of the authors. I want to thank them here
for their crucial input: Dick Canary, Thomas Delzant, Olivier Guichard, Fanny
Kassel, Shahar Mozes, Hee Oh, Pierre Pansu, Andrea Seppi, Jérémy Toulisse and
Mike Wolf.

##  1 Counting geodesics and equidistribution

When $(M,h)$ is a negatively curved manifold, there is a one-to-one
correspondence between conjugacy classes of elements of $\pi_{1}(M)$ and
closed geodesics. Even though there are infinitely many closed geodesics, we
can count them “asymptotically”. Equivalently, this will give an asymptotic
count of the conjugacy classes of elements of $\pi_{1}(M)$, or to start a
point of view that we shall pursue later, the set of free homotopy classes of
maps of $S^{1}$ in $M$.

We review here some important results that will be useful in our discussion
and serve as a motivation.

###  1.1 Entropy and asymptotic counting of geodesics

Let $(M,h)$ be a closed manifold of negative curvature. Fixing a positive
constant $T$, there are only finitely many closed geodesics of length less
than $T$. Let us define

$\Gamma_{h}(T)\mathrel{\mathop{:}}=\\{\hbox{geodesic
}\gamma\mid\operatorname{length}(\gamma)\leqslant T\\}\ .$

The following limit, when it is defined,

${\rm
h}_{top}(M,h)\mathrel{\mathop{:}}=\lim_{t\to\infty}\frac{1}{T}\log\left(\sharp\Gamma_{h}(T)\right)\
,$

is called the topological entropy of $M$. We will see it is always defined in
negative curvature. It measures the exponential growth of the number of
geodesics with respect to the length. The topological entropy is related to
the volume entropy of $M$ defined by

${\rm
h}_{vol}(M,h)=\liminf_{t\to\infty}\frac{1}{R}\log\bigl{(}\operatorname{Vol}(B(x,R))\bigr{)}\
,$

where $B(x,R)$ is the ball of radius $R$ in the universal cover $\tilde{M}$ of
$M$, $x$ any point in $\tilde{M}$. The volume entropy does not depend on the
choice of the point $x$ and we have Let $(M,h)$ be a closed negatively curved
manifold.

  1. (i)

The topological ${\rm h}_{top}(M,h)$ is well-defined. When $h_{0}$ is
hyperbolic222that is when the curvature is constant and equal to $-1$,

${\rm h}_{top}(M,h_{0})=\dim(M)-1\ .$

  2. (ii)

We have

${\rm h}_{top}(M,h)={\rm
h}_{vol}(M,h)=\lim_{R\to\infty}\frac{\log\left(\sharp\\{\gamma\in\pi_{1}(M)\mid
d_{M}(\gamma.x,x)\leqslant R\\}\right)}{R}\ .$

The first item is a celebrated result by [Bowen:1972] and [Margulis:1969ve].
The second item is due to [Manning:1979vk].

####  1.1.1 Rigidity of the entropy

We have several rigidity theorems for the entropy. First in the presence of an
upper bound on the curvature, a metric on closed manifold has curvature less
than $-1$, then

${\rm h}_{vol}(M,h)\geqslant\dim(M)-1\ ,$

with equality if and only if $h$ is hyperbolic. For deeper results in the
presence of upper bounds on the curvature, see [Pansu:1989ug] and
[Hamenstadt:1990tx]. As a special case of [Besson:1995um], we have, when we
drop the condition on the curvature Let $(M,h_{0})$ be a hyperbolic manifold
of dimension $m$ and $h$ another metric on $M$, then

${\rm h}_{vol}(M,h)^{m}\operatorname{Vol}(M,h)\geqslant{\rm
h}_{vol}(M,h_{0})^{m}\operatorname{Vol}(M,h_{0})\ .$

The equality implies that $h$ has constant curvature. In this exposé, we will
only use the case of $m=2$, which is due to [Katok:1982uv].

###  1.2 Equidistribution

This asymptotic counting has a counterpart called equidistribution. Let us
first recall that geodesics are solutions of some second order differential
equation, and we may as well consider non closed geodesics in the Riemannian
manifold $M$. Let us consider the phase space $\mathcal{G}$ of this equation
as the space of maps $\gamma$ from $\mathbb{R}$ to $M$, where $\gamma$ is an
arc length parametrized solution of the equation. The precomposition by
translation gives a right action by $\mathbb{R}$, and thus $\mathcal{G}$ is
partitioned into leaves which are orbits of the right action of $\mathbb{R}$.
The space $\mathcal{G}$ canonically identifies with the unit tangent bundle
$\mathsf{U}M$ by the map $\gamma\mapsto(\gamma(0),\dot{\gamma}(0))$, and the
above $\mathbb{R}$-action corresponds to the action of the geodesic flow.

We may thus associate to each closed orbit $\gamma$ of length $\ell$ a unique
probability measure $\delta_{\gamma}$ on $\mathcal{G}=\mathsf{U}M$ supported
on $\gamma$, $\mathbb{R}$-invariant and so that for any function on
$\mathsf{U}M$

$\int_{\mathsf{U}M}f{\rm
d}\delta_{\gamma}\mathrel{\mathop{:}}=\frac{1}{\ell}\int_{0}^{\ell}f(\gamma(s)){\rm
d}s\ .$

When $M$ is hyperbolic, another natural and $\mathbb{R}$-invariant probability
measure comes from the left invariant $\mu_{Leb}$ measure (under the group of
isometries) in the universal cover.

The next result is intimately related to Theorem 1.1 and also due to
[Bowen:1972] and [Margulis:1969ve]. Assume $(M,h_{0})$ is hyperbolic, then

$\lim_{T\to\infty}\frac{1}{\sharp\Gamma_{h_{0}}(T)}\sum_{\gamma\in\Gamma_{h_{0}}(T)}\delta_{\gamma}=\mu_{{Leb}}\
.$

##  2 Totally geodesic submanifolds of higher dimension

As a first attempt of generalization, it is quite tempting to understand what
happens to totally geodesic submanifolds of higher dimension, where by totally
geodesic we mean complete and such that any geodesic in the submanifold is a
geodesic for the ambient manifold.

###  2.1 Closed totally geodesic submanifolds are rare

One easily constructs by arithmetic means hyperbolic manifolds with infinitely
many closed totally geodesic submanifolds, however this situation is
exceptional and we have, as a special case of a beautiful recent theorem by
[Margulis:2020wp] – generalized in [Bader:2019wu]:

If a closed hyperbolic 3-manifold $M$ contains infinitely many closed totally
geodesic subspaces of dimension at least $2$, then $M$ is arithmetic. Thus,
having infinitely many closed totally geodesic submanifolds is quite rare for
hyperbolic $3$-manifolds, and an asymptotic counting as we defined for
geodesics does not yield interesting results in general – see however
[Jung:2019tq] for a result in the arithmetic case and a general upper bound in
[Mohammadi:2020wi, Corollary 1.12].

###  2.2 The set of pointed totally geodesic spaces and Shah’s Theorem

Let $(M,h)$ be an oriented Riemannian $3$-manifold and $G(M)$ be the bundle
over $M$ whose fiber at a point $x$ is the set of oriented $2$-planes in the
tangent space at $x$. Every surface $S$ in $M$ then has a Gauß lift $G(S)$ in
$G(M)$ which consists of the set of tangent spaces to $S$.

####  2.2.1 Totally geodesic hyperbolic planes, the frame bundle and the
$\mathsf{PSL}_{2}(\mathbb{R})$-action

When $h$ is hyperbolic, the space $G(M)$ has a natural foliation $\mathcal{F}$
whose leaves are Gauß lifts of immersed totally geodesic hyperbolic planes. We
can thus interpret $G(M)$ as the space of pointed totally geodesic planes, or
equivalently as the set of (local) totally geodesic embeddings of ${\bf
H}^{2}$ into $M$, equipped with the right action by precomposition of
$\mathsf{PSL}_{2}(\mathbb{R})$.

Let $F_{h_{0}}(M)$ be the frame bundle over $M$ whose fiber at $x$ is the set
of oriented orthonormal frames in the tangent space of $x$. We have a natural
fibration

$S^{1}\to F_{h_{0}}(M)\to G(M)\ .$

The choice of a frame at a point $x$ in $\mathbf{H}^{3}$ identifies
$\operatorname{Isom}(\mathbf{H}^{3})$, the group of orientation preserving
isometries of $\mathbf{H}^{3}$, with $\mathsf{PSL}(2,\mathbb{C})$. Thus
$F_{h_{0}}(\mathbf{H}^{3})$ is interpreted as the space of isomorphisms of
$\operatorname{Isom}(\mathbf{H}^{3})$ with $\mathsf{PSL}(2,\mathbb{C})$ and as
such carries commuting actions of $\operatorname{Isom}(\mathbf{H}^{3})$ on the
left by postcomposition and $\mathsf{PSL}(2,\mathbb{C})$ on the right by
precomposition. Then the foliation of $F_{h_{0}}(\mathbf{H}^{3})$ by the
orbits of the right action of the subgroup $\mathsf{PSL}_{2}(\mathbb{R})$ of
$\mathsf{PSL}(2,\mathbb{C})$, projects to the foliation $\mathcal{F}$ of
$G(\mathbf{H}^{3})$ that we just described, with the corresponding action of
$\mathsf{PSL}_{2}(\mathbb{R})$.

####  2.2.2 Shah and Ratner’s Theorem

From this interpretation of the $\mathsf{PSL}_{2}(\mathbb{R})$-action on
$G(M)$, a theorem by [Shah:1991wr], which is now also a consequence of the
celebrated theorem by [Ratner:1991wl], gives:

Let $(M,h_{0})$ be a closed hyperbolic $3$-manifold. Then any orbit of the
$\mathsf{PSL}_{2}(\mathbb{R})$-action on $F_{h_{0}}(M)$ is either dense or
closed. Moreover any closed set invariant by $\mathsf{PSL}_{2}(\mathbb{R})$ in
$F_{h_{0}}(M)$ is either everything or a finite union of closed orbits.

Recent work by [McMullen:2017ub, Theorem 11.1] give results in the non closed
case, see also [Tholozan:2019uw].

We state this result as part of our promenade in the subject and will not use
it in the proof, as opposed to [Calegari:2020uo]. However, we will use as
special case of the measure classification theorem of [Ratner:1991tu].

Let $(M,h_{0})$ be a closed hyperbolic $3$-manifold. Then any ergodic
$\mathsf{SL}_{2}(\mathbb{R})$-invariant measure on $F_{h_{0}}(M)$ is
$\mu_{Leb}$ or is supported on a closed leaf in $F_{h_{0}}(M)$. In particular,
there are only countably many ergodic $\mathsf{SL}_{2}(\mathbb{R})$-invariant
measures. For a short and accessible proof of Ratner’s theorem in the context
of $\mathsf{SL}_{2}(\mathbb{R})$-action see [Einsiedler:2006wz].

We only need the following corollary that may have a direct proof. Let $\mu$
be an $\mathsf{SL}_{2}(\mathbb{R})$-invariant probability measure on
$F_{h_{0}}(M)$. Let $p$ be the projection of $F_{h_{0}}(M)$ to $M$. Assume
that $p_{*}\mu$ is (up to a constant) the volume form on $M$, then
$\mu=\mu_{Leb}$.

###  2.3 In variable curvature

For a generic metric, there are no totally geodesic surfaces, not even
locally. Thus, even non closed totally geodesic surfaces are rare in variable
curvature. We actually explain now a more precise result of [Calegari:2020uo]
which implies that if a negatively curved manifold has too many hyperbolic
planes, then it is hyperbolic.

We start by discussing briefly the boundary at infinity of negatively curved
manifolds.

####  2.3.1 The boundary at infinity and circles

In the Poincaré ball model, $\mathbf{H}^{3}$ is the interior of a ball. The
boundary of the ball is denoted $\partial_{\infty}\mathbf{H}^{3}$ and carries
an action of the isometry group $\operatorname{Isom}(\mathbf{H}^{3})$ of
$\mathbf{H}^{3}$, which is isomorphic to $\mathsf{PSL}(2,\mathbb{C})$. Under
this isomorphism, $\partial_{\infty}\mathbf{H}^{3}$ identifies as a
homogeneous space with $\mathbf{CP}^{1}$.

Observe that the choice of real plane $P$ in $\mathbb{C}^{2}$ defines a circle
in $\mathbf{CP}^{1}$ which is the set of complex lines intersecting non
trivially the plane $P$. In the ball model of $\mathbf{H}^{3}$, these circles
are boundaries of hyperbolic planes totally geodesically embedded in
$\mathbf{H}^{3}$ .

The boundary at infinity $\partial_{\infty}\mathbf{H}^{3}$ has an intrinsic
definition as the set of equivalence classes of oriented geodesics which are
parallel at infinity, where the equivalence is defined as

$\gamma_{1}\sim\gamma_{2}\ \ \hbox{ if and only }\ \ \
\limsup_{t\to\infty}d(\gamma_{1}(t),\gamma_{2}(t))<\infty\ .$

This notion also makes sense in the case of a nonpositively curved simply
connected manifold $\tilde{M}$ and allow us to define the notion of the
boundary at infinity $\partial_{\infty}\tilde{M}$.

We recall here briefly that $\tilde{M}\sqcup\partial_{\infty}\tilde{M}$ admits
a topology and becomes so a compactification of $\tilde{M}$ as a closed ball.
Moreover, if $\tilde{M}$ is the universal cover of a closed manifold that
admits a hyperbolic metric, we have an identification of
$\partial_{\infty}\tilde{M}$ with $\partial_{\infty}\mathbf{H}^{3}$.

If $M$ has dimension at least 3, by Mostow rigidity, the above identification
of $\partial_{\infty}\tilde{M}$ with $\mathbf{CP}^{1}$ is unique up to the
action of $\mathsf{PSL}(2,\mathbb{C})$ and thus circles make perfect sense in
$\partial_{\infty}\tilde{M}$.

####  2.3.2 A characterization of hyperbolic metrics

######  Proposition 1.

Assume that the curvature of the closed $3$-manifold $M$ is less than $-1$.
Assume that every circle at infinity bounds a totally geodesic hyperbolic
surface. Then $M$ is hyperbolic.

######  Proof 2.1 (Sketch of a proof).

Since the curvature is negative, a circle at infinity cannot bound more than
one totally geodesic surface. Let $\mathcal{T}$ be the space of triples of
pairwise distinct elements in $\partial_{\infty}\mathbf{H}^{3}$. Every such
triple $\tau$ defines a unique circle $\gamma$ and thus a unique totally
geodesic oriented hyperbolic plane $\mathbf{H}$ in $\tilde{M}$, the universal
cover of $M$. Since $\gamma$ is the boundary at infinity of $\mathbf{H}$, a
triple of points $\tau=(a,b,c)$ define a frame $(x,u,v)$ where $x$ is a point
in $\mathbf{H}$ and $(u,v)$ an orthonormal basis of $\mathsf{T}_{x}\mathbf{H}$
by the following procedure: $x$ is on the geodesic joining $a$ to $c$, $u$ is
the vector tangent at $x$ in the direction of $b$ and $v$ the tangent vector
in the direction of $c$. Let us define

$\beta(\tau)=(x,u,v,n)\ ,\hbox{ such that $(u,v,n)$ is an oriented orthonormal
basis of $\mathsf{T}_{x}\tilde{M}$}\ .$

Then $\beta$ is $\pi_{1}(M)$-equivariant from $F_{h_{0}}(\mathbf{H}^{3})$ to
$F_{h}(M)$. One can show that $\beta$ has degree $1$ and is surjective. It
follows in particular that there is a totally geodesic hyperbolic plane
through any tangent plane in $M$. Thus $M$ has constant curvature $-1$.

##  3 Surface subgroups in hyperbolic $3$-manifolds

Let us move to topological questions. The natural question is, after we have
spent some time in the first sections studying homotopy classes of circles in
negatively curved manifolds, to understand conjugacy classes of fundamental
groups of surfaces in $3$-manifolds. We will spend some time recalling
classical facts about quasi-Fuchsian surface groups, the surface subgroup
theorem by Kahn and Marković, and finally asymptotic counting of those surface
subgroups by the genus.

To be explicit, a surface group is the fundamental group of a compact
connected orientable surface of genus greater than or equal to $2$.

Any such group can be represented as a Fuchsian subgroup, that is a discrete
cocompact subgroup of the isometry group $\mathsf{PSL}_{2}(\mathbb{R})$ of the
hyperbolic plane ${\bf H}^{2}$.

We concentrate our discussion first on discrete surface subgroups of
$\operatorname{Isom}(\mathbf{H}^{3})$.

###  3.1 Quasi-Fuchsian groups and quasicircles

Let $S$ be a closed connected oriented surface $S$ of genus greater than $2$.
Let $\rho_{0}$ be a faithful representation of the fundamental group
$\pi_{1}(S)$ in $\operatorname{Isom}({\bf H}^{2})$ whose image is a cocompact
lattice $\Gamma$. Seeing ${\bf H}^{2}$ sitting as a geodesic plane in
$\mathbf{H}^{3}$, gives rise to an embedding of $\operatorname{Isom}({\bf
H}^{2})$ in $\operatorname{Isom}(\mathbf{H}^{3})$. The corresponding morphism
of $\Gamma<\operatorname{Isom}({\bf H}^{2})$ in
$\operatorname{Isom}(\mathbf{H}^{3})$ is called a Fuchsian representation and
its image a Fuchsian group333we warn the reader our definitions are slightly
non standard here.

We saw in paragraph 2.3.1 that such a Fuchsian group preserves a circle in
$\partial_{\infty}\mathbf{H}^{3}$. This motivates the following definitions.

  1. (i)

a quasi-Fuchsian representation is a morphism $\rho$ from a cocompact lattice
$\Gamma$ of $\operatorname{Isom}({\bf H}^{2})$ in
$\operatorname{Isom}(\mathbf{H}^{3})$, such that there exists a continuous
injective map $\Lambda$ from $\partial_{\infty}{\bf H}^{2}$ to
$\partial_{\infty}\mathbf{H}^{3}$ which is $\rho$-equivariant.

  2. (ii)

The map $\Lambda$ is called the limit map of the quasi-Fuchsian morphism.

  3. (iii)

a quasi-Fuchsian group is the image of a quasi-Fuchsian representation.

  4. (iv)

a quasi-Fuchsian manifold is the quotient of the hyperbolic space by a quasi-
Fuchsian group.

  5. (v)

The limit set $\partial_{\infty}\Gamma$ of a quasi-Fuchsian group $\Gamma$ is
the image of its limit map.

In this definition, observe that the choice of a quasi-Fuchsian representation
depends on the choice of a realization of the surface group as a lattice in
$\mathsf{PSL}_{2}(\mathbb{R})$. Similarly, the limit map of a quasi-Fuchsian
representation depends on the choice of a lattice in
$\mathsf{PSL}_{2}(\mathbb{R})$, while $\partial_{\infty}\Gamma$ only depends
on the quasi-Fuchsian group $\Gamma$.

Quasi-Fuchsian manifolds are not compact: they are homeomorphic to
$S\times\mathbb{R}$, if the quasi-Fuchsian group is isomorphic to
$\pi_{1}(S)$. However, they keep some cocompactness feature:

######  Proposition 2.

Let $\partial_{\infty}\Gamma$ be the limit set of a quasi-Fuchsian surface
group $\Gamma$ and $\operatorname{Env}(\partial_{\infty}\Gamma)$ be the convex
hull (for instance in the projective Klein model) of $\partial_{\infty}\Gamma$
in $\mathbf{H}^{3}$. Then

  1. (i)

the distance to $\operatorname{Env}(\partial_{\infty}\Gamma)$ is convex444A
function is convex its restriction to any geodesic is convex. In nonpositive
curvature, any distance to a convex set is convex,

  2. (ii)

the group $\Gamma$ acts cocompactly on
$\operatorname{Env}(\partial_{\infty}\Gamma)$.

The quotient $\operatorname{Env}(\partial_{\infty}\Gamma)/\Gamma$ is called
the convex core of the quasi-Fuchsian manifold associated to $\Gamma$.

Quasi-Fuchsian groups are plentiful, and in particular

######  Proposition 3.

Any small deformation of a Fuchsian group is quasi-Fuchsian.

The limit map of a quasi-Fuchsian group has many remarkable properties:

######  Proposition 4.

Given the limit map $\Lambda$ of a quasi-Fuchsian group, there exists a
constant $K$, such that for any quadruple of pairwise distinct points
$(x,y,z,t)$ in $\partial_{\infty}{\bf H}^{2}\simeq\bf{RP}^{1}$, then

$\bigl{|}[\Lambda(x),\Lambda(y),\Lambda(z),\Lambda(w)]\bigr{|}\leqslant
K\bigl{|}[x,y,z,w]\bigr{|}\ ,$ (1)

where $[a,b,c,d]$ denotes the cross-ratio of the quadruple $(a,b,c,d)$ in
either $\bf{RP}^{1}$ or $\bf{CP}^{1}$.

More generally a map $\Lambda$ from $\bf{RP}^{1}$ to $\bf{CP}^{1}$ is called
$K$-quasisymmetric if it satisfies inequality (1). The image of a
$K$-quasisymmetric map is called a $K$-quasicircle. The above proposition can
be strengthened as

######  Proposition 5.

The limit map of any quasi-Fuchsian representation is $K$-quasisymmetric for
some $K$. If $K=1$, then the group is actually Fuchsian.

Accordingly, a surface group is $K$-quasi-Fuchsian if it admits a
$K$-quasisymmetric limit map. The constant $K$ gives a feeling of how far a
quasi-Fuchsian group is from being Fuchsian.

Not all discrete surface groups in $\operatorname{Isom}(\mathbf{H}^{3})$ are
quasi-Fuchsian. We shall see an example of that in the next paragraph.

###  3.2 Surface subgroups in fundamental groups of closed hyperbolic
$3$-manifolds

Solving a crucial conjecture of Thurston, Kahn and Marković proved that
fundamental groups of closed hyperbolic $3$-manifolds contain surface groups.
The amazing proof, in [Kahn:2009wh], uses mixing and equidistribution of the
geodesic flow and we shall have to extract further information from it.

Kahn–Marković surface subgroup theorem states the existence of many surface
groups which are “more and more” Fuchsian in some precise way

[Kahn–Marković surface subgroup theorem] Let $M$ be a closed hyperbolic
$3$-manifold. Let $\varepsilon$ be any positive constant. Then there exists a
quasi-Fuchsian subgroup in $\pi_{1}(M)$ whose limit map is a
$(1+\varepsilon)$-quasicircle. This result was explained in a Bourbaki exposé
by [Bergeron2013]. The quantitative part of the result plays a crucial role in
the proof of Agol’s Virtual Haken Theorem by [Agol:2013ts] stating that any
hyperbolic $3$-manifold has a finite covering which is a surface bundle over
the circle. Quite interestingly, in those manifolds fibering over the circle
the fundamental group of the fiber is not quasi-Fuchsian.

To add a little perspective that will come up later, recall that surface
groups and fundamental groups of hyperbolic manifolds are prototypes of
Gromov-hyperbolic groups. Gromov has broadened Thurston’s conjecture in the
following question.

######  Question 3.1.

Does any one-ended Gromov-hyperbolic group contain a surface group?

###  3.3 Counting surface subgroups

A classical theorem in geometric group theory says

A Gromov-hyperbolic group contains only finitely many conjugacy classes of
surface groups of a given genus. This result is suggested in [Gromov:1987tk],
a proof and a generalization is given in [Delzant:1995un].

The special case of the fundamental group of a hyperbolic $3$-manifold is due
to [Thurston:1997ux] and his beautifully simple proof works in general for
fundamental groups of negatively curved manifolds. We prove it in theorem 4.2
after our discussion of minimal surfaces.

For a hyperbolic manifold $M$, let $S(M,g)$ be the number of conjugacy classes
of surface subgroups in $\pi_{1}(M)$ of genus $g$. Thurston already gave a
crude estimate of an upper bound for $S(M,g)$, later on improved by
[Masters:2005vi] and [Soma:1991tr]. A crucial improvement of this count is
made in [Kahn:2010uo]. Let $M$ be a hyperbolic $3$-manifold, then there exist
constants $c_{1}$ and $c_{2}$ so that for $g$ large enough

$(c_{1}g)^{2g}\leqslant S(M,g)\leqslant(c_{2}g)^{2g}\ ,$ (2)

where $c_{2}$ only depends on the injectivity radius of $M$. The previous
upper bound by Masters was of the form $g^{c_{2}g}$. To get the lower bound,
it is actually enough —and distressing— to have the existence of one surface
group, and counts its covers —see proposition 18. However, Kahn and Marković
also have the same estimates in the harder case when one counts
commensurability classes.

We deduce,

$\lim_{g\to\infty}\frac{\log(S(M,g))}{2g\log(g)}=1\ .$ (3)

Kahn and Marković conjecture a more precise asymptotic555also for
commensurability classes:

######  Conjecture 6.

Let $M$ be a hyperbolic $3$-manifold, then there exists a constant $c(M)$ only
depending on $M$, so that

$\lim_{g\to\infty}\frac{1}{g}\left(S(M,g)\right)^{\frac{1}{2g}}=c(M)\ .$

Observe that this counting is purely topological, the results do not make any
reference to the underlying Riemannian structure of the manifold.

To summarize this discussion, according to [Kahn:2009wh], there are many
surface groups in $\pi_{1}(M)$ and we have a purely topological asymptotic of
the growth of the numbers of those when the genus goes to infinity.
Calegari–Marques–Neves article also addresses the question of counting those
subgroups but with a geometric twist. Before explaining their result, let us
review some fundamental results on minimal surfaces.

##  4 Minimal surfaces in $3$-manifolds

In order to recover the flexibility that we lost when considering closed
totally geodesic submanifolds, let us now introduce minimal immersions which
will allow us to extend our discussion about geodesics. We first spend some
time recalling some basic properties and definition of minimal immersions,
before actually addressing the question of counting surface subgroups.

###  4.1 Minimal immersions

Let $(M,g)$ be a Riemannian manifold. We denote in general by ${\rm
d}\operatorname{vol}(h)$, the volume density of a metric $h$. An immersion $f$
from a compact manifold $N$ into $M$ is a minimal immersion if $f$ is a
critical point for the volume functional

$\operatorname{Vol}(f)\mathrel{\mathop{:}}=\int_{N}\ {\rm
d}\operatorname{vol}{f^{*}g}\ .$

More precisely, this means that for any family of smooth deformations
$\\{f_{t}\\}_{t\in]-1,1[}$ with $f_{0}=f$ we have

$\left.\frac{{\rm d}}{{\rm d}t}\right|_{t=0}\operatorname{Vol}(f_{t})=0\ .$
(4)

To the family of deformations $\\{f_{t}\\}_{t\in]-1,1[}$ is associated the
infinitesimal deformation vector $\xi$ which is the section of
$f^{*}(\mathsf{T}M)$ given by

$\xi(x)=f^{*}\Bigl{(}\Bigl{.}\frac{{\rm d}}{{\rm
d}t}\Bigr{|}_{t=0}f_{t}(x)\Bigr{)}\ .$ (5)

####  4.1.1 The first variation formula

One can now compute effectively the left hand side of equation (4) by a
classical computation which is called the first variation formula. Let us
introduce the second fundamental form $\rm{II}$ which is the symmetric tensor
with values in the normal bundle given by

$\rm{II}(X,Y)=p(\nabla_{X}Y)\,$

where $X$ and $Y$ are tangent vectors to $N$, $\nabla$ is the Levi-Civita
connection of $g$ pulled back on $f^{*}(\mathsf{T}M)$, and $p$ is the
orthogonal projection of $f^{*}(\mathsf{T}M)$ on the normal bundle of $N$.
Then the first variation formula reads

$\left.\frac{{\rm d}}{{\rm
d}t}\right|_{t=0}\operatorname{Vol}(f_{t})=\int_{N}\braket{\xi\mid H}{\rm
d}\operatorname{vol}(f^{*}g)\ .$

where $H$ is the mean curvature vector defined as the trace of $\rm{II}$.

Thus being a minimal immersion is equivalent to the fact that the mean
curvature vanishes identically. As an important corollary, we have two useful
properties

  1. (i)

The restriction of a convex function to a minimal submanifold is subharmonic.

  2. (ii)

The curvature of a minimal surface at a point is less than the ambient
curvature of its tangent plane.

One then defines a minimal immersion from a (possibly non compact, possibly
with boundary) manifold as one for which the mean curvature vanishes
everywhere. Equivalently, one can show that those are the immersions $f$ for
which for any variation $\\{f_{t}\\}_{t\in]-1,1[}$ with $f_{0}=f$, constant on
the boundary as well as outside a bounded open set $U$, we have

$\Bigl{.}\frac{{\rm d}}{{\rm d}t}\Bigr{|}_{t=0}\int_{U}{\rm
d}\operatorname{vol}(f_{t}^{*}g)=0\ .$

When the dimension of $N$ is $1$, minimal immersions are exactly
parameterizations of geodesics.

####  4.1.2 The second variation formula

The misleading terminology “minimal immersions” or “minimal surfaces” tends to
suggest that minimal surfaces are not only critical point of the area
functional but actual minima. This not always the case.

In order to understand whether the immersion is actually a local minimum of
the volume functional, we need —as in the case for geodesics— to study the
second variation formula of the volume.

Let us assume for simplicity that the source is compact, since our goal is
only to present the subject.

Let us denote by $\xi$ an infinitesimal variation of $f$ as in equation (5).
We may as well assume that this infinitesimal variation is normal since
tangent deformation do not affect the volume. Then the second variation
formula is given by

${\rm
D}^{2}_{f}\operatorname{Vol}(\xi,\xi)\mathrel{\mathop{:}}=\Bigl{.}\frac{{\rm
d^{2}}}{{\rm d}t^{2}}\Bigr{|}_{t=0}\int_{N}{\rm
d}\operatorname{vol}(f_{t}^{*}g)=2\int_{N}(R_{\xi}+a_{\xi}-b_{\xi}){\rm
d}\operatorname{vol}(f^{*}g)\ ,$ (6)

where $R_{\xi}$, $a_{\xi}$ and $b_{\xi}$ are the trace —with respect to the
induced metric— of the symmetric tensors defined by respectively

$\displaystyle R_{\xi}(X,Y)$
$\displaystyle\mathrel{\mathop{:}}=\braket{R(\xi,X)\xi\mid Y}\ ,$
$\displaystyle a_{\xi}(X,Y)$
$\displaystyle\mathrel{\mathop{:}}=\braket{p(\nabla_{X}\xi)\mid
p(\nabla_{Y}\xi)}\ ,$ $\displaystyle b_{\xi}(X,Y)$
$\displaystyle\mathrel{\mathop{:}}=\braket{B(X)\xi\mid B(Y)\xi}\ ,$

where $X$ and $Y$ are tangent vectors and $\xi$ is normal, $R$ is the
curvature tensor of the Levi-Civita curvature $\nabla$ of the ambient
manifold, $B$ is the shape operator defined by

$\braket{B(X)\xi\mid Y}=\braket{\rm{II}(X,Y)\mid\xi}\ .$

Since our ultimate goal is to understand the sign of ${\rm
D}^{2}\operatorname{Vol}$, we now comment on the sign of these quantities:

  1. (i)

$a_{\xi}$ is nonnegative.

  2. (ii)

$b_{\xi}$ is nonnegative, but vanishes when the submanifold is totally
geodesic.

  3. (iii)

$R_{\xi}$ is nonnegative when the ambient curvature is nonpositive. When the
ambient manifold is hyperbolic, $R_{\xi}=2\|\xi\|^{2}$.

In particular, when the ambient curvature is nonpositive and the submanifold
is totally geodesic then $D_{f}^{2}\operatorname{Vol}(\xi,\xi)$ is nonnegative
and the minimal immersion is a local minimum. This covers for instance the
case of geodesics in nonpositive curvature.

However, in general one cannot expect that just controlling the sign of the
curvature would guarantee that the minimal immersion is an actual local
minimum. Nevertheless, we shall see that under some other additional
assumptions the minimal immersion will be a local minimum. We now introduce
the standard terminology:

A minimal immersion $f$ is stable666The terminology is unstable here: some
people call this condition semistable if for any compactly supported
infinitesimal deformation $\xi$, ${\rm
D}^{2}_{f}\operatorname{Vol}(\xi,\xi)\geqslant 0$. Thus a totally geodesic
surface in a nonpositively curved manifold is stable. A famous result proved
independently in [Pogorelov:1981uq], [Carmo:1979uo], and [Fischer-
Colbrie:1980vl] states The plane is the only stable embedded complete minimal
surface in $\mathbb{R}^{3}$.

As a standard phenomenon, we will explain later on, that such a rigidity
result implies a compactness property for the space of stable minimal surfaces
(proposition 9).

###  4.2 Minimal surfaces in $3$-manifolds

Let us now focus on minimal immersions of surfaces in $3$-manifolds. The
volume is then called the area of the immersion, and denoted by
$\operatorname{Area}(f)$.

Two important results by [Schoen:1979] and [Sacks:1982] guarantee the
existence of minimal surfaces from some topological data. Here is a special
case of their result.

Say a continuous map between two connected manifolds is incompressible if it
is injective at the level of fundamental groups.

Let $f$ be a continuous incompressible map from a closed surface to a compact
negatively curved $3$-manifold. Then there exists a minimal immersion,
homotopic to $f$, which is minimal and achieves the minimum of the area
amongst all possible maps homotopic to $f$.

In particular, surface groups in fundamental groups of compact hyperbolic
manifolds can be represented by minimal surfaces, albeit not necessarily
uniquely as we shall see.

As an application, let us now give a hint of the proof by [Thurston:1997ux] of
theorem 3.3 in the case of the fundamental groups of a hyperbolic manifold as
was explained to us by Delzant. The fundamental group of a hyperbolic manifold
only contains finitely many conjugacy classes of surface groups of a given
genus.

######  Proof 4.1.

Let $S$ be a minimal surface in $M$ representing a surface group. The
curvature of the minimal surface $S$ is bounded from above by the curvature of
$M$, and thus the area $\operatorname{Area}(S)$ of $S$ is bounded from above
by $4\pi(g-1)$, where $g$ is the genus of $S$.

Moreover since the surface is incompressible, the injectivity radius $i_{S}$
of $S$ is bounded from below by the injectivity radius $i_{M}$ of $M$. Let
then $a(i_{M})$ be a lower bound of the area of a ball of radius $i_{M}/2$ in
$S$ and observe that by comparison theorems, we can have an explicit formula
for $a(i_{M})$ in terms of $i_{M}$.

Thus we can cover $S$ by $\operatorname{Area}(S)/a(i_{M})$ balls of radius
$i_{M}/2$. Hence $\pi_{1}(S)$ is generated by curves of length (in $S$) less
than $2i_{M}\operatorname{Area}(S)/a(i_{M})$ and hence less than $8\pi
i_{M}(g-1)/a(i_{M})$. The same holds a fortiori for the length of those curves
in $M$. This implies that there is only finitely many possibilities for
conjugacy classes of surface groups.

Observe that Thurston originally used pleated surfaces rather than minimal
ones: we only use the fact that the surface representing the surface group has
curvature no greater than $-1$.

####  4.2.1 Back to quasi-Fuchsian manifolds

The work of Schoen and Yau carries on immediately in the context of quasi-
Fuchsian manifolds, due to the existence of a convex hull which traps minimal
surfaces:

######  Proposition 7.

Let $M$ be a $3$-manifold of curvature less than $-1$. Let $\rho$ be a
representation of $\pi_{1}(S)$ in the isometries of $M$. Let $C$ be a convex
set in $M$ invariant by the action of $\rho(\Gamma)$. Let $f$ be a minimal
immersion of the universal cover $\tilde{S}$ of $S$, equivariant under $\rho$,
then $f(S)$ is a subset of $C$.

######  Proof 4.2.

The distance function to the convex set $C$ is convex and strictly convex for
positive values. Hence by corollary 4.1.1, its pullback on $S$ is strictly
subharmonic for positive values and $\pi_{1}(S)$-invariant, hence vanishes
identically.

However the number of those surfaces is not a priori bounded, as it follows
from results of [Anderson:1983cm] and [Huang:2015wl]

For any given positive integer $N$, there exists a quasi-Fuchsian manifold
that contains at least $N$ distinct (closed, incompressible and embedded)
minimal surfaces.

For a survey about the use of minimal surfaces in $3$-manifolds, see
[Hass:2005tr].

###  4.3 Compactness results

The set of minimal surfaces enjoy compactness properties. In particular we
have the standard fact valid in all dimensions.

######  Proposition 8.

Let $\\{{(M_{m},h_{m},x_{m})}\\}_{m\in\mathbb{N}}$ be a sequence of pointed
Riemannian manifolds converging to a Riemannian manifold
$(M_{\infty},h_{\infty},x_{\infty})$. For each $m$, let $S_{m}$ be a complete
minimal surface without boundary in $M_{m}$, so that $x_{m}$ belongs to
$S_{m}$. Assume that, for every $R$, the second fundamental form of $S_{m}$ is
bounded independently on $m$, on every ball in $S_{m}$ containing $x_{m}$ and
radius $R$.

Then the sequence of pointed minimal surfaces
$\\{{(S_{m},x_{m})}\\}_{m\in\mathbb{N}}$ converges uniformly on every compact
to a pointed minimal surface $(S_{\infty},x_{\infty})$ in $M_{\infty}$

The following is essentially contained in [Fischer-Colbrie:1980vl] and is a
consequence of Theorem 4.1.2. Let $\lambda$ be the function on a minimal
surface defined as the positive eigenvalue of the shape operator.

######  Proposition 9.

Let $M$ be a $3$-manifold with a metric $h$ with curvature bounded from above
by $-1$. Then there is a positive $K$ only depending on $M$, such that for any
stable minimal disk $D$ embedded in the universal cover $\tilde{M}$, we have
$\lambda(D)\leqslant K$.

We sketch a proof to emphasize the standard philosophy in geometric analysis
that a rigidity result yields a compactness result.

######  Proof 4.3 (Sketch of a proof).

Let us give a proof by contradiction and assume that there exists a sequence
of stable complete minimal disks $\\{D_{m}\\}_{m\in\mathbb{N}}$ and a point
$x_{m}$ in $D_{m}$ so that $\\{{\lambda_{m}(x_{m})}\\}_{m\in\mathbb{N}}$ goes
to infinity. Here we denote by $\lambda_{m}$ the function $\lambda$ on
$D_{m}$. We can assume using the cocompact group $\pi_{1}(M)$ in
$\operatorname{Isom}(\tilde{M})$ that $x_{m}$ lies in a compact fundamental
domain for $\pi_{1}(M)$.

The optimum would be to find a point $x_{m}$ in $D_{m}$ where $\lambda_{m}$
achieves its maximum value $\Lambda_{m}$, which happens for instance if the
disk projects to a closed surface in $M$. However, since this is not
necessarily the case, we use a classical trick in geometric analysis.

Let then $K_{m}$ be the maximum of $\lambda_{m}$ on the ball of center $x_{m}$
and radius $10$. According to the $\Lambda$-maximum lemma as in
[Gromov:1991uy, Paragraph 1.D], and assuming $\lambda_{m}(x_{m})\geqslant 1$
there exists $y_{m}$, such that

$\Lambda_{m}\mathrel{\mathop{:}}=\lambda_{m}(y_{m})\geqslant\sup\Bigl{\\{}K_{m},\frac{1}{2}\lambda_{m}(z)\mid
d(z,y_{m})\leqslant\frac{1}{2\sqrt{\lambda_{m}(y_{m})}}\Bigr{\\}}\ .$ (7)

Observe that $\\{\Lambda_{m}\\}_{m\in\mathbb{N}}$ also goes to infinity. We
now consider the metric $h_{m}=\Lambda_{m}h$ on $M$, associated to a distance
$d_{m}$, and observe that $(\tilde{M},h_{m},y_{m})$ converges smoothly on
every compact to a Euclidean space. The new eigenvalue function
$\tilde{\lambda}_{m}$ is now equal to $\lambda_{m}/\Lambda_{m}$.

Thus we obtain from the assertion (7), that

$\tilde{\lambda}_{m}(z)\leqslant 2,\hbox{ if
}d_{m}(z,y_{m})\leqslant\frac{\sqrt{\Lambda_{m}}}{2}\ .$

Thus the sequence of minimal surface $\\{{(S_{m},y_{m})}\\}_{m\in\mathbb{N}}$
has bounded second fundamental form on larger and larger balls, and thus, by
proposition 8, the sequence converges (on every compact) to a minimal surface
$(S_{\infty},y_{\infty})$ in $(M,h_{\infty},y_{\infty})$. Now $S_{\infty}$ is
a stable minimal surface in the Euclidean $3$-space $(M,h_{\infty})$. By
Theorem 4.1.2, $S_{\infty}$ is a plane and thus its second fundamental form is
zero. Hence, $\tilde{\lambda}_{m}$ converges to zero uniformly on every
compact and this contradicts $\tilde{\lambda}_{m}(y_{m})=1$. This proves by
contradiction that $\\{\Lambda_{m}\\}_{m\in\mathbb{N}}$ is bounded. It follows
that $\\{K_{m}\\}_{m\in\mathbb{N}}$ — and in particular
$\\{{\lambda_{m}(x_{m})}\\}_{m\in\mathbb{N}}$ — is bounded.

###  4.4 Almost Fuchsian minimal surfaces, Uhlenbeck’s result and the
asymptotic Plateau problem

The examples constructed by Anderson, then by Huang and Wang are far from
being Fuchsian but the situation improves when we are close to being Fuchsian.

####  4.4.1 Almost Fuchsian minimal surfaces

Let us go back to the second variation formula (6) for surfaces in the case of
hyperbolic $3$-manifolds. We saw that the Hessian of the volume at a minimal
immersion $f$ is given by

${\rm
D}^{2}_{f}\operatorname{Vol}(\xi,\xi)=2\int_{N}(R_{\xi}+a_{\xi}-b_{\xi}){\rm
d}\operatorname{vol}(f^{*}g)\ ,$ (8)

where $a_{\xi}\geqslant 0$, $R_{\xi}=2\|\xi\|^{2}$ and

$b_{\xi}=\operatorname{Trace}\braket{B(X)\xi\mid
B(Y)\xi}=2\lambda^{2}\|\xi\|^{2}\ ,$

where $\lambda$ is the positive eigenvalue of $B$. Thus if we assume that
$\lambda<1$ we can guarantee that $b_{\xi}\leqslant R_{\xi}$ and thus that
${\rm D}^{2}\operatorname{Vol}(\xi,\xi)>0$, for a non vanishing $\xi$, hence
that $S$ is stable.

This suggest the following definition, where the term almost Fuchsian was
coined by [Krasnov:2007wd].

  1. (i)

A nearly geodesic minimal surface is a complete minimal surface $S$ in
$\mathbf{H}^{3}$ with $\lambda(S)<1$. If the nearly geodesic surface is
invariant under a quasi-Fuchsian group, we say the quasi-Fuchsian group is
almost Fuchsian and the nearly geodesic surface almost Fuchsian.

  2. (ii)

An almost Fuchsian manifold is a quasi-Fuchsian manifold that contains an
almost Fuchsian minimal surface.

Using our freshly minted terminology, we can rephrase the previous discussion
as the first part of the proposition

######  Proposition 10.

A nearly geodesic minimal surface in a hyperbolic $3$-manifold is stable.

Then, as suggested by this stability result, [Uhlenbeck:1983wl] proved the
following, which is a simple application of the maximum principle.

An almost Fuchsian manifold contains a unique minimal embedded incompressible
surface, which is then stable.

####  4.4.2 Asymptotic Plateau problem

Let us quit the realm of equivariant minimal surfaces. As a special case of a
theorem of [Anderson:1982vw] we have Given any embedded circle $C$ in
$\partial_{\infty}\mathbf{H}^{3}$, there exists a minimal embedded surface in
$\mathbf{H}^{3}$ bounded by $C$ in the Poincaré ball model. Let us then say
that An embedded minimal surface $S$ is solution of the asymptotic Plateau
problem defined by the embedded circle $C$ in
$\partial_{\infty}\mathbf{H}^{3}$ if in the ball model the closure
$\overline{S}$ of $S$ is $S\sqcup C.$ Alternatively we say that $C$ is the
boundary at infinity of $S$, that $S$ is bounded by $C$ and write
$C=\partial_{\infty}S$.

One naturally hopes there should be a correspondence between $K$-quasicircle
and almost Fuchsian minimal surfaces. This is indeed obtained as a consequence
of a theorem of [Seppi:2016ut, Theorem A], while the second part follows from
an extension of [Guo:2010wk], where the result is only stated for almost
Fuchsian surfaces.

There exist constants $K_{0}$ and $C_{0}$ such that if $S$ is a complete
embedded minimal surface whose boundary at infinity is a $K$-quasicircle, with
$K$ less than $K_{0}$, then

$\lambda(S)\leqslant C_{0}\log(K)\ .$

Conversely, there exists $\lambda_{0}$, such that if
$\lambda(S)\leqslant\lambda\leqslant\lambda_{0}$, then the surface $S$ is
embedded in $\mathbf{H}^{3}$ and the boundary at infinity is a
$K(\lambda)$-quasicircle, with

$\lim_{\lambda\to 0}K(\lambda)=1\ .$

For a survey in the asymptotic Plateau problem for minimal surfaces see
[Coskunuzer:2014uk], for results when the target is negatively curved see
[Lang:2003we].

###  4.5 In variable curvature

Assume now that the closed hyperbolic manifold $(M,h_{0})$ is also equipped
with a metric $h$ of curvature less than $-1$.

For any set $\Lambda$ in $\partial_{\infty}\tilde{M}$, let
$\operatorname{Env}_{h}(\Lambda)$ be the convex hull of $\Lambda$ that is the
intersection of all convex subsets of $\tilde{M}$ whose closure in
$\tilde{M}\sqcup\partial_{\infty}\tilde{M}$ contains $\Lambda$.

Observe again that thanks to Mostow rigidity $K$-quasicircles in the boundary
at infinity of the universal cover of $(M,h)$ is a topological notion. We
denote by $\tilde{M}$ the universal cover of $M$.

For the paper being discussed, the authors need to obtain a control between
minimal surfaces for both $(M,h)$ and $(M,h_{0})$ given in [Calegari:2020uo,
Theorem 3.1.]. This result follows from results of [Bowditch:1995wo].

[Morse Lemma for minimal surfaces] There exists a positive constant $R$, such
that if $S$ and $S_{0}$ are incompressible minimal surfaces in $M$, for $h$
and $h_{0}$, having the same fundamental group, then

$d_{0}(S_{0},S)\leqslant R\ ,$

where $R$ only depends on $h$, $h_{0}$ and $\lambda(S_{0})$, and $d_{0}$ is
the distance with respect to $h_{0}$. Correctly extended this result also
makes sense for other minimal surfaces than equivariant ones, and could be
understood as a Morse Lemma for minimal surfaces. Since $S$ and $S_{0}$ are
trapped in $\operatorname{Env}_{h}(\Lambda)$ and
$\operatorname{Env}_{h_{0}}(\Lambda)$ respectively by proposition 7, it is
enough to prove

######  Proposition 11.

For any set $\Lambda$ in
$\partial_{\infty}\tilde{M}=\partial_{\infty}\mathbf{H}^{3}$, we have

$d_{h}(\operatorname{Env}_{h}(\Lambda),\operatorname{Env}_{h_{0}}(\Lambda))\leqslant
R\ ,$ (9)

for some constant only depending and $h$ and $h_{0}$.

######  Proof 4.4 (Indication of the proof).

We prove that by introducing the following set: for any $p$, let

$\operatorname{Clo}^{h}_{p}(\Lambda)\mathrel{\mathop{:}}=\\{\gamma(t)\mid t>0\
,\gamma\hbox{ geodesic for $h$ with }\gamma(0)=p,\gamma(+\infty)\in\Lambda\\}\
.$

Then, by a result of [Bowditch:1995wo, proposition 2.5.4], there is some
positive constant $R_{1}(h)$ only depending on $h$, so that for any $p$ in
$\operatorname{Env}_{h}(\Lambda)$,

$d_{h}(\operatorname{Env}_{h}(\Lambda),\operatorname{Clo}^{h}_{p}(\Lambda))\leqslant
R_{1}(h)\ .$ (10)

Take now geodesics $\gamma$ and $\gamma_{0}$ joining two points of $\Lambda$,
then by the Morse Lemma for geodesics, we can find points $p$ and $p_{0}$ in
$\gamma$ and $\gamma_{0}$ respectively so that

$d_{h}(p,p_{0})\leqslant R_{2}\ ,$ (11)

where $R_{2}$ only depends on $h$ and $h_{0}$. As a final ingredient observe
that for any $p$ and $q$,

$d_{h}(\operatorname{Clo}^{h}_{p}(\Lambda),\operatorname{Clo}^{h}_{q}(\Lambda))\leqslant
R_{3}+d_{h}(p,q)\ ,$ (12)

where $R_{3}$ only depends on $h$. Observing that $p$ belongs to
$\operatorname{Env}_{h}(\Lambda)$ while $p_{0}$ belongs to
$\operatorname{Env}_{h_{0}}(\Lambda)$, and combining inequalities (10), (11)
and (12), we get the desired inequality (9) with

$R=R_{2}+R_{2}+R_{1}(h)+R_{1}(h_{0})\ .$

####  4.5.1 Minimal surfaces and quasi-isometries

Let again $(M,h_{0})$ be a closed hyperbolic $3$-manifold and $h$ another
metric on $M$ of curvature less than $-1$.

As a consequence of proposition 4.5, a priori bounds on the curvature of
minimal surfaces given by proposition 9, Theorem 4.4.2 that gives this result
in the hyperbolic case, and classical arguments about quasi-isometries, we
have There exist positive constants $\varepsilon_{0}$ and $K$ so that the
following holds. Assume $h$ is close enough to a hyperbolic metric $h_{0}$.
Let $S$ be an area minimizing minimal incompressible surface in $(M,h)$, such
that the boundary at infinity of $\pi_{1}(S)$ is
$(1+\varepsilon_{0})$-quasicircle, then

  1. (i)

the conformal minimal parametrization $\phi$ from ${\bf H}^{2}$ to $S$, is, as
a map to the universal cover of $M$, a $K$-quasi-isometric embedding,

  2. (ii)

$\phi$ admits an extension to $\mathbf{RP}^{1}$ which is a $K$-quasi-symmetric
map with values in
$\partial_{\infty}\tilde{M}=\partial_{\infty}\mathbf{H}^{3}$.

We recall that a map is a $K$-quasi-isometric embedding if the image of every
geodesic is a $K$-quasi-geodesic.

###  4.6 The case of fibered manifolds

To conclude our promenade in minimal surfaces in hyperbolic manifolds, and
after discussing almost Fuchsian manifolds, let us say a word about manifolds
fibering over the circle, even though none of this will be used further on.

By Agol’s Virtual Haken Theorem, any hyperbolic $3$-manifold has a finite
cover that fibers over the circle. The fibers of these fibrations are not
quasi-Fuchsian, but nevertheless can be represented by minimal surfaces by
theorem 4.2.

This fibration is taut by a result of [Sullivan:1979vg], which means one can
realize this foliation by minimal surfaces for some metric.

A long standing question was whether this fibration could be realized by a
minimal fibration in the hyperbolic metric. The answer to this question is no:
there exists $3$-manifolds fibering over the circle, so that the fibers of
this fibration cannot be all minimal surfaces. This is a result of
[Hass:2015we] —see also [Huang:2019tp].

##  5 Equidistribution in the phase space of minimal surfaces

###  5.1 A phase space for stable minimal surfaces

Dealing with solutions of ordinary differential equations, for instance
geodesics, we introduced the phase space of the problem, which can be
identified the space of pairs $(x,L)$777One has to be careful of what we call
“orbit” to avoid the space to be non Hausdorff, where $x$ is a point in the
orbit $L$ of the ordinary differential equation. One can generalize this
construction to solutions of partial differential equations as was done in
[Gromov:1991uv] for minimal surfaces and harmonic mappings and studied in
[Labourie:2005b] for surfaces with constant Gaußian curvature in negatively
curved $3$-manifolds. We will do so for stable minimal surfaces and describe
measures on this space.

Let $M$ be a closed manifold equipped with a metric $h$ of curvature less than
$-1$, and $\tilde{M}$ its universal cover.

In this section ${\bf H}^{2}$ will be the upper half plane model of the
hyperbolic plane, which comes with a canonical identification of
$\partial_{\infty}{\bf H}^{2}$ with $\mathbf{RP}^{1}$ and
$\operatorname{Isom}({\bf H}^{2})$ with $\mathsf{PSL}_{2}(\mathbb{R})$. We say
a minimal immersion from ${\bf H}^{2}$ to $M$ is conformal if the pullback
metric is in the conformal class of the hyperbolic metric.

[Conformal minimal lamination] Let us fix some small $\varepsilon_{0}$ and
large constant $K$ so that Theorem 4.5.1 holds.

  1. (i)

Let $\mathcal{F}_{h}(\tilde{M})$ be the space of stable minimal conformal
immersions of ${\bf H}^{2}$ in $\tilde{M}$ which are $K$-quasi-isometric
embeddings, equipped with the topology of uniform convergence on every
compact, and
$\mathcal{F}_{h}(M)\mathrel{\mathop{:}}=\mathcal{F}_{h}(\tilde{M})/\pi_{1}(M)$.

  2. (ii)

For $\varepsilon\leqslant\varepsilon_{0}$, let
$\mathcal{F}_{h}(\tilde{M},\varepsilon)$ be the set of those $\phi$ in
$\mathcal{F}_{h}(\tilde{M})$ so that $\phi(\partial_{\infty}{\bf H}^{2})$ is a
$(1+\varepsilon)$-quasicircle. Similarly, let
$\mathcal{F}_{h}(M,\varepsilon)\mathrel{\mathop{:}}=\mathcal{F}_{h}(\tilde{M},\varepsilon)/\pi_{1}(M)$.

The space $\mathcal{F}_{h}(M)$ together with the action of
$\mathsf{PSL}_{2}(\mathbb{R})$ by precomposition is called the conformal
minimal lamination of $M$.

Finally denote by $\mathcal{Q}(K)$ the space of $K$-quasicircles in
$\partial_{\infty}\mathbf{H}^{3}$ equipped with the Gromov–Hausdorff topology.
Then

  1. (i)

The map from $\mathcal{F}_{h}(\tilde{M})$ to $\tilde{M}$, given by
$\phi\mapsto\phi(i)$ is a proper map.

  2. (ii)

The action of $\mathsf{PSL}_{2}(\mathbb{R})$ by precomposition on
$\mathcal{F}_{h}(M)$ is continuous and proper.

  3. (iii)

Moreover, the map $\partial$ from $\mathcal{F}_{h}(\tilde{M})$ to
$\mathcal{Q}(1+\varepsilon)$, which maps $\phi$ to $\phi(\mathbf{RP}^{1})$ is
continuous and $\mathsf{PSL}_{2}(\mathbb{R})$-invariant.

We may assume that $(1+\varepsilon)\leqslant K$, and we will consider from now
on $Q(1+\varepsilon)$ as a subset of $Q(K)$ to lighten the notation.

######  Proof 5.1.

The first point is a rephrasing of proposition 9 and 8. The second point and
third point also follow from the first and from the fact that any element of
$\mathcal{F}_{h}(M)$ is a $K$-quasi-isometric embedding.

Here is a corollary, using the constants that appear in the previous theorem.

The map $\partial$ gives rise to a continuous map —also denoted $\partial$—
from $\mathcal{F}_{h}(M)/\mathsf{PSL}_{2}(\mathbb{R})$ to $\mathcal{Q}(K)$.

A recent preprint of [Lowe:2020vu] states —in this language— that upon small
deformation $h$ of the hyperbolic metric the projection from
$\mathcal{F}_{h}(M,0)$ to $G(M)$ is a homeomorphism. This is a special case of
a theorem by [Gromov:1991uv].

###  5.2 Laminar measures and conformal currents

This paragraph is an extension of the theory of geodesic currents and
invariant measures as in [Bonahon:1997tl].

  1. (i)

A laminar measure on $\mathcal{F}_{h}(\tilde{M})/\pi_{1}(M)$ is a
$\mathsf{PSL}_{2}(\mathbb{R})$-invariant finite measure.

  2. (ii)

A conformal current is a $\pi_{1}(M)$-invariant locally finite measure on
$\mathcal{Q}(K)$.

Here are two examples which are the analogues of the situation for closed
geodesics. Let $\Gamma$ be a Fuchsian group acting on ${\bf H}^{2}$. Let $U$
be a fundamental domain of the action of $\Gamma$ on
$\mathsf{PSL}_{2}(\mathbb{R})$. Let $\rho$ be a representation of $\Gamma$
into $\pi_{1}(M)$.

######  Proposition 12.

For $\varepsilon$ small enough, let $\phi$ be a an element of
$\mathcal{F}_{h}(\tilde{M})$, equivariant under a representation $\rho$ from
$\Gamma$ to $\pi_{1}(M)$, such that $\rho$ is injective and its boundary at
infinity is a $(1+\varepsilon)$-quasicircle $\Lambda_{0}$.

  1. (i)

Let $\delta^{h}_{\phi}$ be the measure on $\mathcal{F}_{h}(M)$ defined by

$\int_{\mathcal{F}_{h}(M)}f\ {\rm
d}\delta^{h}_{\phi}=\frac{1}{\operatorname{Vol}(U)}\int_{U}f(\phi\circ g)\
{\rm d}\mu(g),$

where $\mu$ is the bi-invariant measure on $\mathsf{PSL}_{2}(\mathbb{R})$.
Then $\delta^{h}_{\phi}$ is a $\mathsf{PSL}_{2}(\mathbb{R})$-invariant
probability measure.

  2. (ii)

Let $\delta_{\rho}$ be the measure on $\mathcal{Q}(K)$ defined by

$\delta_{\rho}=\sum_{\gamma\in\pi_{1}(M)/\rho(\Gamma)}\gamma_{*}\delta_{\Lambda_{0}}\
,$

where $\delta_{\Lambda_{0}}$ is the Dirac measure supported on $\Lambda_{0}$.
Then $\delta_{\rho}$ is a locally finite $\pi_{1}(M)$-invariant measure on
$\mathcal{Q}(K)$.

The only non-trivial point is the fact that $\delta_{\rho}$ is a locally
finite measure. This is checked at the end of the proof of the next
proposition.

The next proposition is crucial

######  Proposition 13.

There exist some positive constant $\varepsilon$ and a continuous map
$\pi_{h}$ from the space of laminar measures (up to multiplication by a
constant) to the space of conformal currents (up to multiplication by a
positive constant) so that

$\pi_{h}(\delta_{\phi}^{h})=\delta_{\rho}\ ,$

if $\phi$ is an element of $\mathcal{F}_{h}(M,\varepsilon)$ equivariant under
a representation $\rho$.

Moreover the support of $\pi_{h}(\mu)$ is the image by $\partial$ of the
support of $\mu$.

######  Proof 5.2.

Let us fix a nonnegative function $\Xi$ supported on a bounded neighborhood of
the identity in $\mathsf{PSL}_{2}(\mathbb{R})$.

Let $\mu$ be a laminar measure on $\mathcal{F}_{h}(M)$. Let us lift $\mu$ to a
locally finite $\pi_{1}(M)$-invariant measure $\tilde{\mu}$ on
$\mathcal{F}_{h}(\tilde{M})$. Let $\Lambda$ be a
$(1+\varepsilon)$-quasicircle. Let $\mathcal{T}$ be the space of triples of
pairwise distinct points of $\partial_{\infty}$. Let $U$ be a small
neighborhood of $\Lambda$ in $\mathcal{Q}(K)$. Let ${\bf F}$ be a continuous
map from $U$ to $\mathcal{T}$ so that if $(a_{0},a_{1},a_{\infty})={\bf
F}(\Lambda)$, then $a_{0}$, $a_{1}$, $a_{\infty}$ belong to $\Lambda$.

For any $\phi$ in $\mathcal{F}_{h}(M)$ so that $\partial\phi=\Lambda$ is in
$U$, let $g_{\phi}$ be the unique element of $\mathsf{PSL}_{2}(\mathbb{R})$ so
that $\phi\circ g_{\phi}(0,1,\infty)={\bf F}(\Lambda)$. Let then $\xi_{\bf F}$
be the function defined on $\partial^{-1}U$, by

$\xi_{\bf F}(\phi)=\Xi(g_{\phi})\ .$

Finally let us define, when $f$ is supported in $U$,

$\int_{\Lambda}f\ {\rm
d}\pi_{h}(\mu)\mathrel{\mathop{:}}=\int_{\mathcal{F}_{h}(\tilde{M})}\xi_{\bf
F}\cdot(f\circ\partial)\ {\rm d}\tilde{\mu}.$ (13)

Since $\xi_{\bf F}\cdot(f\circ\partial)$ is compactly supported, the left hand
side is a well-defined finite real number.

Then, one sees that the left hand side does not depend on the choice of ${\bf
F}$ since $\tilde{\mu}$ is $\mathsf{PSL}_{2}(\mathbb{R})$-invariant. By
construction $\pi(\mu)$ is locally finite since $\tilde{\mu}$ is, similarly
$\pi(\mu)$ is invariant under the action of $\pi_{1}(M)$ since $\tilde{\mu}$
is. Finally the formula shows that $\pi$ is continuous in the weak topology.

We leave the reader check the equality $\pi(\delta_{\phi}^{h})=\delta_{\rho}$.

###  5.3 Equidistribution

We can now explain the equidistribution result that follows from the
techniques in [Kahn:2009wh].

[Equidistribution] Let $M$ be a closed hyperbolic $3$-manifold. There exists a
sequence $\\{\delta^{0}_{m}\\}_{m\in\mathbb{N}}$ of laminar measures on
$\mathcal{F}_{h_{0}}(M,1/m)$, such that $\delta^{0}_{m}$ is supported on
finitely many closed leaves and so that the sequence of
$\\{\delta^{0}_{m}\\}_{m\in\mathbb{N}}$ converges to $\mu_{Leb}$.

We will call in the sequel the sequence of measures obtained in this theorem a
Kahn–Marković sequence. This result is an extended version of
[Calegari:2020uo, Theorem 4.2].

######  Proof 5.3 (Sketch of the proof).

We use a different geometric presentation than [Hamenstadt:2015wa] and
[Kahn:2009wh], developed in [Kahn:2018wx]. The following convention will hold
through this sketch

  1. (i)

All references in this sketch are from [Kahn:2018wx].

  2. (ii)

$K_{i}$ will be constants only depending on the closed hyperbolic manifold
$M={\mathbf{H}^{3}}/\pi_{1}(M)$.

  3. (iii)

$o(m)$ will denote a function that converges to $0$ when $m$ goes to infinity.

  4. (iv)

$\alpha^{-}$ and $\alpha^{+}$ are the repulsive and attractive fixed points of
the element $\alpha$ in $\pi_{1}(M)$, while $\ell(\alpha)$ is the length of
the associated geodesic.

  5. (v)

$\varepsilon$ will be a (small) positive constant and $R$ be a (large)
positive constant.

A tripod is a triple of pairwise distinct points in
$\mathbf{CP}^{1}\simeq\partial_{\infty}\mathbf{H}^{3}$. Let $\mathcal{T}$ be
the space of tripods. The space $\mathcal{T}$ is canonically identified with
the frame bundle $F_{h_{0}}(\mathbf{H}^{3})$ and carries a canonical metric.
Every point $x$ in $\mathcal{T}$ also defines an ideal triangle $\Delta_{x}$
in $\mathbf{H}^{3}$ and we denote by $b(x)$ the barycenter of this triangle.
We see the barycentric map $x\mapsto b(x)$ as a projection from $\mathcal{T}$
to $\mathbf{H}^{3}$

We remark that there is an open subset $\Delta$ in
$\mathsf{PSL}_{2}(\mathbb{R})$, invariant by the right action of $S^{1}$, so
that

$b(\Delta(x))=\Delta_{x}\ ,$

and thus for any function defined on $\mathbf{H}^{3}$, we have

$\frac{1}{\operatorname{Vol}(\Delta)}\int_{\Delta}g\circ b\circ u(x)\ {\rm
d}\mu(u)=\frac{1}{\pi}\int_{\Delta_{x}}g\ {\rm d}\operatorname{area}\ ,$ (14)

where ${\rm d}\mu$ is the bi-invariant measure in
$\mathsf{PSL}_{2}(\mathbb{R})$.

A triconnected pair of tripods [definition 10.1.1] is a quintuple
$(t,s,c_{0},c_{1},c_{2})$ so that $t$ and $s$ are points in
$\mathcal{T}/\pi_{1}(M)$ and $c_{i}$ are three homotopy classes of paths from
$t$ to $s$. We denote by $\pi$ the projection from $\mathcal{T}$ to
$\mathcal{T}/\pi_{1}(M)$.

Let also define $\pi^{0}$ and $\pi^{1}$ as the forgetting maps taking values
in the frame bundle

$\pi^{0}\colon(t,s,c_{0},c_{1},c_{2})\mapsto t\ \ \ ,\ \ \
\pi^{1}\colon(t,s,c_{0},c_{1},c_{2})\mapsto s\ .$

The space of triconnected pair of tripods tripods carries a measure
$\mu_{\varepsilon,R}$ [definition 12.2.3] satisfying the following property,
property which is established by a suitable closing lemma [Theorem
10.3.1][Theorem 9.2.2]:

Let $\varepsilon$ be small enough, then $R$ large enough. If
$(t,s,c_{0},c_{1},c_{2})$ is in the support of $\mu_{\varepsilon,R}$, there
exists three elements $\alpha$, $\beta$ and $\gamma$ of $\pi_{1}(M)$ so that,

  1. (i)

let $t_{0}=(\alpha^{-},\beta^{-},\gamma^{-})$ and
$s_{0}=(\alpha^{-},\alpha(\gamma^{-}),\beta^{-})$, then $\pi(t_{0})$ and
$\pi(s_{0})$ are $K_{1}\frac{\varepsilon}{R}$ close to $t$ and $s$
respectively,

  2. (ii)

$\alpha$, $\beta$ and $\gamma$ are in the homotopy classes of $c_{0}\cdot
c_{1}^{-1}$, $c_{2}\cdot c_{0}^{-1}$ and $c_{1}\cdot c_{2}^{-1}$ respectively.

  3. (iii)

The complex cross-ratio of
$(\alpha^{-},\alpha(\gamma^{-}),\beta^{-},\gamma^{-})$ is
$K_{1}\frac{\varepsilon}{R}$ close to $R$.

  4. (iv)

The complex length of $\alpha$, $\beta$, $\gamma$ is
$K_{2}\frac{\varepsilon}{R}$ close to $2R$.

Observe that $\alpha\gamma\beta=1$. Moreover, gluing the two ideal triangles
$T_{0}\mathrel{\mathop{:}}=\Delta_{t_{0}}$ and $S_{0}=\Delta_{s_{0}}$, then
taking the quotient by $\pi_{1}(M)$ one gets a pleated pair of pants $P$ in
$M$, whose fundamental group is generated by $(\alpha,\beta,\gamma)$.

Conversely, given three elements in $\pi_{1}(M)$, $\alpha$, $\gamma$, and
$\beta$ so that $\alpha\gamma\beta=1$, one gets a unique triconnected pair of
tripods (called exact) by setting $t=(\alpha^{-},\beta^{-},\gamma^{-})$ and
$s=(\alpha^{-},\alpha(\gamma^{-}),\beta^{-})$ and $c_{0}$, $c_{1}$, $c_{2}$
the obvious paths.

We define a triple $(\alpha,\beta,\gamma)$ satisfying the last two items
$(iii)$ and $(iv)$ as an $(\varepsilon,R)$-pair of pants. the sequence of
measures $\pi^{0}_{*}\mu_{\varepsilon,R}$ and $\pi^{1}_{*}\mu_{\varepsilon,R}$
converges to $\mu_{Leb}$ by a mixing argument as $R$ goes to infinity, when
$\varepsilon$ is fixed.

More precisely, we can choose a sequence $\\{R_{m}\\}_{m\in\mathbb{N}}$ going
to $\infty$, so that setting $\mu_{m}=\mu_{\frac{1}{m},R_{m}}$, then for any
continuous function $f$ on $F_{h_{0}}(M)$, then we have [proposition
10.2.6][equation 93]:

$\lim_{m\to\infty}\int(f\circ\pi^{0})\ \ {\rm
d}\mu_{m}=\lim_{m\to\infty}\int(f\circ\pi^{1})\ \ {\rm d}\mu_{m}=\int f{\rm
d}\mu_{Leb}\ .$ (15)

According to [proposition 18.0.3], and using mixing again, the proof goes by
showing that one can approximate $\mu_{m}$ by measures $\nu_{m}$ with finite
support $\\{P^{m}_{1},\ldots,P^{m}_{N_{m}}\\}$, on exact triconnected pair of
tripods and rational weights, with the following property. The sequence of
measures $\pi^{0}_{*}\nu_{m}$ and $\pi^{1}_{*}\nu_{m}$ converges to
$\mu_{Leb}$ by a mixing argument as $R$ goes to infinity. In other words, for
any function $f$,

$\lim_{m\to\infty}\frac{1}{N_{m}}\sum_{i=1}^{N_{m}}f(\pi_{0}(P^{m}_{i}))=\lim_{m\to\infty}\frac{1}{N_{m}}\sum_{i=1}^{N_{m}}f(\pi_{1}(P^{m}_{i}))=\int
f{\rm d}\mu_{Leb}\ .$ (16)

Without loss of generality, we assume that the weight of each $P^{i}_{m}$
—that may appear with multiplicity— is $1/{N_{m}}$: in other words the $P_{i}$
are counted with multiplicities to have the same weights.

Let us assume now assume that $f=g\circ b$, where $b$ is the barycentric map
from $F_{h_{0}}(M)$ to on $M$ and $g$ is defined on $M$. Using the invariance
of $\mu_{Leb}$ under the left action of $\mathsf{PSL}_{2}(\mathbb{R})$ we get
that

$\lim_{m\to\infty}\frac{1}{2\pi
N_{m}}\Bigl{(}\sum_{i=1}^{N_{m}}\int_{\Delta_{0}(P^{i}_{m})}g\ {\rm
d}\operatorname{area}+\sum_{i=1}^{N_{m}}\int_{\Delta_{1}(P^{i}_{m})}g\ {\rm
d}\operatorname{area}\Bigr{)}=\int f{\rm d}\mu_{Leb}\ ,$ (17)

where $\Delta_{0}(P)\mathrel{\mathop{:}}=\Delta_{\pi_{0}(P)}$ and
$\Delta_{1}(P)\mathrel{\mathop{:}}=\Delta_{\pi_{1}(P)}$ are ideal triangles:
Indeed we apply equation (16), to $f\circ u$, for all $u$ in $\Delta$ and
apply formula (14).

Moreover, the measure $\nu_{m}$ satisfies a matching condition ( [lemma
14.2.1]) that we now describe. Let $\\{P^{m}_{1},\ldots,P^{m}_{N_{m}}\\}$ be
the support of $\nu_{m}$.

The matching condition is that for each $m$, we can find by (see [definition
14.1.1][Theorem 16.3.1]) a family of closed surfaces
$S_{m}=(S_{1}^{m},\ldots,S^{m}_{M_{m}})$ in $M$ obtained by gluing the pleated
pair of pants $\\{P^{m}_{1},\ldots,P^{m}_{N_{m}}\\}$ in $M$ such that every
pair of pants appears exactly once. In particular, we get from equation (17)
that

$\lim_{m\to\infty}\frac{1}{2\pi\chi(S_{m})}\int_{S_{m}}g{\rm
d}\operatorname{area}=\int f{\rm d}\mu_{Leb}\ .$ (18)

For each $S^{i}_{m}$, let $\Sigma^{i}_{m}$ be the associated minimal surface
and $\Sigma_{m}$ the union of all $\Sigma^{i}_{m}$. As part of the
construction, each $S^{i}_{m}$ is $(1+o(m))$-almost Fuchsian and thus the
projection $p_{m}$ from the pleated surface $S^{i}_{m}$ to the minimal surface
$\Sigma_{i}$ is $(1+o(m))$-bi-Lipsichitz and satisfies
$d(x,p^{i}_{m}(x))\leqslant o(m)$.

It follows from equation (18) that

$\lim_{m\to\infty}\frac{1}{2\pi\chi(\Sigma_{m})}\int_{\Sigma_{m}}g{\rm
d}\operatorname{area}=\int f{\rm d}\mu_{Leb}\ .$

In other words, setting $\delta^{0}_{m}$ the measure on
$\mathcal{F}_{h_{0}}(M)$ supported on $\Sigma_{m}$, we have

$\lim_{m\to\infty}p_{*}\delta^{0}_{m}=p_{*}\mu_{Leb}\ .$

We can now conclude using corollary 2.2.2.

##  6 Calegari–Marques–Neves asymptotic counting

The article of [Calegari:2020uo] deals with the counting of surface subgroups.
Let $\Sigma$ be the set of conjugacy classes of surface subgroups in
$\pi_{1}(M)$ for a manifold $M$. For $\Pi$ an element of $\Sigma$ and $h$ a
Riemannian metric on $M$, we define

$\operatorname{MinArea}_{h}(\Pi)\mathrel{\mathop{:}}=\inf\\{\operatorname{Area}(S)\mid
S\hbox{ is an incompressible surface in $M$ with $\pi_{1}(S)\in\Pi$}\\}\ .$

Assume now that $M$ is a $3$-manifold that admits a hyperbolic metric. For a
conjugacy class of surface group $\Pi$, we define $\varepsilon(\Pi)$ as

$\varepsilon(\Pi)\leqslant\varepsilon\ ,$

if $\Pi$ is quasi-Fuchsian and the limit circle is a
$(1+\varepsilon)$-quasicircle. Let then

$\displaystyle S_{h}(M,T,\varepsilon)$
$\displaystyle\mathrel{\mathop{:}}=\\{\Pi\in\Sigma\mid\operatorname{MinArea}_{h}(\Pi)\leqslant
T\,,\ \varepsilon(\Pi)\leqslant\varepsilon\\}\ ,$ (19) $\displaystyle
E_{\varepsilon}(M,h)$
$\displaystyle\mathrel{\mathop{:}}=\frac{1}{4\pi}\liminf_{T\to\infty}\frac{\log\left(\sharp
S_{h}(M,T,\varepsilon)\right)}{T\log T}\ ,$ (20) $\displaystyle E(M,h)$
$\displaystyle\mathrel{\mathop{:}}=\lim_{\varepsilon\to
0}E_{\varepsilon}(M,h)\ .$ (21)

The main result of [Calegari:2020uo] is then Let $(M,h_{0})$ be a hyperbolic
$3$-manifold

  1. (i)

for any Riemannian metric $h$ on $M$, we have $E(M,h)\leqslant 2{\rm
h}_{vol}(M,h)^{2}$,

  2. (ii)

assume furthermore that $h$ has curvature less than $-1$ then $E(M,h)\geqslant
2$, with equality if only if the metric $h$ is hyperbolic.

This result is only for counting conjugacy classes of surface groups. Except
for the equality case for which the method does not apply, the theorem also
holds for counting commensurability classes. We believe the same result holds
for commensurability classes after a suitable adaptation of theorem 1.2.

Thus we have an exact analogue to [Bowen:1972] and [Margulis:1969ve] for the
first assertion as well as a rigidity result analogue to [Hamenstadt:1990tx]
for the second assertion.

The equality case involves a mix of analytical and dynamical properties.

Let us now explain a proof of their result insisting on the use of the phase
space of stable minimal surfaces, which synthesizes some of the proofs in
[Calegari:2020uo].

###  6.1 Counting surfaces: from genus to area

The next proposition is an easy exercise on counting, inclusion and
inequalities and is used several times in the proof.

######  Proposition 14.

Let $\mathcal{S}$ be a set of conjugacy classes of surface subgroups. Let
$\mathcal{S}(g)$ be the set of those elements of $\mathcal{S}$ of genus less
than $g$,

  1. (i)

assume that we have a positive constant $c$, so that

$\sharp{\mathcal{S}(g)}\leqslant(cg)^{2g},$

as well as constants $K_{0}$ and $K_{1}$, so that for any $\Pi$ in
$\mathcal{S}$ of genus $g$, we have

$g-K_{1}\leqslant K_{0}\operatorname{MinArea}_{h}(\Pi)\ .$

Then

$\limsup_{T\to\infty}\frac{\log\left(\sharp\left\\{\Pi\in\mathcal{S}\mid\operatorname{MinArea}_{h}(\Pi)\leqslant
T\right\\}\right)}{T\log(T)}\leqslant 2K_{0}\ .$

  2. (ii)

Assume that we have a constant $c$, so that

$\sharp{\mathcal{S}(g)}\geqslant(cg)^{2g},$

as well as constants $K_{0}$ and $K_{1}$ so that for any $\Pi$ in
$\mathcal{S}$ of genus $g$, we have

$g-K_{1}\geqslant K_{0}\operatorname{MinArea}_{h}(\Pi)\ .$

Then

$\limsup_{T\to\infty}\frac{\log\left(\sharp\left\\{\Pi\in\mathcal{S}\mid\operatorname{MinArea}_{h}(\Pi)\leqslant
T\right\\}\right)}{T\log(T)}\geqslant 2K_{0}\ .$

###  6.2 The upper bound in variable curvature

Our first proposition is

######  Proposition 15.

Let $h$ be any Riemannian metric on a manifold $M$ admitting a hyperbolic
metric. Then for all $\varepsilon$, we have $E(M,h,\varepsilon)\leqslant 2{\rm
h}_{top}(M)^{2}$.

######  Proof 6.1.

Recall that

${\rm h}_{top}(M)={\rm
h}_{vol}(M)=\lim_{R\to\infty}\frac{\log\left(\sharp\\{\gamma\in\pi_{1}(M)\mid
d_{M}(\gamma.x,x)\leqslant R\\}\right)}{R}\ .$

where $x$ is a point in the universal cover $\tilde{M}$ of $M$ and $d_{M}$ the
distance in this universal cover. If $S$ is an incompressible surface in $M$
lifting to a disk in $\tilde{M}$, we have

$\\{\gamma\in\pi_{1}(S)\mid d_{S}(\gamma.x,x)\leqslant
R\\}\subset\\{\gamma\in\pi_{1}(M)\mid d_{M}(\gamma.x,x)\leqslant R\\}\ ,$

and thus

${\rm h}_{vol}(S)\leqslant{\rm h}_{vol}(M)\ .$

Combining with Theorem 1.1 as in [Katok:1982uv], we get

${\rm h}_{vol}(M)^{2}\operatorname{Area}(S)\geqslant{\rm
h}_{vol}(S)^{2}\operatorname{Area}(S)\geqslant 4\pi(g-1)\ .$

Proposition 14 applied to $\mathcal{S}=\mathcal{S}_{h}(M,\varepsilon)$ —using
the upper bound given by theorem 3.3— yields the result.

###  6.3 The upper bound in constant curvature

When $(M,h_{0})$ is hyperbolic, the previous upper bound gives
$E(M,h_{0})\leqslant 8$, we however have a finer estimate.

######  Proposition 16.

For the hyperbolic metric $h_{0}$, we have $E(M,h_{0})\leqslant 2$.

######  Proof 6.2.

For a closed minimal surface in $\mathcal{S}_{h_{0}}(M,T,\varepsilon)$, recall
that by Theorem 4.4.2

$\lambda(S)\leqslant
C_{0}\log(1+\varepsilon)=\mathrel{\mathop{:}}\eta(\varepsilon)\ .$

Thus by the Gauß–Bonnet formula

$4\pi(g-1)=\int_{S}(1+\lambda(S)^{2}){\rm
d}\operatorname{area}\leqslant(1+\eta(\varepsilon)^{2})\operatorname{Area}(S)\
.$

From proposition 14 applied to
$\mathcal{S}=\mathcal{S}_{h_{0}}(M,\varepsilon)$, we get that

$E_{\varepsilon}(M,h_{0})\leqslant 2(1+\eta(\varepsilon)^{2})\ ,$

and the result follows.

###  6.4 The lower bound

######  Proposition 17.

Let $h$ be any Riemannian metric of curvature less than $-1$ on a $3$-manifold
$M$ then $E(M,h)\geqslant 2$.

######  Proof 6.3.

Since the curvature of a minimal surface $S$ is less than that of the ambient
manifold by Gauß equation, in that case we get that all minimal surfaces have
curvature less than $-1$. Hence by Gauß–Bonnet

$4\pi(g-1)\geqslant\operatorname{Area}(S)\ .$

Applying proposition 14 yields the inequality.

###  6.5 The equality case

The equality case is the rigidity result for the asymptotic counting of
[Calegari:2020uo]. Let $h$ be a metric of curvature less than $-1$. Assume
that $E(M,h)=2$. Then $h$ is hyperbolic. We assume in the sequel that $h$ is a
metric of curvature less than $-1$ with $E(M,h)=2$. We give a proof in the
simpler case when $h$ is close to a hyperbolic metric, so that Theorem 4.5.1
holds.

####  6.5.1 Finding surfaces which are more and more hyperbolic

Let $S$ be a closed surface. Let $G_{S}(g)$ be the set of connected finite
covers of $S$ of genus less than $g$, and $G_{S}$ the set of all connected
finite covers. Then we have

######  Proposition 18.

There is a constant $c_{1}$ only depending on $S$, so that for $g$ large
enough

$\sharp(G_{S}(g))\geqslant(c_{1}g)^{2g}\ .$

######  Proof 6.4.

By [Muller:2002tt], the number of index $n$ subgroups of the fundamental group
$\pi_{1}(S)$ of a genus $g_{0}$ orientable surface grows like
$2n(n!)^{2g_{0}-2}(1+o(1))$. On the other hand, the genus $g$ of a surface
whose fundamental group has index $n$ in $\pi_{1}(S_{0})$ is $g=n(g_{0}-1)+1$.
It follows that

$\sharp(G_{S}(g))\geqslant(c_{1}g)^{2g}\ .$

where $c_{1}$ only depends on $g_{0}$.

######  Proposition 19.

For every positive integer $m$, let $S_{m}$ be a finite union of stable
minimal surfaces $S^{1}_{m},\ldots,S^{p}_{m}$ with
$\varepsilon(S^{i}_{m})\leqslant\frac{1}{m}$. Let $\lambda^{i}_{m}$ be
positive numbers so that

$\sum_{i=1}^{p_{m}}\lambda^{i}_{m}=1\ .$

Then

$\lim_{m\to\infty}\sum_{i=1}^{p_{m}}\lambda^{i}_{m}\frac{\operatorname{Area}(S^{i}_{m})}{4\pi(g^{i}_{m}-1)}=1\
.$ (22)

where $g^{i}_{m}$ is the genus of $S^{i}_{m}$.

######  Proof 6.5.

Since Gauß–Bonnet formula gives
$4\pi(g^{i}_{m}-1)\geqslant\operatorname{Area}(S^{i}_{m})$, we have

$\limsup_{m\to\infty}\sum_{i=1}^{p_{m}}\lambda^{i}_{m}\frac{\operatorname{Area}(S^{i}_{m})}{4\pi(g^{i}_{m}-1)}\leqslant
1\ .$

Assume now that the limit in the equation (22) is smaller than $k$, with
$k<1$, then for arbitrarily large $m$, we can find $i_{m}$ in
$\\{1,\ldots,p_{m}\\}$ so that

$k4\pi(g^{i_{m}}_{m}-1)\geqslant\operatorname{Area}(S^{i_{m}}_{m})\ .$ (23)

As in the beginning of paragraph 6.3, let $G_{m}$ be the set of of connected
finite covers of $S^{i_{m}}_{m}$, and $G_{m}(g)$ the set of connected finite
covers of genus $g$. Obviously inequality (23) holds for all surfaces in
$G_{m}$.

Thus we could apply proposition 14 —using proposition 18— to get

$E_{\varepsilon}(M,h)\geqslant\frac{2}{k}>2\ ,$

and our contradiction.

####  6.5.2 From more and more hyperbolic to more and more totally geodesic

The previous proposition has a consequence for laminar measures

######  Proposition 20.

Let $\\{\mu_{m}\\}_{m\in\mathbb{N}}$ be a sequence of laminar probability
measures on ${\mathcal{F}_{h}(M)}$ converging to a laminar measure
$\mu_{\infty}$. Assume that each $\mu_{m}$ is supported on finitely many
closed leaves of $\mathcal{F}_{h}(M,\frac{1}{m})$. Then $\mu_{\infty}$ is
supported on the set of totally geodesic maps from ${\bf H}^{2}$ to $M$ whose
boundaries are circles.

######  Proof 6.6.

Let us consider the function $F$ on $\mathcal{F}_{h}(M)$ defined by
associating to a conformal minimal immersion $\phi$ of ${\bf H}^{2}$ in $M$,
the conformal factor $F(\phi)$ of $\phi$ at $i$.

Assume that $\phi_{0}$ is equivariant under a representation $\rho$ of a
Fuchsian group $\Gamma$ and if $S$ is the image of $\phi_{0}$ in $M$, we have

$\frac{\operatorname{Area}(S)}{4\pi(g-1)}=\int_{\mathcal{F}_{h}(M)}F{\rm
d}\delta_{\phi_{0}}\ .$

Then, proposition 19 tells us that

$\lim_{m\to\infty}\Bigl{(}\int_{\mathcal{F}_{h}(M)}F\ {\rm
d}\mu_{m}\Bigr{)}=1\ ,\hbox{ hence }\int_{\mathcal{F}_{h}(M)}F\ {\rm
d}\mu_{\infty}=1\ .$

By the Ahlfors–Schwartz–Pick Lemma since the curvature of $S$ is less than
$-1$, we have the inequality $F\leqslant 1$. It follows that $\mu_{\infty}$ is
supported on the set $F=1$. In particular for any $\phi$ in the support of
$\mu_{\infty}$, $\phi$ is an isometric immersion. Thus the curvature of the
image of $S$ is $-1$. Since the curvature of $M$ is less than $-1$, this only
happens when $S$ is a totally geodesic hyperbolic disk.

Finally, $\mu_{\infty}$ is supported on the intersection for all $m$ of
$\mathcal{F}_{h}(M,\frac{1}{m})$. Since the map $\partial$ is continuous
(corollary 5.1), this intersection is $\mathcal{F}_{h}(M,0)$.

Thus the support of $\mu_{\infty}$ is contained in the set of conformal
isometries into totally geodesic disks whose boundaries are circles.

####  6.5.3 Conclusion

In this conclusion, we finally use the restricting hypothesis that $h$ is
close to a hyperbolic metric so that Theorem 4.5.1 holds.

Let $\\{{\delta^{0}_{m}}\\}_{m\in\mathbb{N}}$ be the Kahn–Marković sequence of
laminar measures obtained in Theorem 1.2 for $\mathcal{F}_{h_{0}}(M)$. For
every $m$, let us write

$\delta^{0}_{m}=\sum_{i=1}^{m}\lambda^{i}_{m}\delta_{\phi^{i}_{m}}\ ,\hbox{
with }\sum_{i=1}^{m}\lambda^{i}_{m}=1\ ,$

where the $\phi^{i}_{m}$ are stable conformal immersions in $(M,h_{0})$
equivariant under a cocompact group. Let

$\delta_{m}=\sum_{i=1}^{m}\lambda^{i}_{m}\delta_{\psi^{i}_{m}}\ .$

where $\psi^{i}_{m}$ is a stable conformal immersion in $(M,h)$ equivariant
under a cocompact group and homotopic to $\phi^{i}_{m}$.

Let us extract a subsequence so that $\\{\delta_{m}\\}_{m\in\mathbb{N}}$
converges to $\mu_{\infty}$, while, by the Equidistribution Theorem 1.2,
$\\{{\delta^{0}_{m}}\\}_{m\in\mathbb{N}}$ converges to $\mu_{Leb}$.

Let us consider the projections $\pi_{h}$ and $\pi_{h_{0}}$ as in proposition
13 and observe that

$\pi_{h}(\delta_{m})=\pi_{h_{0}}(\delta^{0}_{m})\ .$

Thus by taking limits and using the continuity of $\pi_{h}$ and $\pi_{h_{0}}$,
we have

$\pi_{h}(\mu_{\infty})=\pi_{h_{0}}(\mu_{Leb})\ ,$

and in particular $\pi_{h}(\mu_{\infty})$ and $\pi_{h_{0}}(\mu_{Leb})$ have
the same support. We conclude by making two observations

  1. (i)

the support of $\pi_{h_{0}}(\mu_{Leb})$ is the set of all circles,

  2. (ii)

any circle in the support of $\pi_{h_{0}}(\mu_{\infty})$ bounds a totally
geodesic hyperbolic plane by proposition 20.

Thus every circle at infinity bounds a totally geodesic hyperbolic disk in
$M$, hence by proposition 1, $h$ is hyperbolic.