数理解析研究所講究録
第 1991 巻 2016 年 117-129

117

VOLUMES OF CONVEX CORES OF HYPERBOLIC 3-MANIFOLDSGREG MCSHANE
ABSTRACT. We discuss recent results on volume of the convex core of an incompressible
hyperbolic 3 manifold. By work of Brock the volume is related to Weil-Pctcrsson translation distance. Krasnov and Schlenker have shown that the volume is “commensurable”
with another quantity -the renormalized volume. We give an exposition of the behaviorof both quantities infinitesimally and on globally.

1. INTRODUCTION
For the last thirty years, since the pioneering work of Jorgensen and Thurston in the, the geometry of hyperbolic 3 manifolds have been the focus of abundant research.
There are three main reasons for this. Firstly, Thurston proved that unless a compact 3
manifold satisfies a certain obstruction condition, specifically it contains an incompressible
torus or a 2-sphere, then it must be hyperbolic. Secondly, by Mostow-Prasad rigidity ahyperbolic structure of finite volume in dimension 3 is unique and geometric invariants
are in fact topological invariants. Thirdly, Thurston and his students have given us avariety of tools, e.g. SNAPPEA, for constructing hyperbolic 3 manifolds and determining. volumes. The behavior of volume as thethese geometric invariants–in particularunderlying manifold changes has been much studied. It was observed by Thurston andJorgenson that given an upper bound on volume and a lower bound on injectivity radius
there are only finitely many compact hyperbolic manifolds satisfying these bounds.
consequence of Thurston’s Dehn surgery theorem is that there such are infinitely many
distinct hyperbolic manifolds of volume below the volume of the Whitehead link (or any
other cusped manifold) see [14] for a more detailed account. Neumann and Zagier [23] gave
a formula for the volume under Dehn filling. More recently volumes have been studied
in relation to the conjecture of Kashaev-Murakami [21]. Thus the structure of the set ofvolumes is rich and quite delicate.
One of Thurston’s constructions concerns the mapping torus of a surface diﬀeomorphism. Let
be an orientable surface of genus with $m$ punctures. We
admits a Riemannian metric of constantwill suppose that $3g-3+m\geq 1$ so that$-1$
, a hyperbolic structure of finite area, which, by Gauss-Bonnet, satisfies
curvature
Area $\Sigma=2\pi|\chi(\Sigma)|=2\pi(2g-2+m)$ with respect to the hyperbolic metric. The isotopy classes of orientation preserving automorphisms of , called mapping classes, wereclassified into three families by Nielsen and Thurston [27], namely periodic, reducible andpseudo-Anosov. Choose a representative of a mapping class , and consider its mapping$70s$$thei_{1}$$A$$\Sigma=\Sigma_{g,m}$$g$$\Sigma$$\Sigma$$h$$\varphi$2010 Mathematics Subject Cassification. Primary $57M27$ , Secondary $37E30,$ $57M5S.$

Key words and phrases. mapping class, entropy, mapping torus, Teichmtiller translation distance,
Weil Petersson translation distance, hyperbolic volume.
The author is partially supported by the ANR $Mo(iGrou\iota)$ .
Received January 24, 2016.

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torus,$\Sigma\cross[0, 1]/(x, 1)\sim(h(x), 0)$.

Since the topology of the mapping torus depends only on the mapping class , we denote
Anosov diﬀeomorphism preserves a. No power of a
its topological type byhomotopy class of essential closed curves on the surface. This means that the mapping
torus does not contain an incompressible torus. Now a theorem of Thurston [28] asserts
that
admits a hyperbolic structure iﬀ is pseudo-Anosov.
$\varphi$$pseudo\ovalbox{\tt\small REJECT}$$N_{\varphi}$$N_{\varphi}$$\varphi$1.1. Quasi-Fuchsian space the proof. There are several proofs of this theorem in thelitterature. The basic idea is to prove a fixed point theorem for the action of the diﬀeomorphism on the space of hyperbolic structures on
the infinite cyclic covering space of. The manifold
is homeomorphic to a productand to each hyperbolic structure on there is a corresponding hyperbolic structure on. By the Uniformization
Theorem the hyperbolic structure on induces an isometry between and the hyperbolic
plane $H$ the group of deck transformations is identified with a discrete subgroup of theisometries of H. Any inclusion of $H$ into 3 dimensional hyperbolic spaceinduces an
inclusion of its group of isometries into isomby theand the quotient of theimage of the deck transformations, is homeomorphic to. Any structure obtained inthis way is called a Fuchsian. Since there is a unique hyperbolic metric in each conformalclass the Fuchsian structures are in 1-1 correspondence with the points of $T$ , the Teichmuller space of the surface. In fact, the set of all hyperbolic structures on
is in 1-1
correspondence with the product $T\cross T$ . One way to see this is to consider the action of
the group on
conformal boundary of. Hyperbolic space is homeomorphic toa disc and its confromal boundary can be identified with the Riemann sphere. The actionof extends toand is identified with a subroup of conformal automorphisms of thesphere. The limit setof , that is the smallest, closed -invariant subset of, isa topological circle which separates the sphere into two discs
The quotient
and
space
, each homeomorphic to . The conformal
is a pair of Rieman surfaces
structures on. It can be proved that any such strucand
defines a point inture is geometrically finite that is the associated group of deck transformations admits afinite sided fundamental region. Some additional work is needed to prove the so-called
Ahlfors-Bers Theorem which asserts that the geometrically finite hyperbolic structures on
are in 1-1 correspondence with. Historically, because of the approach adopted
by Ahlfors-Bers via quasi-conformal deformations of , this set is called quasi-Fbchsian
space and there is a bijection{geometrically finite hyperbolic structures on}$\tilde{N}_{\varphi}$$\Sigma\cross R$$\tilde{N}_{\varphi}$$N_{\varphi}$$\Sigma$$\tilde{N}_{\varphi}$$\tilde{\Sigma}$$\Sigma$$H^{3}$$H^{3}$$(H^{3})$$\Gamma,$$\tilde{N}_{\varphi}$$\tilde{N}_{\varphi}$$H^{3}$$\partial_{\infty}H^{3}$$\Gamma$$\partial_{\infty}H^{3}$$\Gamma$$\Lambda(\Gamma)$$\Gamma$$\partial_{\infty}H^{3}$$\Gamma$$H^{+}$$H^{-}$$X^{\pm}$$H^{\pm}/\Gamma$$X^{+}$$X^{-}$$\tilde{N}_{\varphi}$$\Sigma$$\mathcal{T}\cross \mathcal{T}$$\mathcal{T}\cross \mathcal{T}$$\Gamma$$\mathcal{T}\cross \mathcal{T}$$arrow$$\tilde{N}_{\varphi}$$(X^{+},X^{-}) \mapsto QF((X^{+},X^{-})$ .

To avoid complicating the discussion with considerations of non-compact sets we suppose for the instant that X is compact. Choose a pseudo-Anosov automorphism on X
and choose a marked Riemann surface
on the Teichm\"uller geodesic invariant by. The diﬀeomorphism acts naturally on $T$ by pre-composing
on the marking ofand consider a family of quasi-Fuchsian manifolds
The intuition behind Thurston’s theorem is that, for is suﬃciently large:is (quite close”’ to the infinite cyclic covering space ofthe map induces a map onthat is “quite close” to being an
isometry on a big subset”’$\varphi$$X\in \mathcal{T}$$\varphi$$\varphi^{-1}$$\varphi$$X\in \mathcal{T}$$\{QF(\varphi^{-n}X, \varphi^{n}X);n\in \mathbb{Z}\}.$$n$$\bullet$$QF(\varphi^{-n}X, \varphi^{n}X)$$\bullet$$\varphi$$N_{\varphi}$$QF(\varphi^{-n}X, \varphi^{n}X)$119
By passing to the limit one should obtain a hyperbolic structure on
induced by such that the quotient space is homeomorphic toa hyperbolic structure from the covering map.

$N_{\varphi}$$\varphi$$\varphi_{\infty}$

and an isometry
so that it inherits$\tilde{N}_{\varphi}$$C_{n}:=C(QF(\varphi^{-n}X, \varphi^{n}X))$-bilipschitz$(1b\epsilon)$$\overline{N_{\varphi}}$

FIGURE 1. The sequence of (convex cores) quasi Fuchsian manifolds isnearly isometric to a big subset of the infinite cyclic cover$\tilde{N}_{\varphi}$

We see from the above that, even if we are only really
1.2. Geometry of $QF((X^{+},$ $X$
interested in compact hyperbolic 3 manifolds, it is natural to study hyperbolic structures
on non-compact manifolds. The topology of $QF((X^{+}, X^{-})$ is very simple since it is justthe product of a surface with R. However, the geometry of $QF((X^{+}, X^{-})$ varies subtly$X$
When Thurston and Jorgensen began their workwith the conformal structures
little was known and progress was slow until several breakthroughs in the early $2000s$ andsince then our understanding was greatly improved. The main result to be cited here is
Brock, Canary and Minsky’s solution of the Ending Lamination Conjecture. The ending
lamination conjecture is, in a sense, a generalization of the Mostow Rigidity Theorem
to hyperbolic manifolds of infinite volume. Mostow rigidity theorem asserts that thefundamental group determines the manifold up to isometry when it is of finite volume As
we have seen above, for $QF((X^{+}, X^{-})$ the fundamental group is not enough to determine$X$
the manifold: one also needs to know the conformal structures on the surfaces
Another, more subtle, case is that of non geometrically finite structures the limit set of
into two discs so that we do not have a pair of conformal
does not separate
perhaps none if is a space filling curve–for example for
or
just
one
but
structures. Thurston introduced the notation of ending
the group of deck transformations of$(X^{+},$$(X^{+},$$\partial_{\infty}H^{3}$$\Gamma$$X^{\pm}$$\Gamma$$\Gamma$$\tilde{N}_{\varphi}$

120
laminations for such groups and conjectured that these would determine the hyperbolic
structure.
Very roughly an ending lamination encodes which closed curves are “short” in the an
end of a non-geometrically finite 3 manifold. A geodesic lamination on can be thoughtof as the limit of closed simple curves . For example, consider the hyperbolic structureonand pick any closed simple curve , the systole for example, and define$\varphi$$\gamma_{n}$$\tilde{N}_{\varphi}$$\gamma_{0}$$\gamma_{n}:=\varphi_{\infty}^{n}(\gamma_{0})$.

Now recall thatacts by isometry so that all the curves
have the same length. If
we consider the sequences of closed geodesics on$n>0$ and$n<0$ then they
converge respectively to the stable and unstable laminations for . It is not hard to see
that
has exactly two ends and that the curves$n>0$ and$n<0$ “exit” themanifold via diﬀerent ends. In fact if$n>0$ is any sequence of geodesics inofuniformly bounded lenght exiting by the same end as$n>0$ then it will converge tothe stable lamination too.
$\varphi_{\infty}$$\gamma_{n}$$\Sigma\gamma_{n},$$\gamma_{n},$$\varphi$$\tilde{N}_{\varphi}$$\gamma_{n},$$\gamma_{n},$$\tilde{N}_{\varphi}$$\alpha_{n},$$\gamma_{n},$2. GEOMETRY OF THE CONVEX CORE

FIGURE 2. The quasi-Fuchsian manifold, the surfaces at infinity and theconvex core.
The convex core $C(QF((X^{+}, X of a QF((X^{+}, X^{-})$ is defined to be the smallest(non empty) closed, geodesically convex subset. There is a natural construction for theconvex core as follows: if $QF((X^{+}, X^{-})$ is the quotient of H3 by a discrete group then$C(QF((X^{+}, X is the$ quotient, the convex hull of the limit set, by . If X
is compact then $C(QF((X^{+},$ $X$
is compact and is in fact a compact core in the senseof Scott, that is the inclusion is a homotopy equivalence, so that it carries the topology of$C(QF((X^{+},$ $X$
It is clear that there are many advantages to working with compact
manifolds especially when considering convergence of sequences.
According to a theorem of Thurston the boundary of the convex core $\partial C(QF((X^{+},$ $X$
consists of a pair of surfaces
each homeomorphic to . An important case is whencorresponds to a Fuchsian structure and then $\partial C(QF((X^{+},$ $X$
is totally geodesic(and the convex hull of the limit set is just a copy of H.) Otherwise
are not smoothly
embedded in $QF((X^{+},$ $X$
the set of singular points is a lamination consists of a geodesic
lamination called the pleating lamination. The simplest case of a pleating lamination is a$\Gamma$$of C(\Lambda(\Gamma))$$\Sigma^{\pm}$$\Gamma$$\Sigma$$X^{\pm}$$\Sigma^{\pm}$121single simple closed curve and it is not hard to see that the surface must be “bent along
this curve and that their is a dihedral angle between the support planes that intersect
in this curve. In general the bending occurs along a geodesic lamination with noclosed leaves and the (dihedral angle” is replaced by a transverse bending measure. This
which measures how much the arc deviates
a numberassociates to an arc$X$$QF((X^{+},$
from a geodesic segment in$\lambda$$\alpha C\Sigma^{\pm}$$i(\alpha, \lambda)$

2.1. Comparing boundaries. A very diﬃcult question is to understaI} how the geom. Sullivan began thesc investivaries withetry of $\partial C(QF((X^{+}, X$
from the conformal
gations, conjecturing that the nearest point retraction
boundary equipped with its Poincar\’e metric to the boundary of the convex core with its
induced metric was 2-bi-Lipschitz. Many people worked on this question, notably EpsteinMarden, Epstein-Marden-Markovic and Bridgeman. Although the conjecture is false, ina very elegant treatment, Bridgeman shows that there is a $1+K$-Lipschitz homotopy
inverse for where $K=\pi/\sinh^{-1}(1)$ . To prove this he introduces the notiuon of average. If a is a geodesic arc in $\partial C(QF((X^{+},$ $X$
bending for the pleating lamination
is defined to be the bending per unit length, or specificallythen the average bending$d$$(X^{+}, X^{-})\in \mathcal{T}\cross \mathcal{T}$$r:X^{\pm}arrow\Sigma^{\pm}$$r$$\lambda^{\pm}c\Sigma^{\pm}$$B(\alpha)$$B( \alpha):=\frac{i(\alpha,\lambda)}{l(\alpha)},$is the length of the arc. Bridgeman’sis the intersection number andwhereapproach is based on bounding average bending and he proves the following:$l(\alpha)$$i(\alpha, \lambda)$

Theorem 2.1. Let $K=\pi/\sinh^{-1}(1)$ then for any closed geodesic$\alpha\subseteq\partial C(QF((X^{+},$ $X$$B(\alpha)\leq K.$

3. VOLUME OF THE CONVEX COREAs we noted above, if is compact then the convex core of $QF((X^{+},X^{-})$ is compact
and so its volume well defined. In fact, the notion of geometrical finiteness can be definedin terms of the finiteness of the volume of an -neighborhood of $C\langle QF((X^{+},X^{-}))$ .
There are two questions which come to mind immediately:
How does the volume vary on a smﬃ scale i.e. infinitesimally?
How does the volume vary on a large scale
The first of these was studied by Bonahon and the second by Brock and we describe
there approaches below.
$\Sigma$$\epsilon$$\bullet$$\ovalbox{\tt\small REJECT}$$\bullet$

3.1. Variational formula. In 3-dimensional hyperbolic geometry, the classical Schlafli
formula expresses the variation of the volume of a hyperbolic polyhedron in terms of thelength of its edges and of the variation of its dihedral angles. Bonahon [4] proves anand more generally
analogous formula for the variation of the volume $C(QF((X^{+},$ $X$, as we vary the
3-manifold
hyperbolic
finitegeometricallyfor the convex core of a
hyperbolic metric of M. What is diﬃcult is taking account of the way in which the pleating
lamination varies. Bonahon does this by showing that the variation of the bending of theboundary of the convex core is described by a geodesic lamination with a certain transverse
distribution. He proves that the variation of the volume of the convex core is then equal
to 1/2 the length of this transverse distribution.
Bonahon’s approach is elegant but it seems diﬃcult to extract “large scale” estimatesfor volume from his formula.
$M$

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3.2. Comparison with Weil-Petersson distance. There are two natural metrics on
Teichmuller space $T$:
The Teichmuller metric , which is the solution to an optimisation problem: itis theof the minimal quasiconformal dilation of maps : $Xarrow Y.$. This is a Riemanian metric and to define it one
The Weil-Peterson metricmust first discuss the tangent space to which we do in the Appendix following
the approach of Bers’.
$d_{T}$$\bullet$$f$$\log$$d_{wp}$$\bullet$$\mathcal{T}$

3.2.1. Comparing the two metrics. Both theses metrics are natural in the sense that every
mapping class is an isometry for the metric. The Teichmuller metric is complete whereas
the Weil-Petersson metric though geodesically convex but not complete. In fact there is a
Finsler metric on the tangent space to which induces the Teichmuller distance. Using
this Linch proved in her thesis that there is an inequality relating the two distances are
related:$\mathcal{T}$Theorem 3.1.
$d_{wp}\leq|2\pi\chi(\Sigma)|^{\frac{1}{2}}d_{T}.$3.2.2. Comparing with the pants complex. Much later, Brock showed that

equippedwith the Weil-Petersson metric is quasi-isometric to the pants complex. The pants complex
is just a graph the edges of which are pants decompositions of the surface and theedges are pairs of vertices which are related by an elementary move (see figure). Since it
is a graph the pants complex comes equipped with a simplicial metric . Recall that the
Ber’s constant of a surface is a number $L>0$ such that for any hyperbolic metric on
there is a pair of pants of length less than . Buiding on the ideas of Minsky, Brockdefines a “rough projection”’$\mathcal{T}$$\mathcal{P}$$\Sigma$$d_{\mathcal{P}}$$\Sigma$$L$$\Sigma$$\pi:\mathcal{T}arrow \mathcal{P},$which takes a conformal structure $X$ to a pair of pants $P$ of length less than the Bers’constant for the Poincar\’e metric in the class of $X.$

Theorem 3.2 (Brock). Let
such that for all$\Sigma$be a compact surface then here exists $K_{1}>1,$ $K_{2}>0$$(X^{+}, X^{-})\in \mathcal{T}\cross \mathcal{T}$$\frac{1}{K_{1}}d_{wp}(X^{+}, X^{-})-K_{2}\leq d_{\mathcal{P}}(\pi(X^{+}), \pi(X^{-}))\leq K_{1}d_{wp}(X^{+}, X^{-})+K_{2}$So, on a large scale, Teichmueller space is modelled on the pants complex . With
some additional work he proves the following comparison theorem for distance and volume.
$\mathcal{T}$Theorem 3.3 (Brock). Let
such that for all$\Sigma$$\mathcal{P}$be a compact surface then here exists $K_{1}>1,$ $K_{2}>0$$(X^{+}, X^{-})\in \mathcal{T}\cross \mathcal{T}$$\frac{1}{K_{1}}d_{wp}(X^{+}, X^{-})-K_{2}\leq volC(QF((X^{+}, X \leq K_{1}d_{wp}(X^{+}, X^{-})+K_{2}$It is important to note that though, by applying Linch’s Theorem above, we get upperbound for volume in terms of Teichmuller distance no lower bound is possible.
It is well known that there is a family of pseudo-Anasov automorphisms
such thatare bounded whilst the entropy of , which is just the translation distancefor the Teichm\"uller metric, diverges. For example, if is the once punctured torus the
mapping class group is isomorphic to $PSL(2,$ Consider the sequence of diﬀeomorphisms
defined as follows:$\varphi_{k}$$||\varphi_{k}||_{WP}$$\varphi_{k}$$\Sigma$$\mathbb{Z}$123$\varphi_{k}:=(\begin{array}{lll}1+ k 1k 1\end{array})=(\begin{array}{ll}1 10 1\end{array})(\begin{array}{ll}1 0k 1\end{array})$is pseudo-Anosov and that the
It is easy to see that for $k>0$ the mapping class$karrow\infty$ .
However,the volume of the sequenceassequence of dilatations tends toconverge to a cusped
is bounded. In fact, the mapping toriof mapping tori
hyperbolic 3-manifold.
In light of this example and Brock’s Theorem above one sees that the relationshipbetween volume and Weil-Petersson distance is stronger than that with the Teichm\"uller
distance.
$\varphi_{k}$$\infty$$N_{\varphi_{k}}$$N_{\varphi_{k}}$4. $W$ -VOLUME AND RENORMALIZED VOLUME
The renormalized volume Rvol is a numerical invariant associated to an infinite-volume
Riemannian manifold with some special structure near infinity, extracted from the divergent integral of the volume form. Early instances of renormalized volumes appear in
Henningson-Skenderis for asymptotically hyperbolic Einstein metrics, and in Krasnov for
Schottky hyperbolic 3-manifolds. Renormalized volume of convex cocompact hyperbolic
3-manifolds were studied extensively by Krasnov and Schlenker in [17] using a geometricconstruction due to C. Epstein for studying ends.
From our point of view the interest of renormalized volume is that it is commensurable”
with the usual hyperbolic volume, that is they diﬀer by a bounded quantity only dependingon the topology of the surface, and that it has a very nice Schaﬄi formula. Both theseobservations are due essentially to Krasnov and Schlenker.
$\langle$

FIGURE 3. The ends of a quasi-Fuchsian manifold can be foliated bysurfaces equidistant to the boundary of the convex core.

$C^{1,1}$

is compact. Let $M$ be
4.0.1. Definition. To simplify the exposition we assume that
. Following [17] we say that ahomeomorphic toa quasi-Fuchsian manifold$N\subset
M$ is strongly convex if thecodimension-zero smooth compact convex submanifold
to the boundary at infinity
normal hyperbolic Gauss map fromis a homeomorphism. For example, a closed -neighborhood of the convex core of a quasithen there is a family of surfaces
Fuchsian manifold is strongly convex. Let$\Sigma$$\Sigma\cross \mathbb{R}$$\mathbb{H}^{3}/\Gamma$$\partial N=\partial_{+}N\sqcup\partial_{-}N$$\epsilon$$S_{0}=\partial N$$\Omega_{\Gamma}/\Gamma$

124
equidistant to foliating the ends of $M$ . If denotes the induced metric onthen define a metric at infinity associated to the family
by$\{S_{r}\}_{r\geq 0}$$S_{0}$$g_{r}$$S_{r},$$\{S_{r}\}_{r\geq 0}$$g= \lim_{rarrow\infty}2e^{-2r}g_{r}.$The resulting metric in fact belongs to the conformal class at infinity that is the conformal structure determined by the complex structure on. It is easy to see that if
we start with a strongly convex submanifold bounded byfor some $r_{0}>0$ , then thelimiting metric is. Namely, if we shift the parametrization of an equidistant foliation
by , then the limiting metric changes only by scaling
Conversly, if is a Riemannian metric in the conformal class at infinity, then Theorem
5.8 in [17] shows that there is a unique foliation of the ends of $M$ by equidistant surfaces
with compatible parametrization of leaves starting
so that the associated metric at
infinity is equal to . Notice that the parametrization may have to start with a positive. The construction of a foliation is due to Epstein [11]. Then, a natural functional in
the context of strongly convex submanifolds $N\subset M$ is the $W$-volume defined by$g$$\Omega_{\Gamma}/\Gamma$$S_{r0}$$e^{2r_{0}}g$$e^{2r_{0}}.$$r_{0}$$g$$r_{0}\geq 0$$g$$r_{0}$(4.1)$W(M, g) := volN_{r}-\frac{1}{4}\int_{S_{r}}H_{r}da_{r}+\pi r\chi(\partial M)$,

where the parametrization is induced by
is a strongly convex submanifold boundedby the associated leaf
is the mean curvature of andis the induced area formof . Note that, strictly speaking,is only well defined on a set of full measure as
the surfaces
are notbut onlysee [6] for a discussion. A simple computation$W$
which shows that the -volume depends only on the metric at infinity , justifying thenotation.
The renormalized volume of $M$ is now defined by$r$$S_{r},$$g,$$N_{r}$$H_{r}$$S_{r}$$S_{r}$$da_{r}$$H_{r}$$C^{2}$$S_{r}$$C^{1,1}$$g$

Rvol $(M):= \sup_{g}W(M, g)$ ,where the supremum is taken over all metrics in the conformal class at infinity suchthat the area of each surface at infinity
with respect to is. Section
7 in [17] presents an argument, based on the variational formula, that the metric ofconstant curvature $-1$ is a critical point of the functional $W(M, g)$ and that this is alocal maximum. Guillarmou Moroianu Schlenker study the change of $W(M, g)$ underconformal transformation and prove that there is a unique maximum.
$g$$X^{\pm}$$g$4.0.2. Commensurability. When$2\pi|\chi(\Sigma)|$

is compact Schlenker, using Bridgeman’s bound on
average bending proves that the volume and the renormalized volume arc “commensurable”’ that is they diﬀer by a bounded quantity only depending (in the compact case) on
the topology of the surface. Bridgeman and Canary using a slightly modified approach
reprove this result and we have:

Theorem 4.1. Let(4.2)$\Sigma$$\Sigma$be a compact surface then Then,$volC(QF((X^{+}, X -D\leq$ Rvol Q $F((X^{+}, X^{-})\leq volC(QF((X^{+},$ $X$

with $D=9.185|\chi(\Sigma)|.$The proof of this theorem is interesting and shows why the $\sup$ in the definition isimportant. It hinges on the comparison of two metrics defined on the. The first of$p(z)|dz|$
Poncar\’ethese is themetricand the second the so-called the Thurston metric$X^{\pm}$

125$\tau(z)|dz|$ . According to a result of Herron, Ma and Minda thes metrics are 2 bi Lipschitz
and satisfy$\frac{1}{2}\tau(z)\leq p(z\rangle\leq r(z)$.

one obtains respecWhen one computes the corresponding $W$-volumes of $p(z)$ and$volC(QF((X^{+},
X-\frac{1}{4}L(\lambda)$
is the pleating
where
tively the renormalised volume andlamination, see Schlenker [24] for details.
$\tau(z)$$\lambda$

4.0.3.

$Schaﬄ_{\fbox{Error::0x0000}}i$formula. The tangent space ofBeltrami diﬀerentials

$\mu$

on

$X^{\pm}$$T\cross \mathcal{T}$can be identified with the space of(see appendix for details).

Theorem 4.2. Under an infinitesimal deformation of the complex structure $(X^{+}, X^{-})$
represented by the Beltrami diﬀerential$\mu$$d Rvo1_{(X+,X^{-})}(\mu)=-\frac{1}{2}{\rm Re}(q, \mu)=-\frac{1}{2}\int_{X^{\pm}}{\rm Re}\mu q$

holds where$q$is the Schwarzian associated to the projective structure.

5. MORE VOLUME ESTIMArES
5.1. Variational proof of inequalities. Brock’s Theorem above provides a comparison
theorem between volume of the convex core and distance in the pants complex. Distancein the pants complex is defined in a combinatorial manner and seems diﬃcult to computein general. Using Brock’s quasi-isometry between the pants complex and 7“‘ equipped withdoes not seem to yield much information in particular since the constants in Theorem
3.3 are not explicit. Schlenker applies his SchaMi formula to prove the following:$d_{wp}$

Theorem 5.1. Let$\Sigma$be a compact surface then$volC(QF(X^{+}, X \leq\frac{3}{2}|2\pi\chi(\Sigma\rangle|^{\frac{1}{2}}d_{wp}(X^{+}, X-)+9.185|\chi(\Sigma)|.$Each of the numbers on the right hand side has a geometric interpretation:
The term $9.185|\chi(\Sigma)|$ comes from Bridgeman’s bound on average bending.
comes from Nehari’s bound on the Schwarzian of a univalent map.
is the hyperbolic area of the surface
It is instl.uctive to see how Schlenker proves this formula$\bullet$$e \frac{3}{2}$$\bullet$$\Sigma.$$2\pi\chi(\Sigma)$

Proof. It suﬃces to prove
Ryol $QF(X, Y) \leq\frac{3}{2}|2\pi|\chi(\Sigma)|^{\frac{1}{2}}d_{w}p(X, Y)$ ,(5.1)

Let : $[fJ, d]arrow T$ bee the unit speed Teichmifller geodesic joining $X$ and , so that, inparticular, $Y(O)=X,$ $Y(d)=Y$ and $d=d_{T}(X_{7}Y)$ . If $\{QF(X, Y(t))\}_{0\leq t\leq d}$ denotes theassociated one-parameter family of quasi-Fuchsian manifolds then, applying Shlenker’s
SchaMi formula one has:
$Y$

$Y$

$d$

Rvol $( \dot{X},\dot{Y}(t))=-\frac{1}{2}{\rm Re}((q_{X}(t),\dot{X})+(q_{Y(t)}(t),\dot{Y}(t)))$ ,126
where ,
are the tangent vectors to the deformations of the complex structures oneach boundary. Integrating the variation of Rvol along the path $Y(t)(t\in[0, d])$ andusing the fact that $X(t)=Y(O)$ is constant, we obtain$\dot{X}$$\dot{Y}(t)$Rvol $QF(X, Y)=- \frac{1}{2}{\rm Re}\int_{t=0}^{d}(q_{Y(t)}(t),\dot{Y}(t))dt.$

The renormalised volume is a real number and so it suﬃces to bound the module of theright hand side of this equation. Let denote the hyperbolic metric on and consider,$R$$\rho$$|(q_{Y(t)}(t)) , \dot{Y}(t))|^{2}\leq\int_{R}|q_{Y(t)}(t)|^{2}\rho^{-}2\int_{R}|\dot{Y}(t)|^{2}\rho^{2}$$=( \int_{R}\frac{|q_{Y(t)}(t)|^{2}}{\rho^{4}}\rho^{2})\Vert\dot{Y}(t)\Vert_{wp}^{2}$$\leq\Vert q\Vert_{\infty}^{2}(\int_{R}\rho^{2})\Vert\dot{Y}(t)\Vert_{wp}^{2}$

Nehari’s Theorem [22] allows us to bound the factor
path $Y(t)$ just a Weil Petersson geodesic
The statement follows easily from these observation.

$||q_{Y(t)}(t)||_{\infty}$by$\frac{3}{2}$and since our$\Vert\dot{Y}(t)\Vert_{wp}=1.$$\square$5.2. Estimates from 3 manifolds. A more restricted problem is to fix a pseudo-Anosov
and compare how the volume of the convex core of $C(QF(\varphi^{-n}X, \varphi^{n}X))$ varies with
Kojima-McShane and independantly Brock-Bromberg prove the following:$n.$

Theorem 5.2. Suppose(6.2)$\Sigma$

is compact, then$|volC(QF(\varphi^{-n}X, \varphi^{n}X))-2nvolN_{\varphi}|$is uniformly bounded.

As mentioned in the introduction
should be quite close” toon alarge compact subset $K$ . The proof of the theorem is to estimate the size, an control thegeometry of such a compact set which isbi-Lipschitz to a subset of
Combining this with Schlenker’s estimate above yields :$QF(\varphi^{-n}X, \varphi^{n}X)$$\tilde{N}_{\varphi}$$(1+\epsilon)$

Corollary 5.3. If$\Sigma$$\tilde{N}_{\varphi}.$is compact, then:$\sqrt{2\pi|\chi(\Sigma)|}||\varphi||_{WP}\geq\frac{4}{3}volN_{\varphi}$holds for any pseudo-Anosov , where$\varphi$of$||\cdot||_{WP}$

is the Weil-Petersson translation distance$\varphi.$

Brock and Bromberg [9] observed that one can apply this inequality to get a lower
bound for the diameter of the moduli space of the once punctured torus
Corollary 5.4. The diameter of the moduli space of the once punctured torus is bounded
below by$\frac{1}{6}\sqrt{\frac{2}{\pi}}\mathcal{V}_{8}$where$\mathcal{V}_{8}$is the volume of the regular ideal octahedron in$H^{3}.$

127
The idea is to bound the Weil-Petersson distance between two points in frontier ofTeichmuller space. As explained in Paragraph 3.2.2 the Teichmuller metric and the WeilPetersson metric are not equivalent. Brock and Bromberg consider a sequence of pseudoAnosovs on the punctured torus:$\varphi_{k}:=(\begin{array}{lll}l+ k^{2} kk 1\end{array})=(\begin{array}{ll}l k0 l\end{array})(\begin{array}{ll}1 0k l\end{array}).$Note that the normaliser of$\{N_{\varphi_{k}}\}$

contains$J:=(\begin{array}{l}0-110\end{array}).$They observe that the volume of
converges to. Each of the admits an axis, thatis an invariant Weil-Petersson geodesic on which it acts by translation, $T$ is a proper space
the sequence. Since 7‘ is a proper space, after making an appropriate choice of basepoint
say the fixed point of , the sequence of these axes converges to a Weil-Petersson geodesic
joining two points in the frontiel . One applies Corollary 5.3 to the sequence
to geta iower bound on the length of We note that Wolpert had previously shown that thelength of was at mostObserve now that is invariant under so it double
covers (simple) geodesic in the moduli space and the so diameter is bounded below
the length of $I.$
$2\mathcal{V}_{8}$$N_{\varphi_{k}}$$\varphi_{k}$

$J$

$I$

$\cdot$$N_{\varphi_{k}}$

$I$

$I$

$2\sqrt{30}zr^{\frac{3}{4}}.$

$I$

$J$

5

$a$

6. THE HESSIAN OF THE RENORMALIZED VOLUME.
Ciobotaru-Moroianu [20] and Pallete [25] study the Hessian of the renormalized volume
in relation to the Fuchsian locus. Recalk that the Fuchsian locus in quasi-Fuchsian space isa submanifold such that the boundary of the convex core is totally geodesic. It is easy to
see that the convex core $C(QF((X^{+},X^{-}))is$ a totally geodesic embedding of the surface
and so its hyperbolic volume is zero. From general considerations, we have the following
characterisation of Fuchsian: $QF((X^{+}, X^{-})$ is Fuchsian if and only if the hyperbolic
volume of $C(QF((X^{+}, X is$ zero. $The$ boundary $of the$ convex core $of QF((X^{+}, X^{-})$is pleated or bent along a lamination and it is totally geodesic if and only if the lengthof this lamination is zero. Combining this with Schlenker’s inequality$volC(QF((X^{+}, X -L(\lambda)\leq RvolQF((X^{+}, X \leq volC(QF((X^{+}, X$(6.1)$\lambda$$L(\lambda)$

we see that if renormalized volume of $QF((X^{+}, X^{-})$ is zero when it is Fuchsian. It seems
natural to conjecture that the renormalized volume:
is non negative for all convex co-compact hyperbolic 3-manifolds
is minimal exactly when the boundary of the convex hull is totally geodesic.
Given the context, where the rcnormalized volume arises as a functional in physicalmodels, it is tempting to think of as being analogous to the mass of asymptoticallyEuclidean manifolds thus the first question is an analogue of the positive mass conjecture.
The second question is the analogue of a question of Bonahon for hyperbolizable 3manifolds with incompressible boundary. He conjectured that the volume of a convex
core is at least half the simplicia} volume of the doubled manifold. Storm, using ideas of
Souto and Besson Courtois-Gallot, solved the conjecture:$\bullet$$\bullet$Theorem 6.1 (Storm). If $N$ is a hyperbolic 3-manifold homotopy equivalent to $M$ then$volC(N)\geq\frac{1}{2}$

SimpVol $DM,$128
where $DM$ is the double of $M.$
Moreover, if $vol(CN)=\frac{1}{2}$ SimpVol(DM) $>0$ then $M$ is acylindrical, $N$ is convex
cocompact, andis totally geodesic.
$\partial C(N)$

Some progress has been made for almost Fuchsian manifolds, that is for hyperbolic
structures that are small deformations of a manifold with totally geodesic boundary.
Fuchsian manifolds embedd as the Morianu computed the Hessian (see also Palette [25]and shown that it is positive definite on the (normal bundle”’ to the Euchsian locus
in quasi-Fuchsian space. This means that for manifolds that are nearly Fuchian i.e.
small deformations of Fuchsian structures the renormalised volume is positive. Uhlenbeckgave a more formal definition: An almost-Fuchsian hyperbolic 3-manifold (X, g) is aquasi-Fuchsian hyperbolic 3-manifold containing a closed minimal surface whose principal
curvatures belong to $(-1,1)$ . Ciobotaru-Moroianu prove the following$)$

Theorem 6.2 (Ciobotaru-Moroianu). The renormalized volume of an almost Fuchsian
hyperbolic 3-manifold is non-negative. Further it is zero only at the Fuchsian locus.
It seems likely that the renormalized volume is always positive.

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