Survey on the Canonical Metrics on the Teichmüller Spaces and the Moduli
Spaces of Riemann Surfaces

CHAN, Kin Wai

A Thesis Submitted in Partial Fulfillment

of the Requirements for the Degree of

Master of Philosophy

in

Mathematics

The Chinese University of Hong Kong

September 2010

Thesis Assessment Committee

Professor LEUNG Nai Chung Conan (Chair)

Professor WAN Yau Heng Tom (Thesis Supervisor)

Professor AU Kwok Keung Thomas (Committee Member)

Professor CHEUNG Wing-Sum (External Examiner)

ACKNOWLEDGMENTS

I shall hereby give a sincere gratitude to my supervisor, Prof. Tom Y. H. Wan,
for his continuous support and encouragement to my research life during these
two years. With his aid, I gradually become more interested in and familiar
with geometry. Also, I am really thankful that he suggested me choosing this
fascinating topic as my thesis topic. Working on this topic actually let me
acquire a clearer picture of complex geometry and the structure of moduli
spaces. I would like to thank Prof. Thomas K. K. Au as well, for his useful
advice on my problems, both personal and academic. On the road of applying
foreign graduate schools he helped me a lot. Moreover, I wish to express my
appreciation to Prof. Conan N. C. Leung for his motivative lessons on
different topics in geometry. He is always able to bring us to the front of
the research of geometry, and my knowledge and insight in geometry are thus
broadened quite a lot. Finally I offer my heartfelt thanks to all people who
have helped me in any way, especially for those who gave me a hand for dealing
with the difficulties I met when I was working on this thesis, like Andy
Cheung, John Ma and Lijiang Wu.

Abstract

This thesis results from an intensive study on the canonical metrics on the
Teichmüller spaces and the moduli spaces of Riemann surfaces. The Teichmüller
space $T_{g}$ of Riemann surfaces of genus $g$ can be defined to be the set of
all the conformal equivalence classes of the Riemann surfaces of genus $g$
with quasiconformal mappings. It is well known that the moduli space
$\mathcal{M}_{g}$ of Riemann surfaces of genus g can be viewed as the quotient
of $T_{g}$ by the mapping class group $Mod_{g}$. There are several renowned
classical metrics on $T_{g}$ and $\mathcal{M}_{g}$, including the Weil-
Petersson metric, the Teichmüller metric, the Kobayashi metric, the Bergman
metric, the Carathéodory metric and the Kähler-Einstein metric. The
Teichmüller metric, the Kobayashi metric and the Carathéodory metric are only
(complete) Finsler metrics, but they are effective tools in the study of
hyperbolic property of $\mathcal{M}_{g}$. The Weil-Petersson metric is an
incomplete Kähler metric, while the Bergman metric and the Kähler-Einstein
metric are complete Kähler metrics. However, McMullen[20] introduced a new
complete Kähler metric, called the McMullen metric, by perturbing the Weil-
Petersson metric. This metric is indeed equivalent to the Teichmüller metric.
Recently, Liu-Sun-Yau[17] proved that the equivalence of the Kähler-Einstein
metric to the Teichmüller metric, and hence gave a positive answer to a
conjecture proposed by Yau[33]. Their approach in the proof is to introduce
two new complete Kähler metrics, namely, the Ricci metric and the perturbed
Ricci metric, and then establish the equivalence of the Ricci metric to the
Kähler-Einstein metric and the equivalence of the Ricci metric to the McMullen
metric. The main purpose of this thesis is to survey the properties of these
various metrics and the geometry of $T_{g}$ and $\mathcal{M}_{g}$ induced by
these metrics.

K n

b o g פ夤 A ڱN — Teichmüller Ŷ ξ Ҧ ҪŶ W d ׶q i ‘ J s C 欰$g$ Ҧ Teichmüller Ŷ
$T_{g}$ i H Q w q @ ӥѰt W F @ άM g A ӥB 欰$g$ Ҧ @ ε զ A Ӭ H O 欰$g$ Ҧ ҪŶ
$\mathcal{M}_{g}$ i H Q ѬM g s$Mog_{g}$ $T_{g}$ Ұ ӪŶ C b$T_{g}$
M$\mathcal{M}_{g}$ W n X Ӧ W g ׶q A ] AWeil-Petersson ׶q BTeichmüller ׶q
BKobayashi ׶q BBergman ׶q BCarathéodory ׶q MKähler-Einstein ׶q A 䤤Teichmüller
׶q BKobayashi ׶q MCarathéodory ׶q u O ] ƪ ^ⴵ ǫ׶q A L ̵̫o O ΨӬ
s$\mathcal{M}_{g}$ S ʪ n u CWeil-Petersson ׶q O @ Ӥ ƪ Kähler ׶q A Bergman ׶q
Kähler-Einstein ׶q h O ƪ Kähler ׶q F M ӡAMcMullen[20] H L ZWeil-Petersson ׶q k
޶i F @ ӷs Kähler ׶q A W McMullen ׶q C o ӫ׶q MTeichmüller ׶q O C ̵̪ ALiu-Sun-
Yau[17] ҩ FKähler-Einstein ׶q Teichmüller ׶q A ӥѦ F @ Ӫ֩w פ Yau[33] Ҵ X @ Ӳq Q
C L ҩ i O ޶i ӷs Kähler ׶q GRicci ׶q ηL ZRicci ׶q A M O إ Ricci ׶q MKähler-
Einstein ׶q ʤ Ricci ׶q MMcMullen ׶q ʡC o g פ媺 D n ت O h ҹ o Ǥ P ׶q S ʤΥ ̵̵
$T_{g}$ M$\mathcal{M}_{g}$ a ӤF ƻ ˪ X C

###### Contents

  1. 1 Introduction
  2. 2 Background Knowledge
    1. 2.1 Results from Riemann Surface Theory and Quasiconformal Mappings
      1. 2.1.1 Riemann Surfaces and the Uniformization Theorem
      2. 2.1.2 Fuchsian Groups
      3. 2.1.3 Quasiconformal Mappings and the Beltrami Equation
      4. 2.1.4 Holomorphic Quadratic Differentials
      5. 2.1.5 Nodal Riemann Surfaces
    2. 2.2 Teichmüller Theory
      1. 2.2.1 Teichmüller Spaces
      2. 2.2.2 Teichmüller’s Distance
      3. 2.2.3 The Bers Embedding
      4. 2.2.4 Teichmüller Modular Groups and Moduli Spaces of Riemann Surfaces
      5. 2.2.5 Infinitesimal Theory of Teichmüller Spaces
      6. 2.2.6 Boundary of Moduli Spaces of Riemann Surfaces
    3. 2.3 Schwarz-Yau Lemma
  3. 3 Classical Canonical Metrics on the Teichnmüller Spaces and the Moduli Spaces of Riemann Surfaces
    1. 3.1 Finsler Metrics and Bergman Metric
      1. 3.1.1 Definitions and Properties of the Metrics
      2. 3.1.2 Equivalences of the Metrics
    2. 3.2 Weil-Petersson Metric
      1. 3.2.1 Definition and Properties of the Weil-Petersson Metric
      2. 3.2.2 Results about Harmonic Lifts
      3. 3.2.3 Curvature Formula for the Weil-Petersson Metric
  4. 4 Kähler Metrics on the Teichmüller Spaces and the Moduli Spaces of Riemann Surfaces
    1. 4.1 McMullen Metric
      1. 4.1.1 Definition of the McMullen Metric
      2. 4.1.2 Properties of the McMullen Metric
      3. 4.1.3 Equivalence of the McMullen Metric and the Teichmüller Metric
    2. 4.2 Kähler-Einstein Metric
      1. 4.2.1 Existence of the Kähler-Einstein Metric
      2. 4.2.2 A Conjecture of Yau
    3. 4.3 Ricci Metric
      1. 4.3.1 Definition of the Ricci Metric
      2. 4.3.2 Curvature Formula of the Ricci Metric
    4. 4.4 The Asymptotic Behavior of the Ricci Metric
      1. 4.4.1 Estimates on the Asymptotics of the Ricci Metric
      2. 4.4.2 Estimates on the Curvature of the Ricci Metric
    5. 4.5 Perturbed Ricci Metric
      1. 4.5.1 Definition and the Curvature Formula of the Perturbed Ricci Metric
      2. 4.5.2 Estimates on the Curvature of the Perturbed Ricci Metric
      3. 4.5.3 Equivalence of the Perturbed Ricci Metric and the Ricci Metric
  5. 5 Equivalence of the Kähler Metrics on the Teichmüller Spaces and the Moduli Spaces of Riemann Surfaces
    1. 5.1 Equivalence of the Ricci Metric and the Kähler-Einstein Metric
    2. 5.2 Equivalence of the Ricci Metric and the McMullen Metric

##  Chapter 1 Introduction

This thesis is a survey on the canonical metrics, namely, the Teichmüller
metric $\omega_{T}$, the Kobayashi metric $\omega_{K}$, the Carathéodory
metric $\omega_{C}$, the Bergman metric $\omega_{B}$, the Weil-Petersson
metric $\omega_{WP}$, the McMullen metric $g_{1/l}$, the Kähler-Einstein
metric $g_{KE}$, the Ricci metric $\tau$ and the perturbed Ricci metric
$\tilde{\tau}$, on the Teichmüller spaces $T_{g}$ and the moduli spaces of
Riemann surfaces $\mathcal{M}_{g}$. These metrics will be introduced one by
one, and the geometric properties (say, the curvature bounds) related to some
of these metrics will also be reviewed. The equivalences between these metrics
will also be discussed. In fact, the main part of this thesis consists of the
curvature formulae of the two newest complete Kähler metrics, i.e., the Ricci
metric and the perturbed Ricci metric, and the estimates on their curvatures,
including the holomorphic sectional curvature, bisectional curvature and Ricci
curvature, by Liu-Sun-Yau[17][18]. The importance of these results will be
apparent when we come to the equivalence of the Ricci metric (and the
perturbed Ricci metric) to the Kähler-Einstein metric and to the McMullen
metric respectively. The essential point is that these two equivalences
together solve a conjecture proposed by Yau[33] twenty years ago. Now let me
introduce each of the chapters below, i.e., chapter 2 to chapter 5, in a more
detailed manner.

Chapter 2 is a preparation chapter. In this chapter, some of the background
knowledge which are useful or helpful to the remaining chapters is reviewed.
The background knowledge includes three parts - the first part being the
Riemann surface theory, the second part being the Teichmüller theory and the
third part being the Schwarz-Yau lemma - which are discussed in three
different sections. For the Riemann surface theory, the uniformization theorem
(Theorem 2.1.1) provides a way of modeling the Riemann surfaces. The Fuchsian
groups and the quasiconformal mappings will help define the Teichmüller
spaces, while the Beltrami coefficients and the holomorphic quadratic
differentials play a main role in the infinitesimal theory of Teichmüller
spaces. Moreover, the nodal Riemann surfaces are candidates for the boundary
points of the moduli spaces of Riemann surfaces.

In section 2.2, different equivalent definitions of the Teichmüller spaces are
given. The Teichmüller distance, which turns the Teichmüller spaces into
metric spaces, and the Bers embedding, which provides the Teichmüller spaces
with the complex structures, are briefly reviewed. The moduli spaces of
Riemann surfaces are also defined. With the aid of Bers embedding (and the
Kodaira-Spencer map), the representations of the tangent space and the
cotangent space at some point of a moduli space as the space of harmonic
Beltrami differentials and the space of holomorphic quadratic differentials of
a Riemann surface are obtained (Theorem 2.2.1 and the paragraph below the
theorem). The local behavior of the boundary of moduli spaces is briefly
studied as well. Two versions of the Schwarz-Yau lemma are included in section
2.3 for the later use in the chapter 5.

In chapter 3, the classical canonical metrics, i.e., the Teichmüller metric,
the Kobayashi metric, the Carathéodory metric, the Bergman metric and the
Weil-Petersson metric, are examined. The first three metrics are just Finsler
metrics, while the latter two are Kähler. In section 3.1 the Finsler metrics
and the Bergman metric are introduced. They are all complete, and indeed they
are equivalent to each other. The Teichmüller metric even coincides with the
Kobayashi metric, which is a result from Royden[23]. The equivalence of the
Kobayashi metric to the Carathéodory metric and to the Bergman metric (Theorem
3.1.1) rely on the results of Kobayashi[15] (Lemma 3.1.1). Section 3.2
consists of the description of the Weil-Petersson metric. Since the harmonic
lifts $v_{1},...,v_{n}$ are important in the computation of the curvature
formulae of the Weil-Petersson metric and the Ricci metric, some results
related to them, like the formulae for the Lie derivatives of the harmonic
Beltrami differentials $B_{i}$ with respect to them (Lemma 3.2.4), the
harmonicity of these Lie derivatives (Lemma 3.2.5) and the commutator of
$v_{k}$ and $\overline{v_{l}}$ (Lemma 3.2.6), are included as well. The
definition of Maass operators $K_{p}$ and $L_{p}$ are also given. Moreover,
the curvature formula of the Weil-Petersson metric is quoted (Theorem 3.2.1).

Chapter 4 is the longest chapter of the thesis because it contains numerous
important and useful results of the four ”younger” Kähler metrics, namely, the
McMullen metric, the Kähler-Einstein metric, the Ricci metric and the
perturbed Ricci metric. The first section is an overview of the McMullen
metric, which was constructed by McMullen[20]. A sketch of the proof of the
Kähler hyperbolicity of this metric is given (Theorem 4.1.1). Also, its
equivalence to the Teichmüller metric is proved (Theorem 4.1.3). Section 4.2
is a short review on the existence of the Kähler-Einstein metric (Theorem
4.2.1) and Yau’s conjecture. The last three sections are the main characters
of this chapter. In section 4.3 an intensive study on the Ricci metric due to
Liu-Sun-Yau is included. The commutator operators $\xi_{k}$ are defined so as
to simplify the computation of the curvature formula of the Ricci metric. Some
formulae related to these operators are given (Lemma 4.3.1). The operator
$Q_{k\overline{l}}$, which is indeed the commutator of $\xi_{k}$ and
$\overline{v_{l}}$ (Lemma 4.3.2), is also defined for the purpose of
simplification. The curvature formula of the Ricci metric is then explicitly
worked out (Theorem 4.3.2).

Plenty of estimates on the asymptotics and the curvatures of the Ricci metric
are found in section 4.4. Some $C^{\infty}$ norms on the sections of
$(\kappa\otimes\overline{\kappa}^{-1})^{\frac{p}{2}}$ for $p\geq 0$ are
defined as Wolpert[30] did. Then the Trapani’s expression[28] of the Masur’s
holomorphic quadratic differentials $\psi_{1},...,\psi_{n}$ [19] on the
genuine collars using rs-coordinate provides a tool for the estimates of the
Weil-Petersson metric near a boundary point (Corollary 4.4.1) and the rs-
coordinate expression of the harmonic Beltrami differential $B_{i}$ on the
collars (Lemma 4.4.1). Using this expression of $B_{i}$ the estimates on the
norms of $A_{i}$ and $f_{i\overline{j}}=A_{i}\overline{A_{j}}$ are easily
obtained (Lemma 4.4.2). The approximation functions
$\tilde{e_{i\overline{j}}}$ of
$e_{i\overline{j}}=(\Box+1)^{-1}f_{i\overline{j}}$ are defined and the norm
estimates on them are worked out (Lemma 4.4.3) so that the estimates on the
asymptotics of the Ricci metric can be done (Corollary 4.4.2). As a
preparation for the curvature estimates, the estimates on different norms of
$K_{0}f_{i\overline{j}}$ are given (Lemma 4.4.6). A formula for the term
$\displaystyle\int_{X}Q_{k\overline{l}}(e_{i\overline{j}})e_{\alpha\overline{\beta}}dv$
in the curvature formula of the Ricci metric is provided as well (Lemma
4.4.7). Finally the estimates on the curvatures, especially the holomorphic
sectional curvature, of the Ricci metric are proved (Theorem 4.4.3).

The last section in this chapter is about the perturbed Ricci metric. Using
the curvature formula of the Ricci metric, the curvature formula of the
perturbed Ricci metric is easily obtained (Theorem 4.5.1). By the estimates on
the inverse matrix $(\tilde{\tau}^{i\overline{j}})$ near a boundary point
(Lemma 4.5.2), the estimates on the curvatures of the perturbed Ricci metric
can be done with nearly the same method as that of the Ricci metric (Theorem
4.5.2). The equivalence of the Ricci metric and the perturbed Ricci metric
follows from the estimates on the Weil-Petersson metric and the Ricci metric
(Theorem 4.5.3).

In the last chapter, two essential equivalences of the canonical metrics are
examined. One is the equivalence of the Ricci metric to the Kähler-Einstein
metric proved in the section 5.1. This equivalence (Theorem 5.1.1) follows
from a simple but important linear algebra fact (Lemma 5.1.1) and the Schwarz-
Yau lemma. Another is the equivalence of the Ricci metric to the McMullen
metric in the section 5.2. Using the first derivatives of the length functions
of short geodesics $\partial_{i}l_{j}$ (Lemma 5.2.1) and the Schwarz-Yau
lemma, this equivalence (Theorem 5.2.1) is proved as well. These two
equivalence give a positive answer to the Yau’s conjecture. This remark gives
an end to this thesis.

##  Chapter 2 Background Knowledge

In this chapter some background knowledge about the Riemann surface theory and
the Teichmüller theory will be presented. In addition, we will also take a
look at the Schwarz-Yau Lemma, whose importance will be revealed in chapter 5.

###  2.1 Results from Riemann Surface Theory and Quasiconformal Mappings

Riemann surface theory is a prerequisite for the Teichmüller theory, so we
should begin by a briefing about some essential results related to the Riemann
surfaces.

####  2.1.1 Riemann Surfaces and the Uniformization Theorem

Let us first review the definition of a Riemann surface.

######  Definition 2.1.1.

A Riemann surface $R$ is a connected one dimensional complex manifold, i.e.,
$R$ is a connected Hausdorff topological space with a family
$\\{(U_{\alpha},z_{\alpha})\\}_{\alpha\in A}$ ($A$ is an index set) satisfying
the following three conditions:

  * (i)

$\\{U_{\alpha}\\}_{\alpha\in A}$ is an open covering of $R$.

  * (ii)

Every $z_{\alpha}$ is a homeomorphism from $U_{\alpha}$ onto an open subset
$D_{\alpha}$ of $\mathbb{C}$.

  * (iii)

If $U_{\alpha}\cap U_{\beta}\neq\phi$, the transition mappings
$z_{\alpha\beta}:=z_{\alpha}\circ z_{\beta}^{-1}$ from
$z_{\beta}(U_{\alpha}\cap U_{\beta})$ to $z_{\alpha}(U_{\alpha}\cap
U_{\beta})$ are biholomorphic.

The pairs $(U_{\alpha},z_{\alpha})$ are called charts. Two systems of charts
$(U_{\alpha},z_{\alpha})$ and $(V_{\beta},w_{\beta})$ are called compatible if
$w_{\beta}\circ z_{\alpha}^{-1}$ is a holomorphic mapping from
$z_{\alpha}(U_{\alpha}\cap V_{\beta})$ to $w_{\beta}(U_{\alpha}\cap
V_{\beta})$ for any nonempty $U_{\alpha}\cap V_{\beta}$. Since the composition
of holomorphic mappings is holomorphic and the inverse of holomorphic
bijection is also holomorphic, we can see that compatibility is an equivalence
relation. An equivalent class of compatible systems of charts on $R$ is called
a complex structure or a Riemann surface structure of $R$.

A system of charts on $R$ determines an orientation of $R$ by pulling back the
usual orientation of $D_{\alpha}$ to $U_{\alpha}$ through the mapping
$z_{\alpha}$. On $U_{\alpha}\cap U_{\beta}$ the orientation is consistently
determined as the Jacobian of $z_{\alpha\beta}$ is positive.

Traditionally a compact Riemann surface is called closed while a noncompact
Riemann surface is called open.

A continuous mapping $f$ between two Riemann surfaces $R_{1}$ and $R_{2}$ is
called holomorphic if for any charts $z_{\alpha}$ and $w_{\beta}$ on $R_{1}$
and $R_{2}$ respectively, the mapping $w_{\beta}\circ f\circ z_{\alpha}^{-1}$
is holomorphic. $R_{1}$ and $R_{2}$ are analytically equivalent if there
exists a biholomorphism between $R_{1}$ and $R_{2}$.

The uniformization theorem can be stated in many different ways. The deepest
part of it is called the Koebe’s planarity theorem, which says that a
noncompact planar Riemann surface is analytically equivalent to a domain in
the plane (a surface $R$ is planar if any simple closed curve on $R$ divides
$R$ into two connected components). This result can be combined with the
topological theory of covering surfaces. In particular, $R$ is analytically
equivalent to its universal covering $\tilde{R}$ factored by a deck
transformation group $\Gamma$ which is isomorphic to the fundamental group of
$R$. We are then led to the following theorem, which is also called the
uniformization theorem.

######  Theorem 2.1.1.

Let $R$ be a Riemann surface. Then $R$ is analytically equivalent to one of
the following:

  * (i)

The Riemann sphere
$\hat{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}=\mathbb{C}P^{1}$.

  * (ii)

The complex plane $\mathbb{C}$.

  * (iii)

The punctured plane $\mathbb{C}-\\{0\\}$

  * (iv)

The plane modulo a lattice $\mathbb{C}/L$, where $L$ is isomorphic to
$\mathbb{Z}\times\mathbb{Z}$ and is spanned by two $\mathbb{R}$-linearly
independent vectors in $\mathbb{C}$.

  * (v)

The upper half plane $\mathbb{H}=\\{z|\textrm{Im}z>0\\}$ modulo a properly
discontinuous, torsion free group $\Gamma$ of biholomorphisms of $\mathbb{H}$
(here we allow $\Gamma$ to be a trivial group, in which case the quotient
space is $\mathbb{H}$ itself).

Proofs of this theorem can be found in Ahlfors[2], Ahlfors and Sario[5],
Farkas and Kra[11], and Springer[27]. Notice that any two of the five types of
surfaces above are not analytically equivalent.

####  2.1.2 Fuchsian Groups

Due to the uniformization theorem, groups of holomorphic homeomorphisms of the
upper half plane $\mathbb{H}$ which are torsion free and act properly
discontinuously on $\mathbb{H}$ play a central role in Riemann surface theory.
Let’s drop the torsion free condition and consider the notion of a Fuchsian
group:

######  Definition 2.1.2.

A group of holomorphic homeomorphisms of the upper half plane $\mathbb{H}$
which acts properly discontinuously on $\mathbb{H}$ is called a Fuchsian
group.

The structure of the Fuchsian groups is indeed known. More precisely, we have
the following facts:

######  Fact 2.1.1.

Suppose $A:\mathbb{H}\rightarrow\mathbb{H}$ is s bijective holomorphic
mapping. Then there exist real numbers $a,b,c,d$ such that $ad-bc=1$ and

$A(z)=\dfrac{az+b}{cz+d}.$

The numbers $a,b,c,d$ are uniquely determined up to a $-1$ multiple to all of
them. Hence a Fuchsian group is a subgroup of $\textrm{PSL}(2,\mathbb{R})$.

######  Fact 2.1.2.

A subgroup of $\textrm{PSL}(2,\mathbb{R})$ is discrete if and only if it acts
properly discontinuously on $\mathbb{H}$. Thus any discrete subgroup of
$\textrm{PSL}(2,\mathbb{R})$ is a Fuchsian group.

For the proofs of these facts, one can consult Gardiner[12].

Note that for $B\in\textrm{PSL}(2,\mathbb{R})$ and $\Gamma$ being a subgroup
of $\textrm{PSL}(2,\mathbb{R})$, we also call the group $B\circ\Gamma\circ
B^{-1}$ a Fuchsian group. This group would instead have $B(\mathbb{H})$ as the
invariant domain.

######  Definition 2.1.3.

An elliptic point $p$ for $G$ acting on $X$ is a point on $X$ whose isotropy
group is nontrivial.

We then have a theorem about the elliptic points for a Fuchsian group and the
quotient space $\mathbb{H}/\Gamma$.

######  Theorem 2.1.2.

Let $\Gamma$ be a Fuchsian group acting on $\mathbb{H}$. Then
$\mathbb{H}/\Gamma$ is a complete metric space with the quotient metric given
by

$\overline{d}(\overline{p},\overline{r})=\inf_{A\in G}d(A(p),r)$

where $d$ is the Poincaré metric on $\mathbb{H}$. The set of elliptic points
for $\Gamma$ is a discrete set. Moreover, there is a unique complex structure
on $\mathbb{H}/\Gamma$ making $\mathbb{H}/\Gamma$ into a Riemann surface and
the mapping $\pi:\mathbb{H}\rightarrow\mathbb{H}/\Gamma$ into a holomorphic
mapping. At nonelliptic points $\pi$ is a local homeomorphism, and at elliptic
points $\pi$ is locally an $n$-to-$1$ mapping, where $n$ is the order of the
isotropy group of the elliptic point.

Its proof also appears in Gardiner[12].

####  2.1.3 Quasiconformal Mappings and the Beltrami Equation

A Jordan region on a Riemann surface $R$ is a connected and simply connected
open subset of $R$ whose boundary is a simple closed curve in $R$. This
motivates us to consider the notion of a generalized quadrilateral.

######  Definition 2.1.4.

A generalized quadrilateral $Q$ on $R$ is a Jordan region on $R$ together with
two disjoint closed arcs $\beta_{1}$ and $\beta_{2}$ on the boundary of $Q$.
The module of $Q$, denoted as $m(Q)$, is determined by the conformal mapping
of $Q$ onto a rectangle which takes $\beta_{1}$ and $\beta_{2}$ onto the
vertical sides of the rectangle in this way: if the rectangle has width $a$
and height $b$, then $m(Q)=\dfrac{a}{b}$.

Notice that $m(Q)$ possesses the following property: if $Q^{*}$ is the same
Jordan region as $Q$ but with two disjoint closed arcs $\alpha_{1}$ and
$\alpha_{2}$ complementary in the boundary of $Q$ (except for common
endpoints), then $m(Q^{*})=\dfrac{1}{m(Q)}$.

Now we are ready to introduce the geometric definition of a quasiconformal
mapping.

######  Definition 2.1.5.

Let $f$ be an orientation preserving homeomorphism from a region $\Omega$ to a
region $\Omega^{\prime}$. Then $f$ is $K$-quasiconformal if for every
quadrilateral $Q$ in $\Omega$, $m(f(Q))\leq Km(Q)$. The smallest possible
value of $K$ for which the inequality holds for all quadrilateral $Q$ is
called the dilatation of $f$.

It is obvious that $m(f(Q^{*}))\leq Km(Q^{*})$ and $f(Q)^{*}=f(Q^{*})$, so by
definition we know that

$K^{-1}m(Q)\leq m(f(Q))\leq Km(Q)$

for all quadrilateral $Q$. Using this definition we can easily obtain the
property that if $f_{1}$ and $f_{2}$ are $K_{1}$\- and $K_{2}$-quasiconformal
respectively, then $f_{1}\circ f_{2}$ is $K_{1}K_{2}$-quasiconformal.

The analytic definition of a quasiconformal mapping require the notion of
absolute continuity on lines (ACL). A function $f(z)=u(x,y)+iv(x,y)$ is ACL if
for every rectangle in $\Omega$ with sides parallel to the $x$\- and $y$-axes,
both $u(x,y)$ and $v(x,y)$ are absolutely continuous on almost every
horizontal and almost every vertical line of that rectangle. The function $u$
and $v$ will thus have partial derivatives $u_{x}$, $u_{y}$, $v_{x}$, $v_{y}$
almost everywhere in $\Omega$. Note that the complex partial derivatives are
defined by $f_{z}=\dfrac{1}{2}(f_{x}-if_{y})$ and
$f_{\overline{z}}=\dfrac{1}{2}(f_{x}+if_{y})$.

######  Definition 2.1.6.

Let $f$ be a homeomorphism from a domain $\Omega$ to a domain
$\Omega^{\prime}$. Then $f$ is $K$-quasiconformal if $f$ is ACL in $\Omega$
and $|f_{\overline{z}}|\leq k|f_{z}|$ almost everywhere for
$k=\dfrac{K-1}{K+1}<1$. The minimal possible value of $K$ for which the latter
condition is satisfied is called the dilatation of $f$.

The next theorem provides a bridge connecting two definitions.

######  Theorem 2.1.3.

The geometric and the analytic definitions of $K$-quasiconformality are
equivalent. Moreover, for a quasiconformal mapping $f$, $f_{\overline{z}}$ and
$f_{z}$ are locally square integrable.

The proof of this theorem can be found in Ahlfors[3] or Lehto and
Virtanen[16].

Notice that for a quasiconformal mapping $f$, $K(f)$ is denoted to be the
dilatation of $f$.

We have seen that if $f$ is a topological ACL mapping and if
$\Big{|}\dfrac{f_{\overline{z}}}{f_{z}}\Big{|}\leq k<1$ almost everywhere,
then $f$ is quasiconformal. Let $\mu(z)$ be a measurable complex valued
function defined in a domain $\Omega$ for which
$\left\|\mu\right\|_{\infty}=k<1$. The Beltrami equation is

$f_{\overline{z}}(z)=\mu(z)f_{z}(z)$

where the partial derivatives are assumed to be locally square integrable and
taken in the sense of distributions. The function $\mu$ is called the Beltrami
coefficient of the mapping $f$.

In the theory of Beltrami equation (Ahlfors[3], Lehto and Virtanen[16],
Ahlfors and Bers[4]), a solution $f$ can be expressed as a power series in
$\mu$, where the power series is made up by taking compositions of singular
integral operators. The following theorem asserts the existence of normalized
global solutions to the Beltrami equation on $\hat{\mathbb{C}}$. It also
expresses in a very particular way the analytic dependence of the solution $f$
on the Beltrami coefficient $\mu$. The analyticity of this dependence is
important in the construction of complex structure for the Teichmüller spaces.

######  Theorem 2.1.4.

(Theorem 5, [12]) The Beltrami equation gives a one-to-one correspondence
between the set of quasiconformal homeomorphisms of $\hat{\mathbb{C}}$ which
fix the points $0$, $1$ and $\infty$ and the set of measurable complex valued
functions $\mu$ on $\hat{\mathbb{C}}$ for which
$\left\|\mu\right\|_{\infty}<1$.

Furthermore, the normalized solution $f^{\mu}$ to the equation depends
holomorphically on $\mu$ and for $r>0$ there are $\delta>0$ and $C(r)>0$ such
that

$|f^{\mu}-z-tF(z)|\leq C(r)t^{2}$

for $|z|<r$ and $|t|<\delta$, where

$F(z)=-\dfrac{z(z-1)}{\pi}\iint_{\mathbb{C}}\dfrac{\mu(\xi)d\xi
d\eta}{\zeta(\zeta-1)(\zeta-z)}$

and $\zeta=\xi+i\eta$.

Usually we need solutions $f$ which map the upper half plane $\mathbb{H}$ to
itself and preserve $\hat{\mathbb{R}}$ for arbitrary Beltrami coefficient
$\mu$ with support in $\mathbb{H}$. Let $M(\mathbb{H})$ be the space of
complex valued $L_{\infty}$-functions $\mu$ with support in $\mathbb{H}$ and
$\left\|\mu\right\|_{\infty}<1$. For $\mu$ in $M(\mathbb{H})$, let $\hat{\mu}$
be identical to $\mu$ on $\mathbb{H}$ and equal to
$\overline{\mu(\overline{z})}$ on the lower half plane $\mathbb{H}^{*}$. Then
we solve the Beltrami equation with $\mu$ replaced by $\hat{\mu}$, and let $f$
be the desired normalized solution. Since $\overline{f(\overline{z})}$ has the
same Beltrami coefficient as $f$ and also fixes the same three points on
$\hat{\mathbb{R}}$ as $f$, by uniqueness of the solutions to the Beltrami
equation we know that $\overline{f(\overline{z})}=f(z)$, and $f$ is a
quasiconformal mapping which preserves $\hat{\mathbb{R}}$ in an orientation-
preserving manner. In addition, $f$ preserves $\mathbb{H}$ and
$\mathbb{H}^{*}$.

We hence obtain a corollary to the previous theorem.

######  Corollary 2.1.1.

For every $\mu\in M(\mathbb{H})$, there exists a unique quasiconformal self-
mapping $f$ of $\mathbb{H}$ satisfying the Beltrami equation on $\mathbb{H}$
which extends continuously to the closure of $\mathbb{H}$ and is normalized to
fix $0$, $1$ and $\infty$.

####  2.1.4 Holomorphic Quadratic Differentials

We begin by stating the definition of a holomorphic quadratic differential.

######  Definition 2.1.7.

A holomorphic quadratic differential $\varphi$ on a Riemann surface $R$ is an
assignment of a holomorphic function $\varphi_{1}(z_{1})$ to each local
coordinate $z_{1}$ satisfying the compatibility condition, i.e., if $z_{2}$ is
another local coordinate overlapping $z_{1}$, then
$\varphi_{1}(z_{1})=\varphi_{2}(z_{2})\Big{(}\dfrac{dz_{2}}{dz_{1}}\Big{)}^{2}$.

We usually impose some restrictions on the holomorphic quadratic
differentials. Along a border arc $\alpha$ in the border of $R$ we require
that $\varphi_{1}(\alpha)$ be real if $z_{1}$ is a local coordinate with
$z_{1}(\alpha)$ real. If there exists an isolated boundary point $p$ on $R$,
then we require that $\varphi$ have at most a simple pole at $p$. We do not
permit poles of $\varphi$ to occur on the boundary curves of $R$. The vector
space of holomorphic quadratic differentials with the above conditions is
denoted as $Q(R)$.

Now if the genus of $R$ is $g$ and $R$ is obtained form a compact surface by
deleting $m$ disjoint closed disks and $n$ isolated points outsides the disks,
then we have the following theorem:

######  Theorem 2.1.5.

$dim_{\mathbb{R}}Q(R)=6g-6+3m+2n$

except for cases shown below:

$g$ | $m$ | $n$ | $dim_{\mathbb{R}}Q(R)$
---|---|---|---
$0$ | $0$ | $0,1,2$ | $0$
$0$ | $1$ | $0,1$ | $0$
$1$ | $0$ | $0$ | $2$

In the case when $m=0$, $Q(R)$ is a complex vector space and the same formula
yields

$dim_{\mathbb{C}}Q(R)=3g-3+n$

for any $g\geq 2$, for $g=1$ and $n\geq 1$, and for $g=0$ and $n\geq 3$.

This theorem can be easily deduced from the Riemann-Roch theorem. It can also
be seen as a consequence of the Teichmüller theorem. The proof can be found in
Farkas and Kra[11].

####  2.1.5 Nodal Riemann Surfaces

For the study of the boundary of the Teichmüller spaces or the moduli spaces,
we require the notion of a nodal Riemann surface.

######  Definition 2.1.8.

A nodal Riemann surface $X_{0}$ is a (singular) Riemann surface such that for
every point $p\in X_{0}$, there is a neighborhood of $p$ which is isometric to
$\\{z||z|<1\\}\subset\mathbb{C}$ or
$\\{(z,w)|zw=0,|z|<1,|w|<1\\}\subset\mathbb{C}^{2}$. In the second case $p$ is
called a node. A connected component of the complement of the nodes is called
a part of $X_{0}$.

Suppose $X_{0}$ has nodes $p_{1},...,p_{s}$ and $r\geq 1$ parts
$X_{1},...,X_{r}$, where each $X_{i}$ is a Riemann surface of genus $g_{i}$
with $n_{i}$ punctures. Notice there are exactly two punctures $a_{j}$ and
$b_{j}$ (which may lie on different $X_{i}$) corresponding to the same node
$p_{j}$, and we call them paired punctures. Then we have $\displaystyle
2s=\sum_{i=1}^{r}n_{i}$ and $\displaystyle g=\sum_{i=1}^{r}g_{i}+p+1-r$. In
this thesis we will assume for all $i$, $X_{i}$ is not a sphere with one or
two punctures. Thus each $X_{i}$ is a hyperbolic Riemann surface carrying a
Poincaré metric $\rho$.

Now let us talk about the rs-coordinate on a Riemann surface. This coordinate
was defined by Wolpert[30], and there are two cases: the puncture case and the
short geodesic case.

For the puncture case, let $X$ be a nodal Riemann surface and $p\in X$ be a
node. Also, let $a$ and $b$ be the paired punctures corresponding to $p$.

######  Definition 2.1.9.

A local coordinate chart $(U,u)$ near $a$ is called rs-coordinate if $u(a)=0$
where $u$ maps $U$ to the punctured disk $\\{u|0<|u|<c\\}$ with $c>0$, and the
restriction to $U$ of the Kähler-Einstein metric on $X$ can be written as

$\dfrac{1}{2|u|^{2}(\log|u|)^{2}}|du|^{2}.$

The rs-coordinate $(V,v)$ near $b$ can be defined similarly.

For the short geodesic case, let $X$ be a closed Riemann surface, and let
$\gamma$ be a closed geodesic on $X$ with length $l<c_{*}$, where $c_{*}$ is
the collar constant introduced by Keen[14].

######  Definition 2.1.10.

A local coordinate chart $(U,z)$ is called rs-coordinate at $\gamma$ if
$\gamma\subset U$ where $z$ maps $U$ to the annulus
$\\{z|c^{-1}|t|^{\frac{1}{2}}<|z|<c|t|^{\frac{1}{2}}\\}$ with $c,|t|>0$, and
the restriction to $U$ of the Kähler-Einstein metric on $X$ can be written as

$\dfrac{1}{2}\bigg{(}\dfrac{\pi}{\log|t|}\dfrac{1}{|z|}\csc\dfrac{\pi\log|z|}{\log|t|}\bigg{)}^{2}|dz|^{2}.$

By Keen’s collar theorem [14], we obtain the following lemma:

######  Lemma 2.1.1.

Let $X$ be a closed Riemann surface and let $\gamma$ be a closed geodesic on
$X$ with length $l<c_{*}$. Then there is a collar $\Omega$ on $X$ with
holomorphic coordinate $z$ defined on $\Omega$ such that

  * (i)

$z$ maps $\Omega$ to the annulus
$\\{z|c^{-1}e^{-\dfrac{2\pi^{2}}{l}}<|z|<c\\}$ for $c>0$;

  * (ii)

the Kähler-Einstein metric on $X$ restricted to $\Omega$ is given by

$\dfrac{1}{2}u^{2}r^{-2}\csc^{2}\tau|dz|^{2}$

where $u=\dfrac{l}{2\pi}$, $r=|z|$ and $\tau=u\log r$;

  * (iii)

the geodesic $\gamma$ is given by the equation $|z|=e^{-\dfrac{\pi^{2}}{l}}$.

We call such a collar $\Omega$ a genuine collar.

Notice that the constant $c$ in the lemma has a lower bound such that the area
of $\Omega$ is bounded from below. Moreover, the coordinate $z$ in the lemma
is indeed rs-coordinate.

###  2.2 Teichmüller Theory

In this section we will go through some classical results of the Teichmüller
theory, like the Teichmüller’s distance and the Bers embedding. The
infinitesimal theory of Teichmüller spaces will also be discussed.

####  2.2.1 Teichmüller Spaces

There are basically two kinds of Teichmüller spaces, namely, the Teichmüller
space of a Riemann surface $T(R_{0})$ and the Teichmüller space of a Fuchsian
group $T(\Gamma_{0})$. Now let us see the definition of the Teichmüller space
of a Riemann surface first.

Let $R_{0}$ be a fixed Riemann surface. In fact we can define $T(R_{0})$ in
two different ways. One of them is to define $T(R_{0})$ as a deformation
space.

######  Definition 2.2.1.

Let $\textrm{Def}(R_{0})$ be the set of all pairs $(f,R)$, where $f$ is a
quasiconformal homeomorphism from $R_{0}$ onto a surface $R$. Two pairs
$(f_{1},R_{1})$ and $(f_{2},R_{2})$ in $\textrm{Def}(R_{0})$ are equivalent if
there is a conformal map $c:R_{1}\rightarrow R_{2}$ such that $c\circ f_{1}$
is homotopic to $f_{2}$. Then $T(R_{0})$ is defined as the quotient of
$\textrm{Def}(R_{0})$ by this equivalence relation.

Another method is to define $T(R_{0})$ as an orbit space. Let $M(R_{0})$ be
the open unit ball in the Banach space $L_{\infty}(R_{0})$, where
$L_{\infty}(R_{0})$ is the space of all Beltrami differentials $\mu$ on
$R_{0}$. Note that such a differential $\mu$ is an assignment of a measurable
complex valued function $\mu^{z}$ to each local coordinate $z$ on $R_{0}$ such
that
$\left\|\mu\right\|_{\infty}=\sup\\{\left\|\mu^{z}(z)\right\|_{\infty}|z\textrm{
is a local coordinate}\\}<\infty$, and if $\zeta$ is another local coordinate
overlapping with $z$, then
$\mu^{z}(z)\dfrac{d\overline{z}}{dz}=\mu^{\zeta}(\zeta)\dfrac{d\overline{\zeta}}{d\zeta}$.

Form the theory of Beltrami equation we know that for any $\mu\in M(R_{0})$,
there exists a quasiconformal homeomorphism $w$ from $R_{0}$ onto another
Riemann surface $R^{\mu}$ satisfying $w_{\overline{z}}=\mu w_{z}$. Let
$D_{0}(R_{0})$ be the group of quasiconformal homeomorphic self-mappings of
$R_{0}$ which are homotopic to the identity. Now we can define $T(R_{0})$:

######  Definition 2.2.2.

Consider the group action $D_{0}(R_{0})\times M(R_{0})\rightarrow M(R_{0})$
given by $(h,\mu)\mapsto h^{*}(\mu)$, where

$h^{*}(\mu)=\dfrac{(w\circ h)_{\overline{z}}}{(w\circ
h)_{z}}=\dfrac{\nu(z)+\mu(h(z))\theta(z)}{1+\overline{\nu(z)}\mu(h(z))\theta(z)}$

for which $\nu$ is the Beltrami coefficient of $h$ and
$\theta(z)=\dfrac{\overline{h_{z}}}{h_{z}}$. Then $T(R_{0})$ is defined as
$M(R_{0})/D_{0}(R_{0})$.

Gardiner[12] showed that in fact the above two methods define the same
$T(R_{0})$.

For the definition of the Teichmüller space of a Fuchsian group, let
$\Gamma_{0}$ be a Fuchsian group, and define $B(\Gamma_{0})$ to be the set of
measurable complex valued Beltrami coefficients $\mu$ with support in
$\mathbb{H}$ and which satisfy
$\mu(B(z))\overline{B^{\prime}(z)}=\mu(z)B^{\prime}(z)$ for all
$B\in\Gamma_{0}$. Also, define $M(\Gamma_{0})\subset B(\Gamma_{0})$ such that
$\left\|\mu\right\|_{\infty}<1$ for all $\mu\in M(\Gamma_{0})$. Let $w_{\mu}$
be the unique $qc$-homeomorphism of $\mathbb{H}$ whose extension to
$\hat{\mathbb{R}}$ fixes $0$, $1$ and $\infty$ and satisfies
$w_{\overline{z}}=\mu w_{z}$.

######  Definition 2.2.3.

$T(\Gamma_{0})$ is defined as $M(\Gamma_{0})/\sim$, where $\sim$ is an
equivalence relation such that $\mu\sim\nu$ if $w_{\mu}(x)=w_{\nu}(x)$ for all
$x\in\hat{\mathbb{R}}$.

Again, Gardiner[12] showed that if $R_{0}=\mathbb{H}/\Gamma_{0}$, then
$T(R_{0})$ is canonically isomorphic to $T(\Gamma_{0})$. Notice that by
Imayoshi and Taniguchi[13] we can see that $T(R_{0})$ indeed depends on the
genus $g$ of the (closed) Riemann surface $R_{0}$ instead of the surface
itself. In this thesis, the identification $T_{g}=T(R_{0})=T(\Gamma_{0})$ is
taken.

####  2.2.2 Teichmüller’s Distance

Let $\Gamma$ be a Fuchsian group. We can then define a Teichmüller’s distance
on $T(\Gamma)$:

######  Definition 2.2.4.

For $[\mu],[\nu]\in T(\Gamma)$, the Teichmüller’s distance $d$ on $T(\Gamma)$
is defined as

$d([\mu],[\nu])=\dfrac{1}{2}\inf\log K(w_{\tilde{\mu}}\circ
w_{\tilde{\mu}}^{-1})$

where the infinmum is taken over all $\tilde{\mu}\sim\mu$ and
$\tilde{\nu}\sim\nu$.

The Teichmüller’s distance makes $T(\Gamma)$ a metric space and thus gives it
a topology.

####  2.2.3 The Bers Embedding

Let us first define the Schwarzian derivative, which will be used to define
the Bers embedding.

######  Definition 2.2.5.

For a (quasi)conformal mapping $f$ on a domain in $\mathbb{C}$, the Schwarzian
derivative $\\{f,z\\}$ of $f$ is given by

$\\{f,z\\}=\dfrac{f^{\prime\prime\prime}(z)}{f^{\prime}(z)}-\dfrac{3}{2}\bigg{(}\dfrac{f^{\prime\prime}(z)}{f^{\prime}(z)}\bigg{)}^{2}.$

We can easily verify the Cayley identity: $\\{f\circ
g,z\\}=\\{f,g\\}g^{\prime}(z)^{2}+\\{g,z\\}$ and see that $\\{f,z\\}=0$ if and
only if $f$ is a Möbius transformation.

Let $\Gamma$ be a Fuchsian group such that $\mathbb{H}/\Gamma$ is a closed
Riemann surface with genus $g\geq 2$ and the extensions of all its elements
fix $0$, $1$ and $\infty$. Now for $\mu\in M(\Gamma)$, let
$\varphi_{\mu}(z)=\\{w^{\mu},z\\}$ for $z\in\mathbb{H}^{*}$, where $w^{\mu}$
is the unique $qc$-homeomorphism of $\hat{\mathbb{C}}$ normalized to fix $0$,
$1$ and $\infty$ with Beltrami coefficient $\mu$ on $\mathbb{H}$ and $0$ on
$\mathbb{H}^{*}$. Then we have the following lemma (Lemma 6.4, [13]):

######  Lemma 2.2.1.

Let $\gamma\in\Gamma$, then

$\varphi_{\mu}(\gamma(z))\gamma^{\prime}(z)^{2}=\varphi_{\mu}(z)$

for $z\in\mathbb{H}^{*}$. Furthermore, $\varphi_{\mu}$ can be regarded as a
holomorphic quadratic differential on the Riemann surface
$R^{*}=\mathbb{H}^{*}/\Gamma$, and for $\mu,\nu\in M(\Gamma)$, $\mu\sim\nu$ if
and only if $\varphi_{\mu}=\varphi_{\nu}$ on $\mathbb{H}^{*}$.

Now we can define the Bers embedding.

######  Definition 2.2.6.

The Bers embedding is the mapping $\beta:T(\Gamma)\rightarrow Q(R^{*})$ given
by $\beta([\mu])=\varphi_{\mu}$. Moreover, the Bers projection
$\Phi:M(\Gamma)\rightarrow Q(R^{*})$ is defined by $\Phi(\mu)=\varphi_{\mu}$.

Notice that $Q(R^{*})$ is equipped with the hyperbolic $L^{\infty}$-norm:

$\left\|\varphi\right\|_{\infty,\rho}=\sup_{\mathbb{H}^{*}}\rho^{-2}|\varphi(z)|$

for $\varphi\in Q(R^{*})$, where $ds^{2}=\rho^{2}(z)|dz|^{2}$ is the Poincaré
metric on $\mathbb{H}^{*}$. This makes $Q(R^{*})$ a complex Banach space.
Since the Bers embedding is a continuous injection, $T(\Gamma)$ inherits the
complex structure of $Q(R^{*})$.

####  2.2.4 Teichmüller Modular Groups and Moduli Spaces of Riemann Surfaces

Let $R$ be a closed Riemann surface with genus $g\geq 2$, and let $D_{+}(R)$
be the group of quasiconformal homeomorphic self-mappings of $R$. Then we can
define the Teichmüller modular group of $R$:

######  Definition 2.2.7.

The Teichmüller modular group of $R$, denoted as $Mod(R)$, is given by

$Mod(R)=D_{+}(R)/D_{0}(R).$

Now we consider the moduli space of $R$:

######  Definition 2.2.8.

The moduli space of $R$ is the quotient space of $T(R)$ by $Mod(R)$. It is
denoted as $\mathcal{M}(R)$.

Note that by Imayoshi and Taniguchi[13] again we know that the above
definitions depend on $g$ instead of $R$ itself, so in this thesis the
notations $Mod(R)$ and $\mathcal{M}(R)$ are replaced by $Mod_{g}$ and
$\mathcal{M}_{g}$ respectively.

####  2.2.5 Infinitesimal Theory of Teichmüller Spaces

For $\mu\in M(\Gamma)$ and $\nu\in B(\Gamma)$, we consider the derivative
$\dot{\Phi}_{\mu}[\nu]$ of $\Phi$ in the direction $\nu$ at $\mu$ given by

$\dot{\Phi}_{\mu}[\nu]=\lim_{t\rightarrow\infty}\dfrac{1}{t}(\Phi(\mu_{t})-\Phi(\mu)),$

where the limit is taken with respect to $\left\|\cdot\right\|_{\infty,\rho}$
and $\mu_{t}\in M(\Gamma)$ is defined as $\mu_{t}=\mu+t\nu+t\epsilon(t)$ with
$\left\|\epsilon(t)\right\|_{\infty,\rho}\rightarrow$ as $t\rightarrow 0$. Let
$HB(\Gamma^{\mu})$ be the space of harmonic Beltrami differentials for
$\Gamma^{\mu}=w_{\mu}\circ\Gamma\circ w_{\mu}^{-1}$ on $\mathbb{H}$. Then the
following theorem (Proposition 7.8, [13]) gives us a representation of
$T_{p}T(\Gamma)$ for $p=[\mu]\in T(\Gamma)$:

######  Theorem 2.2.1.

There exists an isomorphism between
$M(\Gamma)/\textrm{Ker}\dot{\Phi}_{\mu}\cong T_{p}T(\Gamma)$ and
$HB(\Gamma^{\mu})\cong T_{0}T(\Gamma^{\mu})$.

Now for $[X]\in\mathcal{M}_{g}$, the identification $T_{X}\mathcal{M}_{g}\cong
HB(X)=H^{1}(X,TX)$ can also be obtained from Kodaira-Spencer theory and Hodge
theory. Moreover, we also have $T_{X}^{*}\mathcal{M}_{g}\cong Q(X)$. Pick
$\mu\in HB(X)$ and $\varphi\in Q(X)$, we can write $\mu$ and $\varphi$ as
$\mu(z)\dfrac{\partial}{\partial z}\otimes d\overline{z}$ and
$\varphi(z)dz^{2}$ in a local coordinate $z$ on $X$. The duality between
$T_{X}\mathcal{M}_{g}$ and $T_{X}^{*}\mathcal{M}_{g}$ is given by

$[\mu;\varphi]=\int_{X}\mu(z)\varphi(z)dzd\overline{z}.$

####  2.2.6 Boundary of Moduli Spaces of Riemann Surfaces

The boundary of $\mathcal{M}_{g}$ is the divisor
$\overline{\mathcal{M}_{g}}-\mathcal{M}_{g}$ in $\overline{\mathcal{M}_{g}}$,
where $\overline{\mathcal{M}_{g}}$ is the Deligne-Mumford compactification of
$\mathcal{M}_{g}$ in [10].

For the local theory of the boundary, we recall the construction of
Wolpert[30]. Let $X_{0,0}$ be a nodal Riemann surface corresponding to a
codimension $m$ boundary point, i.e., $X_{0,0}$ has $m$ nodes
$p_{1},...,p_{m}$. Let
$X_{0}=X_{0,0}-\\{p_{1},...,p_{m}\\}=\cup_{k=1}^{r}X_{k}$. Fix rs-coordinate
charts $(U_{i},\eta_{i})$ and $(V_{i},\zeta_{i})$ at $p_{i}$ for $i=1,...,m$
such that all $U_{i}$ and $V_{i}$ are mutually disjoint. Also, pick an open
set $U_{0}\subset X_{0}$ such that $X_{k}\cap U_{0}\neq\phi$ is relatively
compact and $U_{0}\cap(U_{i}\cup V_{i})$ is empty for all $i$. Now let
$\nu_{m+1},...,\nu_{n}$ be Beltrami differentials supported in $U_{0}$ and
span the tangent space at $X_{0}$ of the deformation space of $X_{0}$. For
$s=(s_{m+1},...,s_{n})$, let
$\displaystyle\nu(s)=\sum_{i=m+1}^{n}s_{i}\nu_{i}$. For small enough $|s|$, we
know that $|\nu(s)|<1$. Let $X_{0,s}$ be the nodal Riemann surface obtained by
solving the Beltrami equation $\overline{\partial}w=\nu(s)\partial w$. Notice
that there are constants $\delta,c>0$ such that when $|s|<\delta$,
$(U_{i},\eta_{i})$ and $(V_{i},\zeta_{i})$ are holomorphic coordinates on
$X_{0,s}$ with $0<|\eta_{i}|<c$ and $0<|\zeta_{i}|<c$. Now we assume
$(t_{1},...,t_{m})$ has small norm. Then we do the plumbing construction on
$X_{0,s}$ to obtain $X_{t,s}$. We remove the disks
$\\{\eta_{i}|0<|\eta_{i}|<\dfrac{|t_{i}|}{c}\\}$ and
$\\{\zeta_{i}|0<|\zeta_{i}|<\dfrac{|t_{i}|}{c}\\}$ for each $i=1,...,m$ from
$X_{0,s}$ and identify $\\{\eta_{i}|\dfrac{|t_{i}|}{c}<|\eta_{i}|<c\\}$ with
$\\{\zeta_{i}|\dfrac{|t_{i}|}{c}<|\zeta_{i}|<c\\}$ by
$\eta_{i}\zeta_{i}=t_{i}$. The surface obtained is $X_{t,s}$. The tuples
$(t_{1},...,t_{m},s_{m+1},...,s_{n})$ are called the (local) pinching
coordinates. Also, the coordinates $\eta_{i}$ (or $\zeta_{i}$) are called the
plumbing coordinates on $X_{t,s}$ and the collar
$\\{\eta_{i}|\dfrac{|t_{i}|}{c}<|\eta_{i}|<c\\}$ (or
$\\{\zeta_{i}|\dfrac{|t_{i}|}{c}<|\zeta_{i}|<c\\}$) is called the plumbing
collar.

###  2.3 Schwarz-Yau Lemma

This lemma was proved by Yau[32] as a generalization of the Schwarz lemma. In
chapter 5 we will use this lemma as a main tool for proving the equivalences
between the canonical metrics.

######  Theorem 2.3.1.

Let $M$ be a complete Kähler manifold with Ricci curvature bounded from below
by a constant $K_{1}$, and let $N$ be another Kähler manifold with holomorphic
sectional curvature bounded from above by a negative constant $K_{2}$. Then if
there is a non-constant holomorphic mapping $f$ from $M$ into $N$, we have
$K_{1}\leq 0$ and

$f^{*}dS_{N}^{2}\leq\dfrac{K_{1}}{K_{2}}dS_{M}^{2}.$

This lemma also has a version for the comparison between volume forms instead
of metrics:

######  Theorem 2.3.2.

Let $M$ be a complete Kähler manifold with scalar curvature bounded from below
by a constant $K_{1}$, and let $N$ be another Hermitian manifold with Ricci
curvature bounded from above by a negative constant $K_{2}$. Suppose the Ricci
curvature of $M$ is bounded form below and $\textrm{dim}(M)=\textrm{dim}(N)$.
Then if there is a non-constant holomorphic mapping $f$ from $M$ into $N$, we
have $K_{1}\leq 0$ and

$f^{*}dV_{N}\leq\dfrac{K_{1}}{K_{2}}dV_{M}.$

##  Chapter 3 Classical Canonical Metrics on the Teichnmüller Spaces and the
Moduli Spaces of Riemann Surfaces

There are a few renowned classical metrics on the Teichnmüller spaces $T_{g}$
and the moduli spaces $\mathcal{M}_{g}$ which have been studied intensively
before. These metrics are important in the study of the geometry and topology
of $T_{g}$ and $\mathcal{M}_{g}$. In this chapter we will briefly review three
(complete) Finsler metrics: the Teichmüller metric, the Kobayashi metric and
the Carathéodory metric, and two Kähler metrics: the Bergman metric and the
Weil-Petersson metric.

###  3.1 Finsler Metrics and Bergman Metric

The definitions and some basic properties of the Teichmüller metric
$\omega_{T}$, the Kobayashi metric $\omega_{K}$, the Carathéodory metric
$\omega_{C}$ and the Bergman metric $\omega_{B}$ will be briefly discussed in
this section.

####  3.1.1 Definitions and Properties of the Metrics

The Teichmüller metric can be defined like this:

######  Definition 3.1.1.

Let $X\in\mathcal{M}_{g}$, and let $\varphi=\varphi(z)dz^{2}\in Q(X)\cong
T_{X}^{*}\mathcal{M}_{g}$. Then the Teichmüller $L^{1}$ norm on
$T_{X}^{*}\mathcal{M}_{g}$ is given by

$\left\|\varphi\right\|_{T}=\int_{X}|\varphi(z)|dzd\overline{z}.$

The Teichmüller norm on $T_{X}\mathcal{M}_{g}$ is thus defined by the duality
as

$\left\|\mu\right\|_{T}=\sup\\{\textrm{Re}[\mu;\varphi]|\left\|\varphi\right\|_{T}=1\\}$

for $\mu\in HB(X)\cong T_{X}\mathcal{M}_{g}$.

Note that $\omega_{T}$ has constant holomorphic sectional curvature equal to
$-1$.

The Kobayashi metric and the Carathéodory metric can indeed be defined for any
complex space. Let $Y$ be a complex manifold of dimension $n$, and let
$\Delta_{R}$ be the disk in $\mathbb{C}$ with radius $R$. Now take
$\Delta=\Delta_{1}$, and suppose $\rho$ is the Poincaré metric on $\Delta$.
Also, define $\textrm{Hol}(Y,\Delta_{R})$ and $\textrm{Hol}(\Delta_{R},Y)$ as
the spaces of holomorphic maps from $Y$ to $\Delta_{R}$ and from $\Delta_{R}$
to $Y$ respectively.

######  Definition 3.1.2.

Let $p\in Y$ and $v\in T_{p}Y$ be a holomorphic tangent vector. The
Carathéodory norm on $T_{p}Y$ is defined to be

$\left\|v\right\|_{C}=\sup_{f\in\textrm{Hol}(Y,\Delta)}\left\|f_{*}v\right\|_{\Delta,\rho},$

and the Kobayashi norm on $T_{p}Y$ is defined as

$\left\|v\right\|_{K}=\inf_{f\in\textrm{Hol}(\Delta_{R},Y),f(0)=p,f^{\prime}(0)=v}\dfrac{2}{R}.$

The Bergman metric can be also be defined for any complex space $Y$, provided
that the Bergman kernel is positive. Let $K_{Y}$ be the canonical bundle of
$Y$ and let $W$ be the space of $L^{2}$ holomorphic sections of $K_{Y}$ in the
sense that if $\sigma\in W$, then

$\left\|\sigma\right\|_{L^{2}}^{2}=\int_{Y}(\sqrt{-1})^{n^{2}}\sigma\wedge\overline{\sigma}<\infty.$

The inner product on $W$ is defined by

$<\sigma,\vartheta>=\int_{Y}(\sqrt{-1})^{n^{2}}\sigma\wedge\overline{\vartheta}$

for $\sigma,\vartheta\in W$.

######  Definition 3.1.3.

Let $\sigma_{1}$, $\sigma_{2}$, … be an orthonormal basis on $W$. The Bergman
kernel form is the non-negative $(n,n)$-form

$B_{Y}=\sum_{j=1}^{\infty}(\sqrt{-1})^{n^{2}}\sigma_{j}\wedge\overline{\sigma_{j}}.$

In local coordinates $z_{1}$, …, $z_{n}$ on $Y$ one can express $B_{Y}$ as

$B_{Y}=BE_{Y}(z,\overline{z})(\sqrt{-1})^{n^{2}}dz_{1}\wedge...\wedge
dz_{n}\wedge d\overline{z_{1}}\wedge...\wedge d\overline{z_{n}}$

where $BE_{Y}(z,\overline{z})$ is called the Bergman kernel function. If
$B_{Y}$ is positive, then the Bergman metric can be defined by

$B_{i\overline{j}}=\dfrac{\partial^{2}\log BE_{Y}(z,\overline{z})}{\partial
z_{i}\partial\overline{z_{j}}}.$

Note that the Bergman metric is well-defined and is non-degenerate if the
elements in $W$ separate points and the first jet of Y, and it becomes a
Kähler metric in this case. Also note that the Bergman metric
$\omega_{B}^{\prime}$ on $\mathcal{M}_{g}$ and the induced Bergman metric
$\omega_{B}$ on $\mathcal{M}_{g}$ from the Bergman metric on $T_{g}$ (by abuse
of notation, let’s still denote it by $\omega_{B}$) are different in the sense
that $\omega_{B}$ is complete while $\omega_{B}^{\prime}$ is incomplete.

####  3.1.2 Equivalences of the Metrics

Royden[23] showed that the Teichmüller metric and the Kobayashi metric in fact
coincide in $T_{g}$ (and in $\mathcal{M}_{g}$ as well). Now we go into the
equivalences between the Kobayashi metric, the Carathéodory metric and the
Bergman metric.

Here are some facts about $\omega_{K}$, $\omega_{C}$ and $\omega_{B}$ which
can be used to prove the equivalence of the Kobayashi metric and the
Carathéodory metric and the equivalence of the Kobayashi metric and the
Bergman metric.

######  Lemma 3.1.1.

(Lemma 2.1, [18]) Let $X$ be a complex space. Then

  * (i)

$\left\|\cdot\right\|_{C,X}\leq\left\|\cdot\right\|_{K,X}$;

  * (ii)

Let $Y$ be another complex space and $f:X\rightarrow Y$ be a holomorphic map.
Let $p\in X$ and $v\in T_{p}X$. Then
$\left\|f_{*}(v)\right\|_{C,Y,f(p)}\leq\left\|v\right\|_{C,X,p}$ and
$\left\|f_{*}(v)\right\|_{K,Y,f(p)}\leq\left\|v\right\|_{K,X,p}$;

  * (iii)

If $X\subset Y$ is a connected open subset, then $BE_{Y}(z)\leq BE_{X}(z)$ for
any local coordinates around $z\in X$;

  * (iv)

If the Bergman kernel of $X$ is positive, then at each $z\in X$, a peak
section $\sigma$ at $z$ exists. This existence is unique up to a constant
factor $c$ with norm 1. Furthermore, we have $BE_{X}(z)=|\sigma(z)|^{2}$ for
any local coordinates around $z$;

  * (v)

If the Bergman kernel of $X$ is positive, then $\left\|\cdot\right\|_{C,X}\leq
2\left\|\cdot\right\|_{B,X}$;

  * (vi)

If $X$ is a bounded convex domain in $\mathbb{C}^{n}$, then
$\left\|\cdot\right\|_{C,X}=\left\|\cdot\right\|_{K,X}$;

  * (vii)

Let $|\cdot|$ be the Euclidean norm and let $B_{r}$ be the open ball with
center $0$ and radius $r$ in $\mathbb{C}^{n}$. Then for any holomorphic
tangent vector $v$ at $0$,

$\left\|v\right\|_{C,B_{r},0}=\left\|v\right\|_{K,B_{r},0}=\dfrac{2}{r}|v|.$

###### Proof.

(i)-(vi) are in fact results of Kobayashi (Proposition 4.2.4, Proposition
4.2.3, Proposition 3.5.18, Proposition 4.10.4, Proposition 4.10.3, Theorem
4.10.18, Theorem 4.8.13, [15]). The last claim follows form (ii) and (vi) by
considering the maps $i:\Delta_{r}\rightarrow B_{r}$ and
$j:B_{r}\rightarrow\Delta_{r}$ given by $i(z)=(z,0,...,0)$ and
$j(z_{1},...,z_{n})=z_{1}$. ∎

The following theorem proved by Liu-Sun-Yau (Theorem 2.1, [18]) tells us the
equivalence of the Kobayashi metric and the Carathéodory metric and the
equivalence of the Kobayashi metric and the Bergman metric on $T_{g}$. Its
proof is quite a direct comparison between the mentioned metrics, so it is
omitted here.

######  Theorem 3.1.1.

There is a positive constant $C$ only depending on $g$ such that for each
$X\in T_{g}$ and each $v\in T_{X}T_{g}$, we have

$C^{-1}\left\|v\right\|_{K,T_{g},X}\leq\left\|v\right\|_{C,T_{g},X}\leq
C\left\|v\right\|_{K,T_{g},X}$

and

$C^{-1}\left\|v\right\|_{K,T_{g},X}\leq\left\|v\right\|_{B,T_{g},X}\leq
C\left\|v\right\|_{K,T_{g},X}.$

###  3.2 Weil-Petersson Metric

In this section the Weil-Petersson metric $\omega_{WP}$ will be introduced.
Some of the basic properties and the curvature formula of $\omega_{WP}$ will
also be given.

####  3.2.1 Definition and Properties of the Weil-Petersson Metric

Let $\mathcal{M}_{g}$ be the moduli space of a Riemann surface of genus $g$
with $g\geq 2$. Also, let $\mathcal{X}$ be the total space and
$\pi:\mathcal{X}\rightarrow\mathcal{M}_{g}$ be the projection map. Before
defining the Weil-Petersson metric, we have to fix some notations first.

Let $s_{1},...,s_{n}$ be holomorphic coordinates near a regular point
$s\in\mathcal{M}_{g}$ and assume that $z$ is a holomorphic local coordinate on
the fiber $X_{s}=\pi^{-1}(s)$. For the holomorphic vector fields
$\dfrac{\partial}{\partial s_{1}},...,\dfrac{\partial}{\partial s_{n}}$, there
are vector fields $v_{1},...,v_{n}$ on $\mathcal{X}$ such that

  * (i)

$\pi_{*}(v_{i})=\dfrac{\partial}{\partial s_{i}}$ for $i=1,...,n$;

  * (ii)

$\overline{\partial}v_{i}$ are harmonic $TX_{s}$-valued $(0,1)$-forms for
$i=1,...,n$.

The vector fields $v_{1},...,v_{n}$ are called the harmonic lifts of
$\dfrac{\partial}{\partial s_{1}},...,\dfrac{\partial}{\partial s_{n}}$, whose
existence was proved by Siu[26]. In fact, Schumacher[25] gave the following
explicit construction of the harmonic lift:

Let $\lambda=\dfrac{\sqrt{-1}}{2}\lambda(z,s)dz\wedge d\overline{z}$ be the
Kähler-Einstein metric on the fiber $X_{s}$. For convenience, let
$\partial_{z}=\dfrac{\partial}{\partial z}$ and
$\partial_{i}=\dfrac{\partial}{\partial s_{i}}$. Notice that the Kähler-
Einstein condition gives us
$\lambda=\partial_{z}\partial_{\overline{z}}\log\lambda$. Now let
$a_{i}=-\lambda^{-1}\partial_{i}\partial_{\overline{z}}\log\lambda$ and
$A_{i}=\partial_{\overline{z}}a_{i}$. Then we have

######  Lemma 3.2.1.

(Lemma 2.1, [17]) The harmonic horizontal lift of $\partial_{i}$ is
$v_{i}=\partial_{i}+a_{i}\partial_{z}$. In particular,

$B_{i}=A_{i}\partial_{z}\otimes d\overline{z}\in H^{1}(X_{s},TX_{s})$

is harmonic. Furthermore, the lift $\partial_{i}\mapsto B_{i}$ gives the
Kodaira-Spencer map $T_{s}\mathcal{M}_{g}\rightarrow H^{1}(X_{s},TX_{s})$.

Now we are ready to define $\omega_{WP}$.

######  Definition 3.2.1.

The Weil-Petersson metric on $\mathcal{M}_{g}$ is defined as

$h_{i\overline{j}}=\int_{X_{s}}B_{i}\cdot\overline{B_{j}}dv=\int_{X_{s}}A_{i}\overline{A_{j}}dv$

where $dv=\dfrac{\sqrt{-1}}{2}\lambda dz\wedge d\overline{z}$ is the volume
form on the fiber $X_{s}$.

Notice that $\omega_{WP}$ is incomplete and the Ricci curvature of
$\omega_{WP}$ is negative.

####  3.2.2 Results about Harmonic Lifts

The harmonic lifts $v_{1},...,v_{n}$ will paly an essential role in
calculating the curvature formulae of the Weil-Petersson metric and the Ricci
metric (which will be discussed in the next chapter), so in this section some
useful results related to $v_{1},...,v_{n}$ will be provided.

Let us review several operators which will be used for the simplifications of
the computations afterwards in this thesis.

Define an $(1,1)$-form on the total space $\mathcal{X}$ by

$G=\dfrac{\sqrt{-1}}{2}\partial\overline{\partial}\log\lambda=\dfrac{\sqrt{-1}}{2}(G_{i\overline{j}}ds_{i}\wedge
d\overline{s_{j}}-\lambda a_{i}ds_{i}\wedge
d\overline{z})-\lambda\overline{a_{i}}dz\wedge d\overline{s_{i}}+\lambda
dz\wedge d\overline{z}.$

Then introduce

$e_{i\overline{j}}=\dfrac{2}{\sqrt{-1}}G(v_{i},\overline{v_{j}})=G_{i\overline{j}}-\lambda
a_{i}\overline{a_{j}}$

to be a global function. Also, let $f_{i\overline{j}}=A_{i}\overline{A_{j}}$.
Schumacher showed the following lemma:

######  Lemma 3.2.2.

We have

$f_{i\overline{j}}=(\Box+1)e_{i\overline{j}}$

where $\Box=-\lambda^{-1}\partial_{z}\partial_{\overline{z}}$.

It will be useful to study the action of $v_{i}$ on $e_{i\overline{j}}$.

######  Lemma 3.2.3.

(Lemma 3.2, [17]) We have

$v_{k}(e_{i\overline{j}})=v_{i}(e_{k\overline{j}}).$

###### Proof.

By applying $dG$ to $v_{i}$, $v_{k}$ and $\overline{v_{j}}$, and noting that
$dG=0$, $g(v_{i},v_{k})=0$ and all $[v_{k},\overline{v_{j}}]$,
$[v_{i},\overline{v_{j}}]$, $[v_{i},v_{k}]$ are tangent to $X_{s}$ and
perpendicular to $v_{i}$, $v_{k}$, $\overline{v_{j}}$ with respect to $G$
respectively, the lemma follows. ∎

Let us define a new operator which is related to the Maass operators:

######  Definition 3.2.2.

Define

$P:C^{\infty}(X_{s})\rightarrow\Gamma(\Lambda^{1,0}(T^{0,1}X_{s}))$

by $P(f)=\partial_{z}(\lambda^{-1}\partial_{z}f)$. Notice that its dual
operator

$P^{*}:\Gamma(\Lambda^{1,0}(T^{0,1}X_{s}))\rightarrow C^{\infty}(X_{s})$

is given by
$P^{*}(B)=\lambda^{-1}\partial_{z}(\lambda^{-1}\partial_{z}(\lambda B))$.

We also recall the definitions of Maass operators by Wolpert[30]. Let $X$ be a
Riemann surface and $\kappa$ be its canonical bundle. For any integer $p$, let
$S(p)$ be the space of smooth sections of
$(\kappa\otimes\overline{\kappa}^{-1})^{\frac{p}{2}}$. Also, let
$ds^{2}=\rho^{2}(z)|dz|^{2}$ be a conformal metric.

######  Definition 3.2.3.

The Maass operators $K_{p}$ and $L_{p}$ are defined to be the metric
derivative $K_{p}:S(p)\rightarrow S(p+1)$ and $L_{p}:S(p)\rightarrow S(p-1)$
given by

$K_{p}(\sigma)=\rho^{p-1}\partial_{z}(\rho^{-p}\sigma)$

and

$L_{p}(\sigma)=\rho^{-p-1}\partial_{z}(\rho^{p}\sigma)$

for $\sigma\in S(p)$.

By definition we have $P=K_{1}K_{0}$.

Since $\omega_{WP}$ is defined using the integral along fibers, the following
formula will help us to calculate the curvature formula:

$\partial_{i}\int_{X_{s}}\eta=\int_{X_{s}}L_{v_{i}}\eta$

where $\eta$ is a relative $(1,1)$-form on $\mathcal{X}$. Notice that the Lie
derivative is defined by

$L_{v_{i}}\eta=\lim_{t\rightarrow 0}\dfrac{1}{t}((\varphi_{t})^{*}\eta-\eta)$

for a form $\sigma$ on $X_{s}$ and by

$L_{v_{i}}\xi=\lim_{t\rightarrow 0}\dfrac{1}{t}((\varphi_{-t})_{*}\xi-\xi)$

for a vector field $\xi$ on $X_{s}$, where $\varphi_{t}$ is the one parameter
group generated by $v_{i}$. For convenience, denote $L_{v_{i}}$ by $L_{i}$.
Then we have

######  Proposition 3.2.1.

(Proposition 3.1, [17])

$L_{i}\sigma=i(v_{i})d_{1}\sigma+d_{1}i(v_{1})\sigma$

where $d_{1}$ is the differential operator along the fiber $X_{s}$ and

$L_{i}\xi=[v_{i},\xi].$

Now we have the following lemmas (Lemma 3.3-3.4, [17]) concerning the Lie
derivative $L_{i}$ of $B_{j}$:

######  Lemma 3.2.4.

We have

  * (i)

$L_{i}dv=0$;

  * (ii)

$L_{\overline{l}}(B_{i})=-\overline{P}(e_{i\overline{l}})-f_{i\overline{l}}\partial_{\overline{z}}\otimes
d\overline{z}+f_{i\overline{l}}\partial_{z}\otimes dz$;

  * (iii)

$L_{k}(\overline{B_{i}})=-P(e_{k\overline{j}})-f_{k\overline{j}}\partial_{z}\otimes
dz+f_{k\overline{j}}\partial_{\overline{z}}\otimes d\overline{z}$;

  * (iv)

$L_{k}(B_{i})=(v_{k}(A_{i})-A_{i}\partial_{z}a_{k})\partial_{z}\otimes
d\overline{z}$;

  * (v)

$L_{\overline{l}}(\overline{B_{j}})=(\overline{v_{l}}(\overline{A_{j}})-\overline{A_{j}}\partial_{\overline{z}}\overline{a_{l}})\partial_{\overline{z}}\otimes
dz$.

######  Lemma 3.2.5.

$L_{k}(B_{i})\in H^{1}(X_{s},TX_{s})$ is harmonic.

The harmonicity of $L_{k}B_{i}$ is helpful in simplifying the curvature
formula if we consider the normal coordinates by noting the following
corollary (Corollary 3.1, [17]):

######  Corollary 3.2.1.

Let $s_{1},...s_{n}$ be normal coordinates at $s\in\mathcal{M}_{g}$ with
respect to $\omega_{WP}$. Then at $s$ we have

$L_{k}B_{i}=0$

for all $i,k$.

###### Proof.

Note that $B_{1},...,B_{n}$ is a basis of $T_{s}\mathcal{M}_{g}$, by
harmonicity we can express $L_{k}B_{i}$ as

$L_{k}B_{i}=h^{p\overline{q}}(\int_{X_{s}}L_{k}B_{i}\cdot\overline{B_{q}}dv)B_{p}=h^{p\overline{q}}\partial_{k}h_{i\overline{q}}B_{p}$

where the last equality follows form normality. ∎

The following formula for the commutator of $v_{k}$ and $\overline{v_{l}}$ is
quite useful:

######  Lemma 3.2.6.

(Lemma 3.5, [17])

$[\overline{v_{l}},v_{k}]=-\lambda^{-1}\partial_{\overline{z}}e_{k\overline{l}}\partial_{z}+\lambda{-1}\partial_{z}e_{k\overline{l}}\partial_{\overline{z}}.$

###### Proof.

Since we have

$\displaystyle[\overline{v_{l}},v_{k}]$
$\displaystyle=\overline{v_{l}}v_{k}-v_{k}\overline{v_{l}}$
$\displaystyle=(\partial_{\overline{l}}+\overline{a_{l}}\partial_{\overline{z}})(\partial_{k}+a_{k}\partial_{z})$
$\displaystyle\quad-(\partial_{k}+a_{k}\partial_{z})(\partial_{\overline{l}}+\overline{a_{l}}\partial_{\overline{z}})$
$\displaystyle=\partial_{\overline{l}}\partial_{k}+\overline{a_{l}}\partial_{\overline{z}}\partial_{k}+\overline{v_{l}}(a_{k})\partial_{z}+a_{k}\partial_{\overline{l}}\partial_{z}+\overline{a_{l}}a_{k}\partial_{\overline{z}}\partial_{z}$
$\displaystyle\quad-\partial_{k}\partial_{\overline{l}}-a_{k}\partial_{z}\partial_{\overline{l}}-v_{k}(\overline{a_{l}})\partial_{\overline{z}}-\overline{a_{l}}\partial_{k}\partial_{\overline{z}}-a_{k}\overline{a_{l}}\partial_{z}\partial_{\overline{z}}$
$\displaystyle=\overline{v_{l}}(a_{k})\partial_{z}-v_{k}(\overline{a_{l}})\partial_{\overline{z}},$

and by Lemma 3.2.4 we know that

$\overline{v_{l}}(a_{k})=\partial_{\overline{l}}a_{k}+\overline{a_{l}}\partial_{\overline{z}}a_{k}=-\lambda^{-1}\partial_{\overline{z}}e_{k\overline{l}}-\overline{a_{l}}A_{k}+\overline{a_{l}}A_{k}=-\lambda^{-1}\partial_{\overline{z}}e_{k\overline{l}}$

and

$v_{k}(\overline{a_{l}})=\partial_{k}\overline{a_{l}}+a_{k}\partial_{z}\overline{a_{l}}=-\lambda{-1}\partial_{z}e_{k\overline{l}}-a_{k}\overline{A_{l}}+a_{k}\overline{A_{l}}=-\lambda{-1}\partial_{z}e_{k\overline{l}},$

we obtain the desired result. ∎

####  3.2.3 Curvature Formula for the Weil-Petersson Metric

Let us fix some notation for curvature. Let $(M,g)$ be a Kähler manifold. Then
the curvature tensor is given by

$R_{i\overline{j}k\overline{l}}=\dfrac{\partial^{2}g_{i\overline{j}}}{\partial
z_{k}\partial\overline{z_{l}}}-g^{p\overline{q}}\dfrac{\partial
g_{i\overline{q}}}{\partial z_{k}}\dfrac{\partial
g_{p\overline{j}}}{\partial\overline{z_{l}}}.$

Hence the Ricci curvature is given by

$R_{i\overline{j}}=-g^{k\overline{l}}R_{i\overline{j}k\overline{l}}.$

The following curvature formula for $\omega_{WP}$ was proved by Wolpert[31],
Schumacher[25] and Siu[26], and its proof can be found in [17]:

######  Theorem 3.2.1.

The curvature of $\omega_{WP}$ is given by

$R_{i\overline{j}k\overline{l}}=\int_{X_{s}}(e_{i\overline{j}}f_{k\overline{l}}+e_{i\overline{l}}f_{k\overline{j}})dv.$

##  Chapter 4 Kähler Metrics on the Teichmüller Spaces and the Moduli Spaces
of Riemann Surfaces

###  4.1 McMullen Metric

The McMullen Metric is a new Kähler metric on $\mathcal{M}_{g}$ constructed by
McMullen[20]. This metric is equivalent to the Teichmüller metric and is
Kähler hyperbolic, which were also proved by McMullen.

####  4.1.1 Definition of the McMullen Metric

For a closed geodesic $\gamma$ on a Riemann surface of genus $g$, let
$l_{\gamma}(X)$ be the length of the corresponding hyperbolic geodesic on
$X\in T_{g}=T(R)$. Mumford[22] showed that a sequence
$X_{n}\in\mathcal{M}_{g}$ tends to infinity if and only if
$\inf_{\gamma}l_{\gamma}(X_{n})\rightarrow 0$. This behavior motivates
McMullen to use the reciprocal length functions $\dfrac{1}{l_{\gamma}}$ to
define the McMullen metric $g_{1/l}$.

######  Definition 4.1.1.

The McMullen metric $g_{1/l}$ on the Teichmüller space $T_{g}$ is defined by
$g_{1/l}(v,w)=\omega_{1/l}(v,\sqrt{-1}w)$ for $v,w\in T_{X}T_{g}$, where
$\omega_{1/l}$ is the Kähler form of $g_{1/l}$ given by

$\omega_{1/l}=\omega_{WP}-\sqrt{-1}\delta\sum_{l_{\gamma}(X)<\epsilon}\partial\overline{\partial}Log\dfrac{\epsilon}{l_{\gamma}}$

for suitable choices of small constants $\delta,\epsilon>0$ and
$Log:\mathbb{R}_{+}\rightarrow[0,\infty)$ being a smooth function such that

$Log(x)=\left\\{\begin{array}[]{ll}\log(x)&\text{if }x\geq 2;\\\ 0&\text{if
}x\leq 1.\end{array}\right.$

The sum above is taken over primitive short geodesics $\gamma$ on $X$, which
has at most $\dfrac{3g}{2}$ terms.

######  Remark 4.1.1.

Notice that $g_{1/l}$ is invariant under the action of $Mod_{g}$, so $g_{1/l}$
is also defined on the moduli space $\mathcal{M}_{g}$.

####  4.1.2 Properties of the McMullen Metric

As mentioned before, McMullen metric is Kähler hyperbolic. Before we go into
the definition of Kähler hyperbolic metric, we should first understand what
$d(bounded)$ is.

######  Definition 4.1.2.

An $n$-form $\alpha$ on a Kähler manifold is $d(bounded)$ if there exists a
bounded $(n-1)$-form $\beta$ such that $\alpha=d\beta$

Now we can define the Kähler hyperbolic metric.

######  Definition 4.1.3.

Let $(M,g)$ be a Kähler manifold. $(M,g)$ is Kähler hyperbolic if the
following conditions are satisfied:

  * (i)

On the universal covering manifold $(\tilde{M},\tilde{g})$, the Kähler form
$\tilde{\omega}$ of the pulled-back metric $\tilde{g}$ is $d(bounded)$;

  * (ii)

$(M,g)$ is complete and of finite volume;

  * (iii)

The sectional curvatures of $(M,g)$ are bounded from above and below;

  * (iv)

The injectivity radius of $(\tilde{M},\tilde{g})$ is bounded from below.

Note that the last three conditions would be automatic when $M$ is compact. A
Kähler metric $g$ on $M$ is said to be Kähler hyperbolic if $(M,g)$ is Kähler
hyperbolic.

The following theorem is a collection of the McMullen’s results (Corollary
5.2, Corollary 7.3 and Theorem 8.1-8.2, [20]) which were used to prove that
$g_{1/l}$ is Kähler hyperbolic.

######  Theorem 4.1.1.

The McMullen metric $g_{1/l}$ satisfies the following properties:

  * (i)

$g_{1/l}$ is complete;

  * (ii)

The Kähler form $\omega_{1/l}$ of the McMullen metric $g_{1/l}$ on $T_{g}$ is
$d(bounded)$ with primitive

$\theta_{1/l}=\theta_{WP}-\delta\sum_{l_{\gamma}(X)<\epsilon}\partial
Log\dfrac{\epsilon}{l_{\gamma}}$

satisfying $d(\sqrt{-1}\theta_{1/l})=\omega_{1/l}$. Note that
$\sqrt{-1}\theta_{WP}$ is a primitive of the Weil-Petersson Kähler form
defined by $\theta_{WP}(X)=-\beta_{X}(Y)$ for a fixed $Y$ in the conjugate
Teichmüller space $T_{g}^{conj}$, where $\beta_{X}:T_{g}^{conj}\rightarrow
Q(X)$ is the Bers embedding;

  * (iii)

$(M_{g},g_{1/l})$ has finite volume;

  * (iv)

The sectional curvatures of $g_{1/l}$ are bounded form below and above over
$T_{g}$, and the injectivity radius of $g_{1/l}$ is uniformly bounded from
below.

Hence $g_{1/l}$ is Kähler hyperbolic.

######  Sketch of the proof.

(i) follows from the fact that $g_{1/l}$ is equivalent to the Teichmüller
metric (which will be discussed in the next subsection) and the Teichmüller
metric is complete.

To prove that $g_{1/l}$ on $T_{g}$ is $d(bounded)$ in (ii), it suffices to
show that $\theta_{1/l}$ is bounded with respect to
$\left\|\cdot\right\|_{T}$. The first term $\theta_{WP}$ is bounded using
Nehari’s bound (Theorem 2.1, [20]) and Gauss-Bonnet theorem. The remaining
terms are bounded by the estimate for the gradient of geodesic length:
$\left\|\partial l_{\gamma}\right\|_{T}\leq 2l_{\gamma}$.

For (iii), we consider the Deligne-Mumford compactification
$\overline{\mathcal{M}_{g}}$ of $\mathcal{M}_{g}$. Then by the existence of a
covering $\\{V_{\alpha}\\}$ with
$\textrm{vol}(V_{\alpha}\cap\mathcal{M}_{g})<\infty$ in the Kobayashi metric
on $M_{g}$ and the coincidence of the Teichmüller metric and the Kobayashi
metric on $M_{g}$, we know that the Teichmüller volume of $M_{g}$ is finite.
Again using the equivalence of $g_{1/l}$ and the Teichmüller metric we can
prove (iii).

Finally for (iv), we first extend $l_{\gamma}$ and $\theta_{WP}$ to
holomorphic functions on the complexification $QF_{g}=T_{g}\times
T_{g}^{conj}$ of $T_{g}$. Then we can use the local uniform bounds on these
holomorphic functions to control all of their derivatives, and thus the
derivatives of $g_{1/l}$ are also controlled, which implies the sectional
curvatures are bounded. The lower bound of the injectivity radius hence
follows from the curvature bounds.

####  4.1.3 Equivalence of the McMullen Metric and the Teichmüller Metric

Let’s fix some notations and present some useful facts before the discussion
of the equivalence of the McMullen metric and the Teichmüller metric. Let
$N=\dfrac{3g}{2}+1$ be a bound on the number of terms of $\omega_{1/l}$, and
let

$\psi_{\gamma}=\partial\log l_{\gamma}=\dfrac{\partial
l_{\gamma}}{l_{\gamma}}$

for a given $X\in T_{g}$. Note that we have

$|\psi_{\gamma}(v)|^{2}=\dfrac{\sqrt{-1}}{2}\dfrac{\partial
l_{\gamma}\wedge\overline{\partial}l_{\gamma}}{l_{\gamma}^{2}}(v,\sqrt{-1}v).$

Now let $\pi:X_{\gamma}\rightarrow X$ be the covering space corresponding to
$<\gamma>\subset\pi_{1}(R)$. Notice that we can identify $X_{\gamma}$ with an
annulus $A(r)=\\{z|r^{-1}<|z|<r\\}$ up to rotations by requiring that the
orientation of $\tilde{\gamma}\subset X_{\gamma}$ agrees with that of
$\mathbb{S}^{1}$. Define $\phi_{\gamma}\in Q(X)$ by

$\phi_{\gamma}=\pi_{*}(\dfrac{dz^{2}}{z}).$

Wolpert (Theorem 3.1, [29]) showed the following important relation between
$\phi_{\gamma}$ and $\psi_{\gamma}$:

$\psi_{\gamma}=-\dfrac{l_{\gamma}(X)}{2\pi^{3}}\phi_{\gamma}.$

Here are the facts about the quadratic differentials which will be used in the
proof of the equivalence of $g_{1/l}$ and the Teichmüller metric (Theorem
4.1-4.4, [20]):

######  Theorem 4.1.2.

For $\epsilon>0$ sufficiently small, any $\phi\in Q(X)$ can be uniquely
expressed as

$\phi=\phi_{0}+\sum_{l_{\gamma}(X)<\epsilon}a_{\gamma}\phi_{\gamma}$

with $\textrm{Res}_{\gamma}(\phi_{0})=0$ for all $\gamma$ in the sum above
such that

  * (i)

$\left\|\phi_{0}\right\|_{T}=O(\left\|\phi\right\|_{T})$ and
$\left\|a_{\gamma}\phi_{\gamma}\right\|_{T}=O(\left\|\phi\right\|_{T})$;

  * (ii)

$\left\|\phi_{0}\right\|_{WP}\leq C(\epsilon)\left\|\phi_{0}\right\|_{T}$ for
some constant $C(\epsilon)$ depending on $\epsilon$.

If $\phi=\phi_{\delta}$ with $\epsilon<l_{\delta}(X)<2\epsilon$, then we have
$\left\|a_{\gamma}\phi_{\gamma}\right\|_{T}=O(l_{\delta}(X)\left\|\phi\right\|_{T})$.
Moreover, $\psi_{\gamma}$ is proportional to $\phi_{\gamma}$ such that
$\left\|\psi_{\gamma}\right\|_{T}\leq 2$ and
$\left\|\psi_{\gamma}\right\|_{T}\rightarrow 2$ as $l_{\gamma}\rightarrow 0$.

The following two lemmas will play the main roles in the proof of the
equivalence as well.

######  Lemma 4.1.1.

There is a Hermitian metric $g$ of the form

$g(v,v)=A(\epsilon)\left\|v\right\|_{WP}^{2}+B\sum_{l_{\gamma}(X)<\epsilon}|\psi_{\gamma}(v)|^{2}$

such that $\left\|v\right\|_{T}^{2}\leq g(v,v)\leq
O(\left\|v\right\|_{T}^{2})$ for all sufficiently small $\epsilon>0$. Hence we
have

$\left\|v\right\|_{T}^{2}\sim\left\|v\right\|_{WP}^{2}+\sum_{l_{\gamma}(X)<\epsilon}|\partial\log
l_{\gamma}(v)|^{2}$

for $\epsilon>0$ sufficiently small and for all $v\in T_{X}T_{g}$.

###### Proof.

By Cauchy-Schwarz inequality and the duality between $Q(X)=T^{*}_{X}T_{g}$ and
$M(X)=T_{X}T_{g}$, we know that
$\left\|v\right\|_{WP}=O(\left\|v\right\|_{T})$ (Proposition 2.4, [20]). Now
by Theorem 4.1.2 we have $|\psi_{\gamma}(v)|\leq 2\left\|v\right\|_{T}$, and
since there are at most $N$ terms in the expression of $g$, we obtain
$g(v,v)\leq O(\left\|v\right\|_{T}^{2})$.

To prove $\left\|v\right\|_{T}^{2}\leq g(v,v)$, we first fix $v\in T_{X}T_{g}$
and pick $\phi\in Q(X)$ such that $\left\|\phi\right\|_{T}=1$ and
$\phi(v)=\left\|v\right\|_{T}$. Then as long as $\epsilon>0$ is small enough,
by Theorem 4.1.2 we can obtain

$\phi=\phi_{0}+\sum_{l_{\gamma}(X)<\epsilon}a_{\gamma}\psi_{\gamma}$

with $\textrm{Res}_{\gamma}(\phi_{0})=0$ and
$\left\|\psi_{\gamma}\right\|_{T}\geq 1$. Moreover we have
$\left\|\phi_{0}\right\|_{T}=O(1)$ and
$\left\|a_{\gamma}\psi_{\gamma}\right\|_{T}=O(1)$ (as
$\left\|\phi\right\|_{T}=1$). Then by duality and Theorem 4.1.2 again we have
$|\phi_{0}(v)|\leq D(\epsilon)\left\|v\right\|_{WP}$ for some constant
$D(\epsilon)$ depending on $\epsilon$. Since
$\left\|\psi_{\gamma}\right\|_{T}\geq 1$ and
$\left\|a_{\gamma}\psi_{\gamma}\right\|_{T}=O(1)$, we know that
$|a_{\gamma}|\leq E$, where $E$ is a constant independent of $\epsilon$. Hence
we have

$\left\|v\right\|_{T}=\phi(v)\leq
D(\epsilon)\left\|v\right\|_{WP}+E\sum_{l_{\gamma}(X)<\epsilon}|\psi_{\gamma}|.$

Notice that $N$ is a bound for the number of terms in the sum above, so by
Cauchy-Schwarz inequality we obtain

$\left\|v\right\|_{T}^{2}\leq
ND(\epsilon)^{2}\left\|v\right\|_{WP}^{2}+NE^{2}\sum_{l_{\gamma}(X)<\epsilon}|\psi_{\gamma}|^{2}.$

Thus taking $A(\epsilon)=ND(\epsilon)$ and $B=NE^{2}$, the result follows. ∎

######  Lemma 4.1.2.

For $\epsilon<l_{\delta}(X)<2\epsilon$, we have

$|\psi_{\delta}(v)|^{2}\leq
D(\epsilon)\left\|v\right\|_{WP}^{2}+O(\epsilon\sum_{l_{\gamma}(X)<\epsilon}|\psi_{\gamma}(v)|^{2}).$

###### Proof.

By Theorem 4.1.2 we can derive that
$\displaystyle\psi_{\delta}=\psi_{0}+\epsilon\sum_{l_{\gamma}(X)<\epsilon}a_{\gamma}\psi_{\gamma}$
with $a_{\gamma}=O(l_{\delta}(X))=O(\epsilon)$ and $|\psi_{0}(v)|\leq
D(epsilon)\left\|v\right\|_{WP}$. Then using similar argument as in Lemma
4.1.1 we obtain the result. ∎

Now we are ready to prove the equivalence.

######  Theorem 4.1.3.

For all sufficiently small $\epsilon>0$, there exists a $\delta>0$ such that
$\omega_{1/l}$ defines a Kähler metric $g_{1/l}$ on $T_{g}$ which is
equivalent to the Teichmüller metric.

###### Proof.

Consider the $(1,1)$-form

$\omega_{1}=(F(\epsilon)+A(\epsilon))\omega_{WP}-B\sum_{l_{\delta}(X)<2\epsilon}\dfrac{\sqrt{-1}}{2}\partial\overline{\partial}Log\dfrac{2\epsilon}{l_{\delta}}$

where

$F(\epsilon)=4NBD(\epsilon)\sup_{[1,2]}|Log^{\prime\prime}(x)|.$

Observe that $\omega_{1}$ and $\omega_{1/l}$ are indeed of the same form (up
to scaling and replacing $\epsilon$ by $\dfrac{\epsilon}{2}$). Therefore, to
prove the theorem it suffices to show that
$g_{1}(v,v)=\omega_{1}(v,\sqrt{-1}v)$ is equivalent to the Teichmüller metric.

Now let $v\in T_{X}T_{g}$ have Teichmüller norm $\left\|v\right\|_{T}=1$. We
first compute

$\partial\overline{\partial}Log\dfrac{2\epsilon}{l_{\delta}}=2\epsilon\bigg{(}Log^{\prime}\dfrac{2\epsilon}{l_{\delta}}\bigg{)}\partial\overline{\partial}\bigg{(}\dfrac{1}{l_{\delta}}\bigg{)}+\dfrac{4\epsilon^{2}}{l_{\delta}^{2}}\bigg{(}Log^{\prime\prime}\dfrac{2\epsilon}{l_{\delta}}\bigg{)}\dfrac{\partial
l_{\delta}\wedge\overline{\partial}l_{\delta}}{l_{\delta}^{2}}.$

Using the fact that the function $\dfrac{1}{l_{\delta}}$ is pluriharmonic
(Theorem 3.1, [20]), we have

$\bigg{|}\partial\overline{\partial}\bigg{(}\dfrac{1}{l_{\delta}}\bigg{)}(v,\sqrt{-1}v)\bigg{|}=O(1).$

Since $Log^{\prime}(x)$ is bounded and
$|\psi_{\gamma}(v)|^{2}=\dfrac{\sqrt{-1}}{2}\dfrac{\partial
l_{\gamma}\wedge\overline{\partial}l_{\gamma}}{l_{\gamma}^{2}}(v,\sqrt{-1}v)$,
we know that

$\dfrac{\sqrt{-1}}{2}\partial\overline{\partial}Log\dfrac{2\epsilon}{l_{\delta}}(v,\sqrt{-1}v)=\dfrac{4\epsilon^{2}}{l_{\delta}^{2}}\bigg{(}Log^{\prime\prime}\dfrac{2\epsilon}{l_{\delta}}\bigg{)}|\psi_{\delta}(v)|^{2}+O(\epsilon).$

Now if $l_{\delta}<\epsilon$, then
$Log^{\prime\prime}\dfrac{2\epsilon}{l_{\delta}}=\log^{\prime\prime}\dfrac{2\epsilon}{l_{\delta}}=-\dfrac{l_{\delta}^{2}}{4\epsilon^{2}}$,
and thus

$-\dfrac{\sqrt{-1}}{2}\partial\overline{\partial}Log\dfrac{2\epsilon}{l_{\delta}}(v,\sqrt{-1}v)=|\psi_{\delta}(v)|^{2}+O(\epsilon).$

If $\epsilon\leq l_{\delta}<2\epsilon$, by Lemma 4.1.2 we have

$\displaystyle\bigg{|}\dfrac{\sqrt{-1}}{2}\partial\overline{\partial}Log\dfrac{2\epsilon}{l_{\delta}}(v,\sqrt{-1}v)\bigg{|}$
$\displaystyle\leq
4\psi_{\delta}(v)|^{2}\sup_{[1,2]}|Log^{\prime\prime}(x)|+O(\epsilon)$
$\displaystyle\leq\dfrac{F(\epsilon)}{NB}\left\|v\right\|_{WP}^{2}+O(\epsilon\sum_{l_{\gamma}(X)<\epsilon}|\psi_{\gamma}(v)|^{2})=O(\epsilon).$

Hence using these two estimates we obtain

$\displaystyle g_{1}(v,v)$ $\displaystyle\geq
A(\epsilon)\left\|v\right\|_{WP}^{2}+B\sum_{l_{\delta}(X)<\epsilon}|\psi_{\delta}(v)|^{2}$
$\displaystyle\quad+B\sum_{\epsilon\leq
l_{\delta}(X)<2\epsilon}\bigg{(}\dfrac{F(\epsilon)}{NB}\left\|v\right\|_{WP}^{2}-\bigg{|}\dfrac{\sqrt{-1}}{2}\partial\overline{\partial}Log\dfrac{2\epsilon}{l_{\delta}}(v,\sqrt{-1}v)\bigg{|}\bigg{)}+O(\epsilon)$
$\displaystyle\geq
A(\epsilon)\left\|v\right\|_{WP}^{2}+B(1+O(\epsilon))\sum_{l_{\delta}(X)<\epsilon}|\psi_{\delta}(v)|^{2}+O(\epsilon).$

Then by Lemma 4.1.1 we have

$g_{1}(v,v)\geq\left\|v\right\|_{T}^{2}+O(\epsilon)=1+O(\epsilon),$

which implies that $\left\|v\right\|_{T}^{2}=O(g_{1}(v,v))$ when $\epsilon$ is
sufficiently small. By similar argument in Lemma 4.1.1 we can also prove
$g_{1}(v,v)=O(\left\|v\right\|_{T}^{2})$. This finishes the proof. ∎

###  4.2 Kähler-Einstein Metric

####  4.2.1 Existence of the Kähler-Einstein Metric

A Kähler-Einstein metric $g$ is a Kähler metric whose Ricci tensor is
proportional to itself, i.e., $R_{i\overline{j}}=cg_{i\overline{j}}$. Cheng-
Yau[9] and Mok-Yau[21] showed the existence of the Kähler-Einstein Metric
$g_{KE}$ on $T_{g}$ with Ricci curvature equal to $-1$. More precisely, the
existence of $g_{KE}$ is ensured by the fact that $T_{g}$ is a bounded domain
of holomorphy (note that Bers-Ehrenpreis[8] showed that $T_{g}$ is a domain of
holomorphy and Bers[6][7] showed that $T_{g}$ is a bounded domain in
$\mathbb{C}^{3g-3}$) and the following theorem (Main Theorem, [21]):

######  Theorem 4.2.1.

Any bounded domain of holomorphy admits a complete Kähler-Einstein metric.

Notice that $g_{KE}$ descends to $\mathcal{M}_{g}$ as a canonical Kähler
metric as well.

####  4.2.2 A Conjecture of Yau

In [33], Yau proposed the following conjecture:

######  Conjecture.

The Kähler-Einstein metric is equivalent to the Teichmüller metric.

Finally it was confirmed by Liu-Sun-Yau[17], and we will review the answer to
this conjecture in chapter 5.

###  4.3 Ricci Metric

The Ricci metric is one of the new complete Kähler metrics (another one being
the perturbed Ricci metric, which will be examined in the next section)
introduced by Liu-Sun-Yau[17]. Their main purpose of introducing these two
metrics was to build up a bridge for proving the equivalence of the Kähler-
Einstein metric to the Teichmüller metric (and McMullen metric).

####  4.3.1 Definition of the Ricci Metric

Ahlfors[1] proved that the Ricci curvature of the Weil-Petersson metric is
negative, and Trapani[28] further showed that the negative of the Ricci
curvature of the Weil-Petersson metric defines a (complete) Kähler metric on
the moduli space $\mathcal{M}_{g}$. This is just the Ricci metric brought by
Yau et. al..

######  Definition 4.3.1.

The Ricci metric $\tau$ on the moduli space $\mathcal{M}_{g}$ is defined by

$\tau_{i\overline{j}}=-R_{i\overline{j}}=h^{\alpha\overline{\beta}}R_{i\overline{j}\alpha\overline{\beta}}.$

Before we step into the next subsection, let’s go through some useful
definitions and a lemma. Here we first define a new operator (which will play
a main role in the curvature formula) acting on functions on the fibers.

######  Definition 4.3.2.

For each $1\leq k\leq n$ and for any $f\in C^{\infty}(X_{s})$, we define the
commutator operator $\xi_{k}$ which acts on $C^{\infty}(X_{s})$ by

$\xi_{k}(f)=\overline{\partial}^{*}(i(B_{k})\partial
f)=-\lambda^{-1}\partial_{z}(A_{k}\partial_{z}f).$

$\xi_{k}$ is called the commutator operator because $\xi_{k}$ is indeed the
commutator of $(\Box+1)$ and $v_{k}$, which we can see from the following
lemma (Lemma 3.6, [17]):

######  Lemma 4.3.1.

As operators acting on $C^{\infty}(X_{s})$, we have

  * (i)

$(\Box+1)v_{k}-v_{k}(\Box+1)=\Box v_{k}-v_{k}\Box=\xi_{k};$

  * (ii)

$(\Box+1)\overline{v_{l}}-\overline{v_{l}}(\Box+1)=\Box\overline{v_{l}}-\overline{v_{l}}\Box=\overline{\xi}_{l};$

  * (iii)

$\xi_{k}(f)=-A_{k}\partial_{z}(\lambda^{-1}\partial_{z}f)=-A_{k}P(f)=-A_{k}K_{1}K_{0}(f).$

Furthermore, we have

$(\Box+1)v_{k}(e_{i\overline{j}})=\xi_{k}(e_{i\overline{j}})+\xi_{i}(e_{k\overline{j}})+L_{k}B_{i}\cdot\overline{B_{j}}$

###### Proof.

Notice that (ii) can be derived from (i) by taking conjugation. To prove (i),
we consider

$\displaystyle(\Box+1)v_{k}-v_{k}(\Box+1)$ $\displaystyle=\Box
v_{k}+v_{k}-v_{k}\Box-v_{k}=\Box v_{k}-v_{k}\Box$
$\displaystyle=-\lambda^{-1}\partial_{z}\partial_{\overline{z}}(a_{k}\partial_{z}+\partial_{k})-(a_{k}\partial_{z}+\partial_{k})(-\lambda^{-1}\partial_{z}\partial_{\overline{z}})$
$\displaystyle=-\lambda^{-1}\partial_{z}(\partial_{\overline{z}}a_{k}\partial_{z}+a_{k}\partial_{\overline{z}}\partial_{z}+\partial_{\overline{z}}\partial_{k})+a_{k}\partial_{z}(\lambda^{-1})\partial_{z}\partial_{\overline{z}}$
$\displaystyle\quad+\lambda^{-1}a_{k}\partial_{z}\partial_{z}\partial_{\overline{z}}+\partial_{k}(\lambda^{-1})\partial_{z}\partial_{\overline{z}}+\lambda^{-1}\partial_{k}\partial_{z}\partial_{\overline{z}}$
$\displaystyle=-\lambda^{-1}\partial_{z}(A_{k}\partial_{z})-\lambda^{-1}\partial_{z}a_{k}\partial_{\overline{z}}\partial_{z}-\lambda^{-1}a_{k}\partial_{z}\partial_{\overline{z}}\partial_{z}-\lambda^{-1}\partial_{z}\partial_{\overline{z}}\partial_{k}$
$\displaystyle-\lambda^{-2}\partial_{z}\lambda
a_{k}\partial_{z}\partial_{\overline{z}}+\lambda^{-1}a_{k}\partial_{z}\partial_{z}\partial_{\overline{z}}-\lambda^{-2}\partial_{k}\lambda\partial_{z}\partial_{\overline{z}}+\lambda^{-1}\partial_{k}\partial_{z}\partial_{\overline{z}}$
$\displaystyle=\xi_{k}-\lambda^{-1}(\partial_{z}a_{k}+\lambda^{-1}\partial_{z}\lambda
a_{k}+\lambda^{-1}\partial_{k}\lambda)\partial_{z}\partial_{\overline{z}}=\xi_{k}.$

Note that we used the fact that
$\partial_{z}a_{k}=-\lambda^{-1}\partial_{z}\lambda
a_{k}-\lambda^{-1}\partial_{k}\lambda$ in the last equality. Now we can prove
(iii). Using the harmonicity of $B_{k}$, we know that
$\overline{\partial}^{*}B_{k}=0$, which implies that $\partial_{z}(\lambda
A_{k})=0$. Hence

$\displaystyle\xi_{k}(f)$
$\displaystyle=-\lambda^{-1}\partial_{z}(A_{k}\partial_{z}f)=-\lambda^{-1}\partial_{z}(\lambda
A_{k}\lambda^{-1}\partial_{z}f)$
$\displaystyle=-\lambda^{-1}\partial_{z}(\lambda
A_{k})\lambda^{-1}\partial_{z}f-\lambda^{-1}\lambda
A_{k}\partial_{z}(\lambda^{-1}\partial_{z}f)$
$\displaystyle=-A_{k}P(f)=-A_{k}K_{1}K_{0}(f).$

The last part can be proved by (i) in this way:

$\displaystyle(\Box+1)v_{k}(e_{i\overline{j}})$
$\displaystyle=v_{k}((\Box+1)(e_{i\overline{j}}))+\xi_{k}(e_{i\overline{j}})=v_{k}(f_{i\overline{j}})+\xi_{k}(e_{i\overline{j}})$
$\displaystyle=L_{k}(B_{i}\cdot\overline{B_{j}})+\xi_{k}(e_{i\overline{j}})=L_{k}(B_{i})\cdot\overline{B_{j}}+B_{i}\cdot
L_{k}(\overline{B_{j}})+\xi_{k}(e_{i\overline{j}})$
$\displaystyle=L_{k}(B_{i})\cdot\overline{B_{j}}-A_{i}P(e_{k\overline{j}})+\xi_{k}(e_{i\overline{j}})=L_{k}(B_{i})\cdot\overline{B_{j}}+\xi_{i}(e_{k\overline{j}})+\xi_{k}(e_{i\overline{j}}).$

∎

Since the terms in the curvature formula will be highly symmetric with respect
to indices, it is more convenient to introduce the symmetrization operators.

######  Definition 4.3.3.

Let $U$ be any quantity which depends on indices
$i,k,\alpha,\overline{j},\overline{l},\overline{\beta}$. The symmetrization
operator $\sigma_{1}$ is defined by taking the summation of all orders of the
triple $(i,k,\alpha)$, i.e.,

$\displaystyle\sigma_{1}(U(i,k,\alpha,\overline{j},\overline{l},\overline{\beta}))$
$\displaystyle=U(i,k,\alpha,\overline{j},\overline{l},\overline{\beta})+U(i,\alpha,k,\overline{j},\overline{l},\overline{\beta})+U(k,i,\alpha,\overline{j},\overline{l},\overline{\beta})$
$\displaystyle\quad+U(k,\alpha,i,\overline{j},\overline{l},\overline{\beta})+U(\alpha,i,k,\overline{j},\overline{l},\overline{\beta})+U(\alpha,k,i,\overline{j},\overline{l},\overline{\beta}).$

In a similar fashion, we define $\sigma_{2}$ to be the symmetrization operator
of $\overline{j}$ and $\overline{\beta}$ and $\tilde{\sigma_{1}}$ to be the
symmetrization operator of $\overline{j},\overline{l},\overline{\beta}$.

####  4.3.2 Curvature Formula of the Ricci Metric

As to compute the curvature of the Ricci metric, we need to compute the first
and second order derivatives of $\tau_{i\overline{j}}$. Let’s compute the
first order derivatives first.

######  Theorem 4.3.1.

(Theorem 3.2, [17])

$\partial_{k}\tau_{i\overline{j}}=h^{\alpha\overline{\beta}}\bigg{\\{}\sigma_{1}\int_{X_{s}}(\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}})dv\bigg{\\}}+\tau_{p\overline{j}}\Gamma^{p}_{ik}$

where $\Gamma^{p}_{ik}$ is the Christoffel symbol of the Weil-Petersson
metric.

###### Proof.

Notice that $\Box+1$ is self-adjoint and by Lemma 3.2.2 we know that
$(\Box+1)e_{i\overline{j}}=f_{i\overline{j}}$. Then using Lemma 4.3.1 and
Theorem 3.2.1 we have

$\displaystyle\partial_{k}R_{i\overline{j}\alpha\overline{\beta}}$
$\displaystyle=\partial_{k}\int_{X_{s}}(e_{i\overline{j}}f_{\alpha\overline{\beta}}+e_{i\overline{\beta}}f_{\alpha\overline{j}})dv$
$\displaystyle=\int_{X_{s}}L_{k}\\{(e_{i\overline{j}}f_{\alpha\overline{\beta}}+e_{i\overline{\beta}}f_{\alpha\overline{j}})dv\\}$
$\displaystyle=\int_{X_{s}}(v_{k}(e_{i\overline{j}})f_{\alpha\overline{\beta}}+e_{i\overline{j}}v_{k}(f_{\alpha\overline{\beta}})+v_{k}(e_{i\overline{\beta}})f_{\alpha\overline{j}}+e_{i\overline{\beta}}v_{k}(f_{\alpha\overline{j}}))dv\qquad\qquad\qquad\
$
$\displaystyle=\int_{X_{s}}(v_{k}(e_{i\overline{j}})(\Box+1)e_{\alpha\overline{\beta}}+e_{i\overline{j}}v_{k}(f_{\alpha\overline{\beta}})+v_{k}(e_{i\overline{\beta}})(\Box+1)e_{\alpha\overline{j}}+e_{i\overline{\beta}}v_{k}(f_{\alpha\overline{j}}))dv$
$\displaystyle=\int_{X_{s}}((\Box+1)v_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}}+e_{i\overline{j}}v_{k}(f_{\alpha\overline{\beta}})+(\Box+1)v_{k}(e_{i\overline{\beta}})e_{\alpha\overline{j}}+e_{i\overline{\beta}}v_{k}(f_{\alpha\overline{j}}))dv$
$\displaystyle=\int_{X_{s}}((v_{k}(\Box+1)+\xi_{k})(e_{i\overline{j}})e_{\alpha\overline{\beta}}+e_{i\overline{j}}v_{k}(f_{\alpha\overline{\beta}})+(v_{k}(\Box+1)+\xi_{k})(e_{i\overline{\beta}})e_{\alpha\overline{j}}$
$\displaystyle\quad+e_{i\overline{\beta}}v_{k}(f_{\alpha\overline{j}}))dv$
$\displaystyle=\int_{X_{s}}(v_{k}(f_{i\overline{j}})e_{\alpha\overline{\beta}}+e_{i\overline{j}}v_{k}(f_{\alpha\overline{\beta}})+v_{k}(f_{i\overline{\beta}})e_{\alpha\overline{j}}+e_{i\overline{\beta}}v_{k}(f_{\alpha\overline{j}}))dv$
$\displaystyle\quad+\int_{X_{s}}(\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}}+\xi_{k}(e_{i\overline{\beta}})e_{\alpha\overline{j}})dv$
$\displaystyle=\int_{X_{s}}((L_{k}B_{i}\cdot\overline{B_{j}})e_{\alpha\overline{\beta}}+(L_{k}B_{\alpha}\cdot\overline{B_{\beta}})e_{i\overline{j}}+(L_{k}B_{i}\cdot\overline{B_{\beta}})e_{\alpha\overline{j}}+(L_{k}B_{\alpha}\cdot\overline{B_{j}})e_{i\overline{\beta}})dv$
$\displaystyle\quad+\int_{X_{s}}((B_{i}\cdot
L_{k}\overline{B_{j}})e_{\alpha\overline{\beta}}+(B_{\alpha}\cdot
L_{k}\overline{B_{\beta}})e_{i\overline{j}}+(B_{i}\cdot
L_{k}\overline{B_{\beta}})e_{\alpha\overline{j}}+(B_{\alpha}\cdot
L_{k}\overline{B_{j}})e_{i\overline{\beta}})dv$
$\displaystyle\quad+\int_{X_{s}}(\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}}+\xi_{k}(e_{i\overline{\beta}})e_{\alpha\overline{j}})dv=I+II+III.$

Notice that $B_{1},...,B_{n}$ is a basis of $T_{s}M_{g}$, so we can simplify
$I$ in this way:

$\displaystyle I$
$\displaystyle=\int_{X_{s}}(L_{k}B_{i}\cdot(\overline{B_{j}}e_{\alpha\overline{\beta}}+\overline{B_{\beta}}e_{\alpha\overline{j}})+L_{k}B_{\alpha}\cdot(\overline{B_{\beta}}e_{i\overline{j}}+\overline{B_{j}}e_{i\overline{\beta}})dv$
$\displaystyle=<L_{k}B_{i},\overline{B_{j}}e_{\alpha\overline{\beta}}+\overline{B_{\beta}}e_{\alpha\overline{j}}>_{WP}+<L_{k}B_{\alpha},\overline{B_{\beta}}e_{i\overline{j}}+\overline{B_{j}}e_{i\overline{\beta}}>_{WP}$
$\displaystyle=h^{p\overline{q}}<L_{k}B_{i},\overline{B_{q}}>_{WP}<B_{p},\overline{B_{j}}e_{\alpha\overline{\beta}}+\overline{B_{\beta}}e_{\alpha\overline{j}}>_{WP}$
$\displaystyle\quad+h^{p\overline{q}}<L_{k}B_{\alpha},\overline{B_{q}}>_{WP}<B_{p},\overline{B_{\beta}}e_{i\overline{j}}+\overline{B_{j}}e_{i\overline{\beta}}>_{WP}$
$\displaystyle=h^{p\overline{q}}\int_{X_{s}}(L_{k}B_{i}\cdot\overline{B_{q}})dv\int_{X_{s}}(B_{p}\cdot(\overline{B_{j}}e_{\alpha\overline{\beta}}+\overline{B_{\beta}}e_{\alpha\overline{j}}))dv$
$\displaystyle\quad+h^{p\overline{q}}\int_{X_{s}}(L_{k}B_{\alpha}\cdot\overline{B_{q}})dv\int_{X_{s}}(B_{p}\cdot(\overline{B_{\beta}}e_{i\overline{j}}+\overline{B_{j}}e_{i\overline{\beta}}))dv$
$\displaystyle=h^{p\overline{q}}\int_{X_{s}}(L_{k}B_{i}\cdot\overline{B_{q}})dv\int_{X_{s}}(f_{p\overline{j}}e_{\alpha\overline{\beta}}+f_{p\overline{\beta}}e_{\alpha\overline{j}})dv$
$\displaystyle\quad+h^{p\overline{q}}\int_{X_{s}}(L_{k}B_{\alpha}\cdot\overline{B_{q}})dv\int_{X_{s}}(f_{p\overline{\beta}}e_{i\overline{j}}+f_{p\overline{j}}e_{i\overline{\beta}})dv$
$\displaystyle=h^{p\overline{q}}\partial_{k}h_{i\overline{q}}R_{p\overline{j}\alpha\overline{\beta}}+h^{p\overline{q}}\partial_{k}h_{\alpha\overline{q}}R_{i\overline{j}p\overline{\beta}}=\Gamma^{p}_{ik}R_{p\overline{j}\alpha\overline{\beta}}+\Gamma^{p}_{\alpha
k}R_{i\overline{j}p\overline{\beta}}.$

By Lemma 3.2.4 and Lemma 4.3.1, we have

$\displaystyle B_{i}\cdot L_{k}\overline{B_{j}}$
$\displaystyle=(A_{i}\partial_{z}\otimes
d\overline{z})\cdot(-\partial_{z}(\lambda^{-1}\partial_{z}e_{k\overline{j}})\partial_{z}\otimes
d\overline{z}-f_{k\overline{j}}\partial_{z}\otimes
dz+f_{k\overline{j}}\partial_{\overline{z}}\otimes d\overline{z})$
$\displaystyle=-A_{i}\partial_{z}(\lambda^{-1}\partial_{z}e_{k\overline{j}})=\xi_{i}(e_{k\overline{j}}).$

Hence $II$ can be simplified as well:

$II=\int_{X_{s}}(\xi_{i}(e_{k\overline{j}})e_{\alpha\overline{\beta}}+\xi_{\alpha}(e_{k\overline{\beta}})e_{i\overline{j}}+\xi_{i}(e_{k\overline{\beta}})e_{\alpha\overline{j}}+\xi_{\alpha}(e_{k\overline{j}})e_{i\overline{\beta}})dv.$

Then using the fact that $\xi_{k}$ is a real symmetric operator, i.e.,
$\displaystyle\int_{X_{s}}\xi_{k}(e_{i\overline{\beta}})e_{\alpha\overline{j}}dv=\int_{X_{s}}\xi_{k}(e_{\alpha\overline{j}})e_{i\overline{\beta}}dv$,
we know that

$\displaystyle\partial_{k}R_{i\overline{j}\alpha\overline{\beta}}$
$\displaystyle=\Gamma^{p}_{ik}R_{p\overline{j}\alpha\overline{\beta}}+\Gamma^{p}_{\alpha
k}R_{i\overline{j}p\overline{\beta}}$
$\displaystyle\quad+\int_{X_{s}}(\xi_{i}(e_{k\overline{j}})e_{\alpha\overline{\beta}}+\xi_{\alpha}(e_{k\overline{\beta}})e_{i\overline{j}}+\xi_{i}(e_{k\overline{\beta}})e_{\alpha\overline{j}}+\xi_{\alpha}(e_{k\overline{j}})e_{i\overline{\beta}})dv$
$\displaystyle\quad+\int_{X_{s}}(\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}}+\xi_{k}(e_{i\overline{\beta}})e_{\alpha\overline{j}})dv$
$\displaystyle=\Gamma^{p}_{ik}R_{p\overline{j}\alpha\overline{\beta}}+\Gamma^{p}_{\alpha
k}R_{i\overline{j}p\overline{\beta}}+\int_{X_{s}}(\xi_{i}(e_{k\overline{j}})e_{\alpha\overline{\beta}}+\xi_{i}(e_{\alpha\overline{j}})e_{k\overline{\beta}})dv$
$\displaystyle\quad+\int_{X_{s}}(\xi_{\alpha}(e_{k\overline{\beta}})e_{i\overline{j}}+\xi_{\alpha}(e_{i\overline{\beta}})e_{k\overline{j}})dv+\int_{X_{s}}(\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}}+\xi_{k}(e_{\alpha\overline{j}})e_{i\overline{\beta}})dv$
$\displaystyle=\Gamma^{p}_{ik}R_{p\overline{j}\alpha\overline{\beta}}+\Gamma^{p}_{\alpha
k}R_{i\overline{j}p\overline{\beta}}+\sigma_{1}\int_{X_{s}}(\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}})dv.$

Finally by the definition of $\tau_{i\overline{j}}$ we obtain

$\displaystyle\partial_{k}\tau_{i\overline{j}}$
$\displaystyle=h^{\alpha\overline{\beta}}\partial_{k}R_{i\overline{j}\alpha\overline{\beta}}+\partial_{k}h^{\alpha\overline{\beta}}R_{i\overline{j}\alpha\overline{\beta}}$
$\displaystyle=h^{\alpha\overline{\beta}}(\partial_{k}R_{i\overline{j}\alpha\overline{\beta}}-R_{i\overline{j}p\overline{\beta}}\Gamma^{p}_{k\alpha})$
$\displaystyle=h^{\alpha\overline{\beta}}(\sigma_{1}\int_{X_{s}}(\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}})dv+\Gamma^{p}_{ik}R_{p\overline{j}\alpha\overline{\beta}})$
$\displaystyle=h^{\alpha\overline{\beta}}\bigg{\\{}\sigma_{1}\int_{X_{s}}(\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}})dv\bigg{\\}}+\tau_{p\overline{j}}\Gamma^{p}_{ik}.$

∎

Before we go into the computation of the second order derivatives of
$\tau_{i\overline{j}}$, let’s first compute the commutator of $\xi_{k}$ and
$\overline{v_{l}}$.

######  Lemma 4.3.2.

(Lemma 3.7, [17]) For any smooth function $f\in C^{\infty}(X_{s})$,

$\overline{v_{l}}(\xi_{k}f)-\xi_{k}(\overline{v_{l}}f)=\overline{P}(e_{k\overline{l}})-2f_{k\overline{l}}\Box
f+\lambda^{-1}\partial_{z}f_{k\overline{l}}\partial_{\overline{z}}f.$

###### Proof.

Let’s fix local holomorphic coordinates and compute locally. We first consider
the commutator of $\overline{v_{l}}$ and $\partial_{z}$:

$\displaystyle\overline{v_{l}}\partial_{z}-\partial_{z}\overline{v_{l}}$
$\displaystyle=\partial_{\overline{l}}\partial_{z}+\overline{a_{l}}\partial_{\overline{z}}\partial_{z}-\partial_{z}\partial_{\overline{l}}-\partial_{z}\overline{a_{l}}\partial_{\overline{z}}-\overline{a_{l}}\partial_{z}\partial_{\overline{z}}$
$\displaystyle=-\partial_{z}\overline{a_{l}}\partial_{\overline{z}}=-\overline{A_{l}}\partial_{\overline{z}}.$

Then we know that the commutator of $\overline{v_{l}}$ and
$\lambda^{-1}\partial_{z}$ is

$\displaystyle\overline{v_{l}}(\lambda^{-1}\partial_{z})-\lambda^{-1}\partial_{z}(\overline{v_{l}})$
$\displaystyle=\overline{v_{l}}(\lambda^{-1})\partial_{z}+\lambda^{-1}(\overline{v_{l}}\partial_{z}-\partial_{z}\overline{v_{l}})$
$\displaystyle=-\lambda^{-2}(\partial_{\overline{l}}\lambda+\overline{a_{l}}\partial_{\overline{z}}\lambda)\partial_{z}-\lambda^{-1}\overline{A_{l}}\partial_{\overline{z}}$
$\displaystyle=\lambda^{-1}\partial_{\overline{z}}\overline{a_{l}}\partial_{z}-\lambda^{-1}\overline{A_{l}}\partial_{\overline{z}}.$

These two formulae imply

$\displaystyle\overline{v_{l}}P-P\overline{v_{l}}$
$\displaystyle=-\overline{v_{l}}(\partial_{z}(\lambda^{-1}\partial_{z}))+\partial_{z}(\lambda^{-1}\partial_{z})\overline{v_{l}}$
$\displaystyle=(\overline{A_{l}}\partial_{\overline{z}}-\partial_{z}\overline{v_{l}})(\lambda^{-1}\partial_{z})+\partial_{z}(\overline{v_{l}}(\lambda^{-1}\partial_{z})-\lambda^{-1}\partial_{\overline{z}}\overline{a_{l}}\partial_{z}+\lambda^{-1}\overline{A_{l}}\partial_{\overline{z}})$
$\displaystyle=\overline{A_{l}}\partial_{\overline{z}}(\lambda^{-1}\partial_{z})-\partial_{z}(\lambda^{-1}\partial_{\overline{z}}\overline{a_{l}}\partial_{z})+\partial_{z}(\lambda^{-1}\overline{A_{l}}\partial_{\overline{z}})$
$\displaystyle=-\lambda^{-2}\partial_{\overline{z}}\lambda\overline{A_{l}}\partial_{z}+\lambda^{-1}\overline{A_{l}}\partial_{\overline{z}}\partial_{z}+\lambda^{-2}\partial_{z}\lambda\partial_{\overline{z}}\overline{a_{l}}\partial_{z}-\lambda^{-1}\partial_{\overline{z}}\overline{A_{l}}\partial_{z}-\lambda^{-1}\partial_{\overline{z}}\overline{a_{l}}\partial_{z}\partial_{z}$
$\displaystyle-\lambda^{-2}\partial_{z}\lambda\overline{A_{l}}\partial_{\overline{z}}+\lambda^{-1}\partial_{z}\overline{A_{l}}\partial_{\overline{z}}+\lambda^{-1}\overline{A_{l}}\partial_{z}\partial_{\overline{z}}.$

Note that the harmonicity gives us
$\partial_{\overline{z}}(\lambda\overline{A_{l}})=0$, so we have
$\partial_{\overline{z}}\overline{A_{l}}=-\lambda^{-1}\partial_{\overline{z}}\lambda\overline{A_{l}}$.
Thus we obtain

$\displaystyle\overline{v_{l}}P-P\overline{v_{l}}$
$\displaystyle=\lambda^{-1}\partial_{\overline{z}}\overline{A_{l}}\partial_{z}-\overline{A_{l}}\Box-\partial_{\overline{z}}\overline{a_{l}}\partial_{z}(\lambda^{-1}\partial_{z})-\lambda^{-1}\partial_{\overline{z}}\overline{a_{l}}\partial_{z}\partial_{z}$
$\displaystyle-\lambda^{-2}\partial_{z}\lambda\overline{A_{l}}\partial_{\overline{z}}+\lambda^{-1}\partial_{z}\overline{A_{l}}\partial_{\overline{z}}-\overline{A_{l}}\Box$
$\displaystyle=-2\overline{A_{l}}\Box-\partial_{\overline{z}}\overline{a_{l}}P-\lambda^{-2}\partial_{z}\lambda\overline{A_{l}}\partial_{\overline{z}}+\lambda^{-1}\partial_{z}\overline{A_{l}}\partial_{\overline{z}}.$

Then since by Lemma 4.3.1 we know that $\xi_{k}=-A_{k}P$, we have

$\displaystyle\overline{v_{l}}(\xi_{k}f)-\xi_{k}(\overline{v_{l}}f)$
$\displaystyle=-\overline{v_{l}}(A_{k})P(f)+A_{k}(\overline{v_{l}}P(f)-P\overline{v_{l}}(f))$
$\displaystyle=-\overline{v_{l}}(A_{k})P(f)-2A_{k}\overline{A_{l}}\Box
f-A_{k}\partial_{\overline{z}}\overline{a_{l}}P(f)-\lambda^{-2}\partial_{z}\lambda
A_{k}\overline{A_{l}}\partial_{\overline{z}}f$
$\displaystyle\quad+\lambda^{-1}A_{k}\partial_{z}\overline{A_{l}}\partial_{\overline{z}}f$
$\displaystyle=-(\overline{v_{l}}(A_{k})+A_{k}\partial_{\overline{z}}\overline{a_{l}})P(f)-2f_{k\overline{l}}\Box
f-\lambda^{-2}\partial_{z}\lambda
A_{k}\overline{A_{l}}\partial_{\overline{z}}f$
$\displaystyle\quad+\lambda^{-1}A_{k}\partial_{z}\overline{A_{l}}\partial_{\overline{z}}f.$

Now we use the fact that
$\overline{v_{l}}(A_{k})+A_{k}\partial_{\overline{z}}\overline{a_{l}}=-\overline{P}(e_{k\overline{l}})$
and the following result from harmonicity: $-\lambda^{-1}\partial_{z}\lambda
A_{k}=\partial_{z}A_{k}$, so we finish the proof:

$\displaystyle\overline{v_{l}}(\xi_{k}f)-\xi_{k}(\overline{v_{l}}f)$
$\displaystyle=\overline{P}(e_{k\overline{l}})P(f)-2f_{k\overline{l}}\Box
f+\lambda^{-1}\partial_{z}A_{k}\overline{A_{l}}\partial_{\overline{z}}f+\lambda^{-1}A_{k}\partial_{z}\overline{A_{l}}\partial_{\overline{z}}f$
$\displaystyle=\overline{P}(e_{k\overline{l}})P(f)-2f_{k\overline{l}}\Box
f+\lambda^{-1}\partial_{z}f_{k\overline{l}}\partial_{\overline{z}}f.$

∎

For convenience we may define the commutator of $\xi_{k}$ and
$\overline{v_{l}}$ as an operator.

######  Definition 4.3.4.

For each $k,l$, we define the operator
$Q_{k\overline{l}}:C^{\infty}(X_{s})\longrightarrow C^{\infty}(X_{s})$ by

$Q_{k\overline{l}}(f)=\overline{P}(e_{k\overline{l}})P(f)-2f_{k\overline{l}}\Box
f+\lambda^{-1}\partial_{z}f_{k\overline{l}}\partial_{\overline{z}}f$

We can now compute the curvature formula of the Ricci metric (Theorem 3.3,
[17]).

######  Theorem 4.3.2.

Let $s_{1},...,s_{n}$ be local holomorphic coordinates at $s\in M_{g}$. Then
at $s$, we have

$\displaystyle\tilde{R}_{i\overline{j}k\overline{l}}$
$\displaystyle=h^{\alpha\overline{\beta}}\bigg{\\{}\sigma_{1}\sigma_{2}\int_{X_{s}}\big{\\{}(\Box+1)^{-1}(\xi_{k}(e_{i\overline{j}}))\overline{\xi}_{l}(e_{\alpha\overline{\beta}})+(\Box+1)^{-1}(\xi_{k}(e_{i\overline{j}}))\overline{\xi}_{\beta}(e_{\alpha\overline{l}})\big{\\}}dv\bigg{\\}}$
$\displaystyle\quad+h^{\alpha\overline{\beta}}\bigg{\\{}\sigma_{1}\int_{X_{s}}Q_{k\overline{l}}(e_{i\overline{j}})e_{\alpha\overline{\beta}}dv\bigg{\\}}$
$\displaystyle\quad-\tau^{p\overline{q}}h^{\alpha\overline{\beta}}h^{\gamma\overline{\delta}}\bigg{\\{}\sigma_{1}\int_{X_{s}}\xi_{k}(e_{i\overline{q}})e_{\alpha\overline{\beta}}dv\bigg{\\}}\bigg{\\{}\tilde{\sigma}_{1}\int_{X_{s}}\overline{\xi}_{l}(e_{p\overline{j}})e_{\gamma\overline{\delta}}dv\bigg{\\}}$
$\displaystyle\quad+\tau_{p\overline{j}}h^{p\overline{q}}R_{i\overline{q}k\overline{l}},$

where $\tilde{R}_{i\overline{j}k\overline{l}}$ is the curvature tensor of the
Ricci metric.

###### Proof.

Let’s compute the second derivatives of $\tau_{i\overline{j}}$ first. By Lemma
4.3.1 and Lemma 4.3.2 we obtain

$\displaystyle\partial_{\overline{l}}\int_{X_{s}}\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}}dv$
$\displaystyle=\int_{X_{s}}(\overline{v_{l}}(\xi_{k}(e_{i\overline{j}}))e_{\alpha\overline{\beta}}+\xi_{k}(e_{i\overline{j}})\overline{v_{l}}(e_{\alpha\overline{\beta}}))dv$
$\displaystyle=\int_{X_{s}}(\xi_{k}(\overline{v_{l}}(e_{i\overline{j}}))e_{\alpha\overline{\beta}}+\xi_{k}(e_{i\overline{j}})\overline{v_{l}}(e_{\alpha\overline{\beta}})+Q_{k\overline{l}}(e_{i\overline{j}})e_{\alpha\overline{\beta}})dv$
$\displaystyle=\int_{X_{s}}(\xi_{k}(e_{\alpha\overline{\beta}})\overline{v_{l}}(e_{i\overline{j}})+\xi_{k}(e_{i\overline{j}})\overline{v_{l}}(e_{\alpha\overline{\beta}})+Q_{k\overline{l}}(e_{i\overline{j}})e_{\alpha\overline{\beta}})dv$
$\displaystyle=\int_{X_{s}}(\Box+1)^{-1}(\xi_{k}(e_{\alpha\overline{\beta}}))(\Box+1)(\overline{v_{l}}(e_{i\overline{j}}))dv$
$\displaystyle\quad+\int_{X_{s}}(\Box+1)^{-1}(\xi_{k}(e_{i\overline{j}}))(\Box+1)(\overline{v_{l}}(e_{\alpha\overline{\beta}}))dv$
$\displaystyle\quad+\int_{X_{s}}Q_{k\overline{l}}(e_{i\overline{j}})e_{\alpha\overline{\beta}}dv$
$\displaystyle=\int_{X_{s}}(\Box+1)^{-1}(\xi_{k}(e_{\alpha\overline{\beta}}))(\overline{\xi}_{l}(e_{i\overline{j}})+\overline{\xi}_{j}(e_{i\overline{l}})+L_{\overline{l}}\overline{B_{j}}\cdot
B_{i})dv$
$\displaystyle\quad+\int_{X_{s}}(\Box+1)^{-1}(\xi_{k}(e_{i\overline{j}}))(\overline{\xi}_{l}(e_{\alpha\overline{\beta}})+\overline{\xi}_{\beta}(e_{\alpha\overline{l}})+L_{\overline{l}}\overline{B_{\beta}}\cdot
B_{\alpha})dv$
$\displaystyle\quad+\int_{X_{s}}Q_{k\overline{l}}(e_{i\overline{j}})e_{\alpha\overline{\beta}}dv.$

From Lemma 3.2.5 we know that $L_{k}B_{i}$ is harmonic. As $B_{1},...,B_{n}$
is a basis of the space of harmonic Beltrami differentials, and notice that
$\displaystyle\int_{X_{s}}L_{k}B_{i}\cdot\overline{B_{q}}dv=\partial_{k}h_{i\overline{q}}$,
we have $L_{k}B_{i}=\Gamma^{s}_{ik}B_{s}$. Hence we can derive that

$\displaystyle\quad\int_{X_{s}}((\Box+1)^{-1}(\xi_{k}(e_{\alpha\overline{\beta}}))(L_{\overline{l}}\overline{B_{j}}\cdot
B_{i})+(\Box+1)^{-1}(\xi_{k}(e_{i\overline{j}}))(L_{\overline{l}}\overline{B_{\beta}}\cdot
B_{\alpha}))dv$
$\displaystyle=\int_{X_{s}}((\Box+1)^{-1}(\xi_{k}(e_{\alpha\overline{\beta}}))(\overline{\Gamma^{t}_{jl}}\overline{A_{t}}A_{i})+(\Box+1)^{-1}(\xi_{k}(e_{i\overline{j}}))(\overline{\Gamma^{t}_{\beta
l}}\overline{A_{t}}A_{\alpha}))dv$
$\displaystyle=\overline{\Gamma^{t}_{jl}}\int_{X_{s}}\xi_{k}(e_{\alpha\overline{\beta}})(\Box+1)^{-1}(A_{i}\overline{A_{t}})dv+\overline{\Gamma^{t}_{\beta
l}}\int_{X_{s}}\xi_{k}(e_{i\overline{j}})(\Box+1)^{-1}(A_{\alpha}\overline{A_{t}})dv$
$\displaystyle=\overline{\Gamma^{t}_{jl}}\int_{X_{s}}\xi_{k}(e_{\alpha\overline{\beta}})e_{i\overline{t}}dv+\overline{\Gamma^{t}_{\beta
l}}\int_{X_{s}}\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{t}}dv.$

Therefore we have

$\displaystyle\partial_{\overline{l}}\int_{X_{s}}\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}}dv$
$\displaystyle=\int_{X_{s}}(\Box+1)^{-1}(\xi_{k}(e_{\alpha\overline{\beta}}))(\overline{\xi}_{l}(e_{i\overline{j}})+\overline{\xi}_{j}(e_{i\overline{l}}))dv$
$\displaystyle\quad+\int_{X_{s}}(\Box+1)^{-1}(\xi_{k}(e_{i\overline{j}}))(\overline{\xi}_{l}(e_{\alpha\overline{\beta}})+\overline{\xi}_{\beta}(e_{\alpha\overline{l}}))dv$
$\displaystyle\quad+\overline{\Gamma^{t}_{jl}}\int_{X_{s}}\xi_{k}(e_{\alpha\overline{\beta}})e_{i\overline{t}}dv+\overline{\Gamma^{t}_{\beta
l}}\int_{X_{s}}\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{t}}dv$
$\displaystyle\quad+\int_{X_{s}}Q_{k\overline{l}}(e_{i\overline{j}})e_{\alpha\overline{\beta}}dv.$

Moreover, we know that

$\displaystyle\partial_{\overline{l}}\Gamma^{p}_{ik}$
$\displaystyle=\partial_{\overline{l}}(h^{p\overline{q}}\partial_{k}h_{i\overline{q}})$
$\displaystyle=-h^{p\overline{\beta}}h^{\alpha\overline{q}}\partial_{\overline{l}}h_{\alpha\overline{\beta}}\partial_{k}h_{i\overline{q}}+h^{p\overline{q}}\partial_{\overline{l}}\partial_{k}h_{i\overline{q}}$
$\displaystyle=h^{p\overline{q}}(\partial_{\overline{l}}\partial_{k}h_{i\overline{q}}-h^{\alpha\overline{\beta}}\partial_{\overline{l}}h_{\alpha\overline{q}}\partial_{k}h_{i\overline{\beta}})$
$\displaystyle=h^{p\overline{q}}R_{i\overline{q}k\overline{l}}.$

Then we can use Theorem 4.3.1 to get the formula for the second derivatives of
$\tau_{i\overline{j}}$:

$\displaystyle\partial_{\overline{l}}\partial_{k}\tau_{i\overline{j}}$
$\displaystyle=(\partial_{\overline{l}}h^{\alpha\overline{\beta}})\bigg{\\{}\sigma_{1}\int_{X_{s}}(\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}})dv\bigg{\\}}+h^{\alpha\overline{\beta}}\bigg{\\{}\sigma_{1}\partial_{\overline{l}}\int_{X_{s}}(\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}})dv\bigg{\\}}$
$\displaystyle\quad+\partial_{\overline{l}}\tau_{p\overline{j}}\Gamma^{p}_{ik}+\tau_{p\overline{j}}\partial_{\overline{l}}\Gamma^{p}_{ik}$
$\displaystyle=-h^{\alpha\overline{t}}\overline{\Gamma^{\beta}_{lt}}\bigg{\\{}\sigma_{1}\int_{X_{s}}(\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{\beta}})dv\bigg{\\}}$
$\displaystyle\quad+h^{\alpha\overline{\beta}}\bigg{\\{}\sigma_{1}\sigma_{2}\int_{X_{s}}(\Box+1)^{-1}(\xi_{k}(e_{i\overline{j}}))(\overline{\xi}_{l}(e_{\alpha\overline{\beta}})+\overline{\xi}_{\beta}(e_{\alpha\overline{l}}))dv\bigg{\\}}$
$\displaystyle\quad+h^{\alpha\overline{\beta}}\bigg{\\{}\sigma_{1}\int_{X_{s}}Q_{k\overline{l}}(e_{i\overline{j}})e_{\alpha\overline{\beta}}dv\bigg{\\}}+h^{\alpha\overline{\beta}}\overline{\Gamma^{t}_{jl}}\bigg{\\{}\sigma_{1}\int_{X_{s}}\xi_{k}(e_{\alpha\overline{\beta}})e_{i\overline{t}}dv\bigg{\\}}$
$\displaystyle\quad+h^{\alpha\overline{\beta}}\overline{\Gamma^{t}_{\beta
l}}\bigg{\\{}\sigma_{1}\int_{X_{s}}\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{t}}dv\bigg{\\}}+h^{\gamma\overline{\delta}}\bigg{\\{}\tilde{\sigma}_{1}\int_{X_{s}}(\overline{\xi}_{l}(e_{p\overline{j}})e_{\gamma\overline{\delta}})dv\bigg{\\}}\Gamma^{p}_{ik}$
$\displaystyle\quad+\tau_{p\overline{q}}\overline{\Gamma^{q}_{jl}}\Gamma^{p}_{ik}+\tau_{p\overline{j}}h^{p\overline{q}}R_{i\overline{q}k\overline{l}}.$

Hence we obtain the desired result:

$\displaystyle\tilde{R}_{i\overline{j}k\overline{l}}$
$\displaystyle=-h^{\alpha\overline{\beta}}\overline{\Gamma^{t}_{\beta
l}}\bigg{\\{}\sigma_{1}\int_{X_{s}}(\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{t}})dv\bigg{\\}}$
$\displaystyle\quad+h^{\alpha\overline{\beta}}\bigg{\\{}\sigma_{1}\sigma_{2}\int_{X_{s}}(\Box+1)^{-1}(\xi_{k}(e_{i\overline{j}}))(\overline{\xi}_{l}(e_{\alpha\overline{\beta}})+\overline{\xi}_{\beta}(e_{\alpha\overline{l}}))dv\bigg{\\}}$
$\displaystyle\quad+h^{\alpha\overline{\beta}}\bigg{\\{}\sigma_{1}\int_{X_{s}}Q_{k\overline{l}}(e_{i\overline{j}})e_{\alpha\overline{\beta}}dv\bigg{\\}}+h^{\alpha\overline{\beta}}\overline{\Gamma^{t}_{jl}}\bigg{\\{}\sigma_{1}\int_{X_{s}}\xi_{k}(e_{\alpha\overline{\beta}})e_{i\overline{t}}dv\bigg{\\}}$
$\displaystyle\quad+h^{\alpha\overline{\beta}}\overline{\Gamma^{t}_{\beta
l}}\bigg{\\{}\sigma_{1}\int_{X_{s}}\xi_{k}(e_{i\overline{j}})e_{\alpha\overline{t}}dv\bigg{\\}}+h^{\gamma\overline{\delta}}\bigg{\\{}\tilde{\sigma}_{1}\int_{X_{s}}(\overline{\xi}_{l}(e_{p\overline{j}})e_{\gamma\overline{\delta}})dv\bigg{\\}}\Gamma^{p}_{ik}$
$\displaystyle\quad+\tau_{p\overline{q}}\overline{\Gamma^{q}_{jl}}\Gamma^{p}_{ik}+\tau_{p\overline{j}}h^{p\overline{q}}R_{i\overline{q}k\overline{l}}$
$\displaystyle-\tau^{p\overline{q}}\bigg{(}h^{\alpha\overline{\beta}}\bigg{\\{}\sigma_{1}\int_{X_{s}}(\xi_{k}(e_{i\overline{q}})e_{\alpha\overline{\beta}})dv\bigg{\\}}+\tau_{p\overline{q}}\Gamma^{p}_{ik}\bigg{)}\bigg{(}h^{\gamma\overline{\delta}}\bigg{\\{}\tilde{\sigma}_{1}\int_{X_{s}}(\overline{\xi}_{l}(e_{p\overline{j}})e_{\gamma\overline{\delta}})dv\bigg{\\}}+\tau_{p\overline{q}}\overline{\Gamma^{q}_{jl}}\bigg{)}$
$\displaystyle=h^{\alpha\overline{\beta}}\bigg{\\{}\sigma_{1}\sigma_{2}\int_{X_{s}}\big{\\{}(\Box+1)^{-1}(\xi_{k}(e_{i\overline{j}}))\overline{\xi}_{l}(e_{\alpha\overline{\beta}})+(\Box+1)^{-1}(\xi_{k}(e_{i\overline{j}}))\overline{\xi}_{\beta}(e_{\alpha\overline{l}})\big{\\}}dv\bigg{\\}}$
$\displaystyle\quad+h^{\alpha\overline{\beta}}\bigg{\\{}\sigma_{1}\int_{X_{s}}Q_{k\overline{l}}(e_{i\overline{j}})e_{\alpha\overline{\beta}}dv\bigg{\\}}$
$\displaystyle-\tau^{p\overline{q}}h^{\alpha\overline{\beta}}h^{\gamma\overline{\delta}}\bigg{\\{}\sigma_{1}\int_{X_{s}}\xi_{k}(e_{i\overline{q}})e_{\alpha\overline{\beta}}dv\bigg{\\}}\bigg{\\{}\tilde{\sigma}_{1}\int_{X_{s}}\overline{\xi}_{l}(e_{p\overline{j}})e_{\gamma\overline{\delta}}dv\bigg{\\}}$
$\displaystyle\quad+\tau_{p\overline{j}}h^{p\overline{q}}R_{i\overline{q}k\overline{l}}.$

∎

In fact the curvature formula of the Ricci metric can be simplified using the
normal coordinates, but for the sake of easier estimates on the asymptotic
behavior of the curvature (since normal coordinates near boundary points are
difficult to be described), the general formula will be used in the upcoming
computations.

###  4.4 The Asymptotic Behavior of the Ricci Metric

Using the formula of $R_{i\overline{j}k\overline{l}}$ (Theorem 3.2.1) we can
obtain the sign of the curvature of the Weil-Petersson metric very easily, but
the formula of $\tilde{R}_{i\overline{j}k\overline{l}}$ (Theorem 4.3.2) does
not help us find out the sign of the curvature of the Ricci metric directly.
Hence we need to do some estimates on the asymptotics of the Ricci metric so
that we can estimate the curvature of the Ricci metric.

####  4.4.1 Estimates on the Asymptotics of the Ricci Metric

Let’s define the $C^{k}$ norm of $\sigma\in S(p)$ as Wolpert[30] did:

######  Definition 4.4.1.

Let $Q$ be an operator which is a composition of operators $K_{*}$ and
$L_{*}$. Denote $|Q|$ to be the number of the factors in the composition. For
any $\sigma\in S(p)$, define

$\left\|\sigma\right\|_{0}=\sup_{X}|\sigma|$

and

$\left\|\sigma\right\|_{k}=\sum_{|Q|\leq k}\left\|Q\sigma\right\|_{0}.$

We can also localize the norm on a subset of $X$. Let $\Omega\subset X$ be a
domain. Then we can define

$\left\|\sigma\right\|_{0,\Omega}=\sup_{\Omega}|\sigma|$

and

$\left\|\sigma\right\|_{k,\Omega}=\sum_{|Q|\leq
k}\left\|Q\sigma\right\|_{0,\Omega}.$

Note that both of the definitions depend on the choice of conformal metric on
$X$. In this section, $X$ will always be equipped with the Kähler-Einstein
metric unless otherwise specified.

We also need the following version of the Schauder estimate proved by Wolpert
(Lemma A.4.2, [30]):

######  Theorem 4.4.1.

Let $X$ be a closed Riemann surface. Let $f$ and $g$ be smooth functions on
$X$ such that $(\Box+1)g=f$. Then for any integer $k\geq 0$, there is a
constant $c_{k}$ such that $\left\|g\right\|_{k+1}\leq
c_{k}\left\|f\right\|_{k}$.

Before estimating the asymptotics of the Ricci metric in the pinching
coordinates, we need to specify some notations first. Let
$(t,s)=(t_{1},...,t_{m},s_{m+1},...,s_{n})$ be the pinching coordinates near
$X_{0,0}$. For $|(t,s)|<\delta$, let $\Omega^{j}_{c}$ be the $j$-th genuine
collar on $X_{t,s}$ which contains a short geodesic $\gamma_{j}$ with length
$l_{j}$. Let $u_{j}=\dfrac{l_{j}}{2\pi}$, $\displaystyle
u_{0}=\sum^{m}_{j=1}u_{j}+\sum^{n}_{j=m+1}|s_{j}|$, $r_{j}=|z_{j}|$ and
$\tau_{j}=u_{j}\log r_{j}$, where $z_{j}$ is the properly normalized rs-
coordinate on $\Omega^{j}_{c}$ such that

$\Omega^{j}_{c}=\\{z_{j}:c^{-1}e^{-\frac{2\pi^{2}}{l_{j}}}<|z_{j}|<c\\}.$

Hence we know that the Kähler-Einstein metric $\lambda$ on $X_{t,s}$
restricted to the collar $\Omega^{j}_{c}$ is given by

$\lambda=\dfrac{1}{2}u_{j}^{2}r_{j}^{-2}\csc^{2}\tau_{j}.$

In addition, let $\displaystyle\Omega_{c}=\cup^{m}_{j=1}\Omega^{j}_{c}$ and
$R_{c}=X_{t,s}-\Omega_{c}$. Note that the constant c may be changed finitely
many times in the following estimates, but this will not affect the result.
Also note that by Wolpert[30], we know that
$u_{j}=\dfrac{l_{j}}{2\pi}\sim-\dfrac{\pi}{\log|t_{j}|}$.

The first step we have to do for the estimates is to find all the harmonic
Beltrami differentials $B_{1},...,B_{n}$ corresponding to the tangent vectors
$\frac{\partial}{\partial t_{1}},...,\frac{\partial}{\partial s_{n}}$. Notice
that Masur[19] constructed $3g-3$ regular holomorphic quadratic differentials
$\psi_{1},...,\psi_{n}$ corresponding to the cotangent $dt_{1},...,ds_{n}$ on
the plumbing collars using plumbing coordinate. However, we need to find them
in rs-coordinates so that we can have an easier estimate of the curvature. Let
us collect Trapani’s results (Lemma 6.2-6.5, [28]) in one theorem using
Masur’s notations and formulation.

######  Theorem 4.4.2.

Let $(t,s)$ be the pinching coordinates on $\overline{\mathcal{M}_{g}}$ near
$X_{0,0}$ which corresponds to a codimension $m$ boundary point of
$\overline{\mathcal{M}_{g}}$. Then there exist constants $M$, $delta>0$ and
$0<c<1$ such that if $(t,s)<delta$, then the $j$-th plumbing collar on
$X_{t,s}$ contains the genuine collar $\Omega^{j}_{c}$. Furthermore, a rs-
coordinate $z_{j}$ can be chosen on the collar $\Omega^{j}_{c}$ properly such
that the holomorphic quadratic differentials $\psi_{1},...,\psi_{n}$
corresponding to the cotangent vectors $dt_{1},...,ds_{n}$ have the form
$\psi_{i}=\varphi_{i}(z_{j})dz_{j}^{2}$ on the genuine collar $\Omega^{j}_{c}$
for $1\leq j\leq m$, where

  * (i)

$\varphi_{i}(z_{j})=\dfrac{1}{z_{j}^{2}}q_{i}^{j}(z_{j})$ if $i\geq m+1$;

  * (ii)

$\varphi_{i}(z_{j})=(-\dfrac{t_{j}}{\pi})\dfrac{1}{z_{j}^{2}}(q_{j}(z_{j})+1)$
if $i=j$;

  * (iii)

$\varphi_{i}(z_{j})=(-\dfrac{t_{i}}{\pi})\dfrac{1}{z_{j}^{2}}q_{i}^{j}(z_{j})$
if $1\leq i\leq m$ and $i\neq j$,

where $\beta_{i}^{j}$ and $\beta_{j}$ are functions of $(t,s)$, $q_{i}^{j}$
and $q_{j}$ are functions of $(t,s,z_{j})$ given by

$q_{i}^{j}(z_{j})=\sum_{k<0}\alpha^{j}_{ik}(t,s)t_{j}^{-k}z_{j}^{k}+\sum_{k>0}\alpha^{j}_{ik}(t,s)z_{j}^{k}$

and

$q_{j}(z_{j})=\sum_{k<0}\alpha_{jk}(t,s)t_{j}^{-k}z_{j}^{k}+\sum_{k>0}\alpha_{jk}(t,s)z_{j}^{k}$

such that

  * (1)

$\displaystyle\sum_{k<0}|\alpha^{j}_{ik}|c^{-k}\leq M$ and
$\displaystyle\sum_{k>0}|\alpha^{j}_{ik}|c^{k}\leq M$ if $i\neq j$;

  * (2)

$\displaystyle\sum_{k<0}|\alpha_{jk}|c^{-k}\leq M$ and
$\displaystyle\sum_{k>0}|\alpha_{jk}|c^{k}\leq M$.

Using this theorem we can derive the following refined version of Masur’s
estimates of the Weil-Petersson metric (Theorem 1, [19]). Note that in the
remaining section, $(t,s)$ will be assumed to have small norm and $X=X_{t,s}$.

######  Corollary 4.4.1.

Let $(t,s)$ be the pinching coordinates. Then

  * (i)

$h^{i\overline{i}}=2\dfrac{|t_{i}|^{2}}{u_{i}^{3}}(1+O(u_{0}))$ and
$h_{i\overline{i}}=\dfrac{1}{2}\dfrac{u_{i}^{3}}{|t_{i}|^{2}}(1+O(u_{0}))$ if
$i\leq m$;

  * (ii)

$h^{i\overline{j}}=O(|t_{i}t_{j}|)$ and
$h_{i\overline{j}}=O(\dfrac{u_{i}^{3}u_{j}^{3}}{|t_{i}t_{j}|})$ if $\leq
i,j\leq m$ and $i\neq j$;

  * (iii)

$h^{i\overline{j}}=O(|t_{i}|)$ and
$h_{i\overline{j}}=O(\dfrac{u_{i}^{3}}{|t_{i}|})$ if $i\leq m$ and $j\geq
m+1$;

  * (iv)

$h^{i\overline{j}}=O(1)$ and $h_{i\overline{j}}=O(1)$ if $i,j\geq m+1$.

###### Proof.

In fact the last three claims coincide with the Schumacher’s formulation of
the Masur’s result (Theorem [M], [24]), so here we just prove (i). However, we
should note that in fact all the claims can be proved using the same method.

Let’s make a useful claim about some integral estimates:

######  Claim.

We have

  * (1)

$\displaystyle\int_{c^{-1}e^{-\frac{\pi}{u_{j}}}}^{c}\dfrac{1}{r_{j}}\sin^{2}\tau_{j}dr_{j}=\dfrac{1}{u_{j}}(\dfrac{\pi}{2}+O(u_{0}));$

  * (2)

$\displaystyle\int_{c^{-1}e^{-\frac{\pi}{u_{j}}}}^{c}r_{j}^{k-1}\sin^{2}\tau_{j}dr_{j}=O(u_{j}^{2})c^{k}$
for $k\geq 1$;

  * (3)

$\displaystyle\int_{c^{-1}e^{-\frac{\pi}{u_{j}}}}^{c}r_{j}^{k-1}\sin^{2}\tau_{j}dr_{j}=O(u_{j}^{2})c^{-k}\Big{(}e^{-\frac{\pi}{u_{j}}}\Big{)}^{k}$
for $k\leq-1$.

###### Proof.

For (1), we consider

$\displaystyle\bigg{|}\int_{c^{-1}e^{-\frac{\pi}{u_{j}}}}^{c}\dfrac{1}{r_{j}}\sin^{2}\tau_{j}dr_{j}\bigg{|}$
$\displaystyle=\dfrac{1}{u_{j}}\bigg{|}\int_{-u_{j}\log c-\pi}^{u_{j}\log
c}\dfrac{1-\cos 2\tau_{j}}{2}d\tau_{j}\bigg{|}$
$\displaystyle=\dfrac{1}{u_{j}}\bigg{|}\bigg{[}\dfrac{\tau_{j}}{2}-\dfrac{\sin
2\tau_{j}}{4}\bigg{]}_{-u_{j}\log c-\pi}^{u_{j}\log c}\bigg{|}$
$\displaystyle\leq\dfrac{1}{u_{j}}\bigg{(}\dfrac{\pi+|2u_{j}\log
c|+|\sin(2u_{j}\log c)|}{2}\bigg{)}$
$\displaystyle=\dfrac{1}{u_{j}}(\dfrac{\pi}{2}+O(u_{0})).$

Then for (2), we have

$\displaystyle\bigg{|}\int_{c^{-1}e^{-\frac{\pi}{u_{j}}}}^{c}r_{j}^{k-1}\sin^{2}\tau_{j}dr_{j}\bigg{|}$
$\displaystyle\leq\dfrac{1}{k}\bigg{|}\bigg{[}r_{j}^{k}\sin^{2}\tau_{j}\bigg{]}_{c^{-1}e^{-\frac{\pi}{u_{j}}}}^{c}\bigg{|}$
$\displaystyle\quad+\dfrac{1}{k}\bigg{|}\int_{c^{-1}e^{-\frac{\pi}{u_{j}}}}^{c}r_{j}^{k}\cdot
2\sin\tau_{j}\cos\tau_{j}\cdot\dfrac{u_{j}}{r_{j}}dr_{j}\bigg{|}$
$\displaystyle=\dfrac{1}{k}(c^{k}-c^{-k}\Big{(}e^{-\frac{\pi}{u_{j}}}\Big{)}^{k})\sin^{2}(u_{j}\log
c)$
$\displaystyle\quad+\dfrac{u_{j}}{k}\bigg{|}\int_{c^{-1}e^{-\frac{\pi}{u_{j}}}}^{c}r_{j}^{k-1}\sin
2\tau_{j}dr_{j}\bigg{|}$ $\displaystyle\leq
O(u_{j}^{2})c^{k}+\dfrac{u_{j}}{k^{2}}\bigg{(}\bigg{|}\bigg{[}r_{j}^{k}\sin
2\tau_{j}\bigg{]}_{c^{-1}e^{-\frac{\pi}{u_{j}}}}^{c}\bigg{|}$
$\displaystyle\quad+\bigg{|}\int_{c^{-1}e^{-\frac{\pi}{u_{j}}}}^{c}r_{j}^{k}\cos
2\tau_{j}\cdot\dfrac{2u_{j}}{r_{j}}dr_{j}\bigg{|}\bigg{)}$ $\displaystyle\leq
O(u_{j}^{2})c^{k}+\dfrac{u_{j}}{k^{2}}(c^{k}-c^{-k}\Big{(}e^{-\frac{\pi}{u_{j}}}\Big{)}^{k})|\sin(2u_{j}\log
c)|$
$\displaystyle\quad+\dfrac{2u_{j}^{2}}{k^{2}}\int_{c^{-1}e^{-\frac{\pi}{u_{j}}}}^{c}r_{j}^{k-1}dr_{j}$
$\displaystyle=O(u_{j}^{2})c^{k}+\dfrac{2u_{j}^{2}}{k^{3}}(c^{k}-c^{-k}\Big{(}e^{-\frac{\pi}{u_{j}}}\Big{)}^{k})$
$\displaystyle=O(u_{j}^{2})c^{k}.$

(3) can be done similarly. ∎

Now we come to the computation of $h^{i\overline{i}}$ for $i\leq m$. Since
$h^{i\overline{j}}$ is given by the formula

$h^{i\overline{j}}=\int_{X}\psi_{i}\overline{\psi_{j}}\lambda^{-2}dv,$

we know that

$h^{i\overline{i}}=\int_{X}|\psi_{i}|^{2}\lambda^{-2}dv=\sum_{l=1}^{m}\int_{\Omega_{c}^{l}}|\varphi_{i}|^{2}\lambda^{-2}dv+\int_{R_{c}}|\psi_{i}|^{2}\lambda^{-2}dv.$

For $l=i$, we have

$\displaystyle\int_{\Omega_{c}^{i}}|\varphi_{i}|^{2}\lambda^{-2}dv$
$\displaystyle=\int_{0}^{2\pi}\int_{c^{-1}e^{-\frac{\pi}{u_{i}}}}^{c}\dfrac{|t_{i}|^{2}}{\pi^{2}r_{i}^{4}}|q_{i}+\beta_{i}|^{2}\cdot\dfrac{2r_{i}^{2}}{u_{i}^{2}}\sin^{2}\tau_{i}r_{i}dr_{i}d\theta_{i}$
$\displaystyle=\dfrac{2|t_{i}|^{2}}{\pi^{2}u_{i}^{2}}\int_{0}^{2\pi}\int_{c^{-1}e^{-\frac{\pi}{u_{i}}}}^{c}\dfrac{1}{r_{i}}\sin^{2}\tau_{i}|q_{i}+1|^{2}dr_{i}d\theta_{i}.$

Now using Wolpert’s estimate on the length of the short geodesic $\gamma_{i}$
(Example 4.3, [30]), we obtain
$l_{i}=-\dfrac{2\pi^{2}}{\log|t_{i}|}(1+O(u_{i}))$, which implies
$-\log|t_{i}|=\dfrac{\pi}{u_{i}}+O(1)$, i.e.,
$-k<-\log|t_{i}|-\dfrac{\pi}{u_{i}}<k$ for some $k>0$. Thus
$\mu<\dfrac{e^{-\frac{\pi}{u_{i}}}}{|t_{i}|}<\mu^{-1}$, so on the collar
$\Omega_{c}^{i}$ we know that

$|q_{i}|\leq\sum_{k<0}|\alpha_{ik}(t,s)||t_{i}|^{-k}(c^{-1}e^{-\frac{\pi}{u_{i}}})^{k}+\sum_{k>0}|\alpha_{ik}(t,s)|c^{k}=O(1).$

Note that

$|q_{i}+1|^{2}=|q_{i}|^{2}+|q_{i}|+|\overline{q_{i}}|+1\leq
O(1)(\sum_{k<0}|\alpha_{ik}(t,s)||t_{i}|^{-k}r_{i}^{k}+\sum_{k>0}|\alpha_{ik}(t,s)|r_{i}^{k})+1.$

Hence along with the claim made before, we have

$\displaystyle\int_{\Omega_{c}^{i}}|\varphi_{i}|^{2}\lambda^{-2}dv$
$\displaystyle\leq\dfrac{2|t_{i}|^{2}}{\pi^{2}u_{i}^{2}}\int_{0}^{2\pi}\bigg{(}O(1)\sum_{k<0}|\alpha_{ik}(t,s)||t_{i}|^{-k}\int_{c^{-1}e^{-\frac{\pi}{u_{i}}}}^{c}r_{i}^{k-1}\sin^{2}\tau_{i}dr_{i}$
$\displaystyle\quad+O(1)\sum_{k>0}|\alpha_{ik}(t,s)|\int_{c^{-1}e^{-\frac{\pi}{u_{i}}}}^{c}r_{i}^{k-1}\sin^{2}\tau_{i}dr_{i}$
$\displaystyle\quad+\int_{c^{-1}e^{-\frac{\pi}{u_{i}}}}^{c}\dfrac{1}{r_{i}}\sin^{2}\tau_{i}dr_{i}\bigg{)}d\theta_{i}$
$\displaystyle=\dfrac{4|t_{i}|^{2}}{\pi
u_{i}^{2}}\bigg{(}O(u_{i}^{2})\sum_{k<0}|\alpha_{ik}(t,s)||t_{i}|^{-k}c^{-k}\Big{(}e^{-\frac{\pi}{u_{j}}}\Big{)}^{k}$
$\displaystyle\quad+O(u_{i}^{2})\sum_{k>0}|\alpha_{ik}(t,s)|c^{k}+\dfrac{1}{u_{j}}(\dfrac{\pi}{2}+O(u_{0}))\bigg{)}$
$\displaystyle=\dfrac{2|t_{i}|^{2}}{u_{i}^{3}}(O(u_{i}^{3})+1+O(u_{0}))=\dfrac{2|t_{i}|^{2}}{u_{i}^{3}}(1+O(u_{0})).$

Similarly for $l\neq i$, we have

$\displaystyle\int_{\Omega_{c}^{l}}|\varphi_{i}|^{2}\lambda^{-2}dv$
$\displaystyle=\dfrac{2|t_{i}|^{2}}{\pi^{2}u_{l}^{2}}\int_{0}^{2\pi}\int_{c^{-1}e^{-\frac{\pi}{u_{l}}}}^{c}\dfrac{1}{r_{l}}\sin^{2}\tau_{l}|q_{i}^{l}|^{2}dr_{l}d\theta_{l}$
$\displaystyle\leq\dfrac{2|t_{i}|^{2}}{\pi^{2}u_{l}^{2}}\int_{0}^{2\pi}\bigg{(}O(1)\sum_{k<0}|\alpha_{ik}^{l}(t,s)||t_{l}|^{-k}\int_{c^{-1}e^{-\frac{\pi}{u_{l}}}}^{c}r_{l}^{k-1}\sin^{2}\tau_{l}dr_{l}$
$\displaystyle\quad+O(1)\sum_{k>0}|\alpha_{ik}^{l}(t,s)|\int_{c^{-1}e^{-\frac{\pi}{u_{l}}}}^{c}r_{l}^{k-1}\sin^{2}\tau_{l}dr_{l}$
$\displaystyle=\dfrac{4|t_{i}|^{2}}{\pi
u_{i}^{2}}\bigg{(}O(u_{i}^{2})\sum_{k<0}|\alpha_{ik}(t,s)||t_{i}|^{-k}c^{-k}\Big{(}e^{-\frac{\pi}{u_{j}}}\Big{)}^{k}\
$ $\displaystyle\quad+O(u_{i}^{2})\sum_{k>0}|\alpha_{ik}(t,s)|c^{k}\bigg{)}$
$\displaystyle=O(|t_{i}^{2}|)).$

Then we use Masur’s result ((iii) in the proof of Theorem 1, [19]) to obtain
$\displaystyle\int_{R_{c}}|\psi_{i}|^{2}\lambda^{-2}dv=O(|t_{i}^{2}|))$.
Therefore the first part of (i) is done. The second part can be proved by the
same argument as that in the proof of Corollary 4.4.2, which will be shown
later. ∎

Then by Theorem 4.4.2 and Corollary 4.4.1, we can obtain the following
expressions for the harmonic Beltrami differentials
$B_{i}=A_{i}\partial_{z}\otimes d\overline{z}$:

######  Lemma 4.4.1.

(Lemma 4.2, [17]) On the genuine collar $\Omega_{c}^{j}$ for $c$ small enough,
the coefficient functions $A_{i}$ of the harmonic Beltrami differentials
$B_{i}$ have the form:

  * (i)

$A_{i}(z_{j})=\dfrac{z_{j}}{\overline{z_{j}}}\sin^{2}\tau_{j}(\overline{p_{i}^{j}(z_{j})}+\overline{b_{i}^{j}})$
if $i\neq j$;

  * (ii)

$A_{j}(z_{j})=\dfrac{z_{j}}{\overline{z_{j}}}\sin^{2}\tau_{j}(\overline{p_{j}(z_{j})}+\overline{b_{j}})$

where

  * (a)

$\displaystyle
p_{i}^{j}(z_{j})=\sum_{k<0}a_{ik}^{j}\rho_{j}^{-k}z_{j}^{k}+\sum_{k>0}a_{ik}^{j}z_{j}^{k}$
if $i\neq j$;

  * (b)

$\displaystyle
p_{j}(z_{j})=\sum_{k<0}a_{jk}\rho_{j}^{-k}z_{j}^{k}+\sum_{k>0}a_{jk}z_{j}^{k}$

such that $\rho_{j}=e^{-\frac{2\pi^{2}}{l_{j}}}$ and the coefficients satisfy
the following conditions:

  * (1)

$\displaystyle\sum_{k<0}|a_{ik}^{j}|c^{-k}=O(u_{j}^{-2})$ and
$\displaystyle\sum_{k>0}|a_{ik}^{j}|c^{k}=O(u_{j}^{-2})$ if $i\geq m+1$;

  * (2)

$\displaystyle\sum_{k<0}|a_{ik}^{j}|c^{-k}=O(u_{j}^{-2})O(\dfrac{u_{i}^{3}}{|t_{i}|})$
and
$\displaystyle\sum_{k>0}|a_{ik}^{j}|c^{k}=O(u_{j}^{-2})O(\dfrac{u_{i}^{3}}{|t_{i}|})$
if $i\leq m$ and $i\neq j$;

  * (3)

$\displaystyle\sum_{k<0}|a_{jk}|c^{-k}=O(\dfrac{u_{j}}{|t_{j}|})$ and
$\displaystyle\sum_{k>0}|a_{jk}|c^{k}=O(\dfrac{u_{j}}{|t_{j}|})$;

  * (4)

$|b_{i}^{j}|=O(u_{j})$ if $i\geq m+1$;

  * (5)

$|b_{i}^{j}|=O(u_{j})O(\dfrac{u_{i}^{3}}{|t_{i}|})$ if $i\leq m$ and $i\neq
j$;

  * (6)

$b_{j}=-\dfrac{u_{j}}{\pi\overline{t_{j}}}+O(\dfrac{u_{j}u_{0}}{|t_{j}|})$.

###### Proof.

From the duality formula between harmonic Beltrami differentials and the
holomorphic quadratic differentials:

$B_{i}=\lambda^{-1}\sum_{l=1}^{n}h_{i\overline{l}}\overline{\psi_{l}},$

we know that $\displaystyle
A_{i}=\lambda^{-1}\sum_{l=1}^{n}h_{i\overline{l}}\overline{\phi_{l}}$. Also,
note that by Wolpert’s estimate of the length of the short geodesic
$\gamma_{j}$ (Example 4.3, [30]) we have
$l_{j}=-\dfrac{2\pi^{2}}{\log|t_{j}|}(1+O(u_{0}))$, so we know that there
exists a constant $0<\mu<1$ such that $\mu|t_{j}|<\rho_{j}<\mu^{-1}|t_{j}|$.
Then by Theorem 4.4.2 (but we need to replace $c$ by $\mu c$) and Corollary
4.4.1, we can compute the desired orders for each cases. ∎

In order to estimate the asymptotics of the Ricci metric, we need to estimate
the norms of the harmonic Beltrami differentials, and the above expressions
for the harmonic Beltrami differentials can help us to do so.

######  Lemma 4.4.2.

Let $\left\|\cdot\right\|_{k}$ be the $C^{k}$ norm defined before. We have

  * (i)

$\left\|A_{i}\right\|_{0,\Omega_{c}^{i}}=O(\dfrac{u_{i}}{|t_{i}|})$ and
$\left\|A_{i}\right\|_{0,X-\Omega_{c}^{i}}=O(\dfrac{u_{i}^{3}}{|t_{i}|})$ if
$i\leq m$;

  * (ii)

$\left\|A_{i}\right\|_{0}=O(1)$ if $i\geq m+1$;

  * (iii)

$\left\|f_{i\overline{i}}\right\|_{0,\Omega_{c}^{i}}=O(\dfrac{u_{i}^{2}}{|t_{i}|^{2}})$
and
$\left\|f_{i\overline{i}}\right\|_{0,X-\Omega_{c}^{i}}=O(\dfrac{u_{i}^{6}}{|t_{i}|^{2}})$
if $i\leq m$;

  * (iv)

$\left\|f_{i\overline{j}}\right\|_{0,\Omega_{c}^{i}}=O(\dfrac{u_{i}u_{j}^{3}}{|t_{i}t_{j}|})$,
$\left\|f_{i\overline{j}}\right\|_{0,\Omega_{c}^{j}}=O(\dfrac{u_{i}^{3}u_{j}}{|t_{i}t_{j}|})$
and
$\left\|f_{i\overline{j}}\right\|_{0,X-(\Omega_{c}^{i}\cup\Omega_{c}^{j})}=O(\dfrac{u_{i}^{3}u_{j}^{3}}{|t_{i}t_{j}|})$
if $i,j\leq m$ and $i\neq j$;

  * (v)

$\left\|f_{i\overline{j}}\right\|_{0,\Omega_{c}^{i}}=O(\dfrac{u_{i}}{|t_{i}|})$
and
$\left\|f_{i\overline{j}}\right\|_{0,X-\Omega_{c}^{i}}=O(\dfrac{u_{i}^{3}}{|t_{i}|})$
if $i\leq m$ and $j\geq m+1$;

  * (vi)

$\left\|f_{i\overline{j}}\right\|_{0}=O(1)$ if $i,j\geq m+1$;

  * (vii)

$|f_{i\overline{i}}|_{L^{1}}=O(\dfrac{u_{i}^{3}}{|t_{i}|^{2}})$ if $i\leq m$;

  * (viii)

$|f_{i\overline{j}}|_{L^{1}}=O(\dfrac{u_{i}^{3}u_{j}^{3}}{|t_{i}t_{j}|})$ if
$i,j\leq m$ and $i\neq j$;

  * (ix)

$|f_{i\overline{j}}|_{L^{1}}=O(\dfrac{u_{i}^{3}}{|t_{i}|})$ if $i\leq m$ and
$j\geq m+1$;

  * (x)

$|f_{i\overline{j}}|_{L^{1}}=O(1)$ if $i,j\geq m+1$.

######  Remark 4.4.1.

Form the lemma, we can also know that
$\left\|f_{i\overline{j}}\right\|_{0}=O(1)\left\|A_{i}\right\|_{0}\left\|A_{j}\right\|_{0}$
for any $i,j$.

###### Proof.

Choose $c$ sufficiently small such that for all $j\leq m$, $\tan(u_{j}\log
c)<-10u_{j}$. Then by some calculus argument we can show that when $1\leq
p\leq 10$, on the collar $\Omega_{c}^{j}$,

$|r_{j}^{k}\sin^{p}\tau_{j}|\leq c^{k}|\log c|^{p}u_{j}^{p}$

if $k\geq 1$;

$|r_{j}^{k}\sin^{p}\tau_{j}|\leq c^{-k}|\log c|^{p}\rho_{j}^{k}u_{j}^{p}$

if $k\leq-1$. The first two claims can be proved similarly, so let’s work out
(i) only. Notice that on $\Omega_{c}^{i}$ we have

$\displaystyle|A_{i}|$
$\displaystyle=\bigg{|}\dfrac{z_{i}}{\overline{z_{i}}}\bigg{|}|\sin^{2}\tau_{i}(\overline{p_{i}}+\overline{b_{i}})|$
$\displaystyle\leq\sum_{k<0}|a_{ik}|\rho_{i}^{-k}r_{i}^{k}\sin^{2}\tau_{i}+\sum_{k>0}|a_{ik}|r_{i}^{k}\sin^{2}\tau_{i}+|b_{i}|$
$\displaystyle\leq(\log
c)^{2}u_{i}^{2}(\sum_{k<0}|a_{ik}|c^{-k}+\sum_{k>0}|a_{ik}|c^{k})+|b_{i}|$
$\displaystyle=O(u_{i}^{2})O(\dfrac{u_{i}}{|t_{i}|})+O(u_{i}^{2})O(\dfrac{u_{i}}{|t_{i}|})+O(\dfrac{u_{i}}{|t_{i}|})=O(\dfrac{u_{i}}{|t_{i}|}).$

Thus $\left\|A_{i}\right\|_{0,\Omega_{c}^{i}}=O(\dfrac{u_{i}}{|t_{i}|})$. In a
similar way we can also show that $|A_{i}|=O(\dfrac{u_{i}^{3}}{|t_{i}|})$ on
$\Omega_{c}^{j}$ with $j\neq i$. Then by Masur’s result[19], the duality