New K\"ahler metric on quasifuchsian space and its curvature properties

TL;DR

This paper introduces a new Kähler metric on quasifuchsian space extending the Weil-Petersson metric, and analyzes its curvature properties, revealing negativity along certain directions, thus advancing understanding of the geometric structure of $QF(S)$.

Contribution

The authors construct a novel mapping class group invariant Kähler metric on quasifuchsian space that extends the Weil-Petersson metric and analyze its curvature properties.

Findings

The new metric extends the Weil-Petersson metric on Teichmüller space.

Curvature calculations show negativity along tautological directions.

The metric is invariant under the mapping class group.

Abstract

Let $QF(S)$ be the quasifuchsian space of a closed surface $S$ of genus $g\geq 2$. We construct a new mapping class group invariant K\"ahler metric on $QF(S)$. It is an extension of the Weil-Petersson metric onthe Teichm\"uller space $\mathcal T(S)\subset QF(S)$. We also calculate its curvature and prove some negativity for the curvature along the tautological directions.

Equations

$${\rho }_f(\gamma )=f\circ \rho (\gamma )\circ f^{-1}.$$

$$\varphi \rho =\varphi (X,Y)=(\varphi X,\varphi Y).$$

$$\pi :\mathcal{P}(S)\to \mathcal{T}(S).$$

$${\sigma }_0:\mathcal{T}(S)\to \mathcal{P}(S).$$

$$S(f)=\left[{\left(\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}\right)}^{\prime }-\frac{1}{2}{\left(\frac{f^{\prime \prime }(z)}{f^{\prime }(z)}\right)}^2\right]d{z}^2$$

$${\beta }_Y:\mathcal{T}(S)\to P(Y)\subset \mathcal{P}(\bar{S}).$$

$$Q(X,Y)\to {\Omega }_{-}/Q(X,Y)$$

$$Q(X,Y)\to {\Omega }_{-}/Q(X,Y)-{\sigma }_0(Y),$$

$$||\varphi |{|}_{\infty }=\underset{Y}{\sup }{\rho }^{-2}|\varphi (z)|$$

$${\int }_Y|\varphi (z){|}^2\rho (z{)}^{-4}\rho (z{)}^2|dz{|}^2\le ||\varphi |{|}_{\infty }^{2}2\pi (2g-2)\le 9\pi (g-1).$$

$$\Vert \mu {\Vert }_{WP}^2={\int }_X|\mu (z){|}^2\rho (z{)}^2|dz{|}^2$$

$$\Phi =\varphi (z)d{z}^2\to \beta ={\beta }_{\Phi }=\frac{\overline{\varphi (z)}}{{\rho }^2(z)}\frac{d\bar{z}}{dz}.$$

$$\Vert \beta {\Vert }^2=||\beta |{|}_{WP}^2={\int }_X\frac{|\varphi (z){|}^2}{{\rho }^4(z)}{\rho }^2(z)|dz{|}^2=||\Phi |{|}^2.$$

$$||w|{|}^2=(w,w)={\int }_Y|\mu (v){|}^2\rho (v{)}^2|dv{|}^2$$

$$\Phi (w)=||w|{|}^2+k{\pi }^{\ast }\psi (w),$$

$$||T|{|}_{\Phi }^2={\bar{\partial }}_T{\partial }_T\Phi (w)=-(R(u,\bar{u})w,w)+({\mathcal{D}}_{u}w+v,{\mathcal{D}}_{u}w+v)+k{\bar{\partial }}_u{\partial }_u\psi >0,$$

$$R_{i\bar{j}\alpha \bar{\beta }}{v}^i{\bar{v}}^j{\xi }^{\alpha }{\bar{\xi }}^{\beta }<0$$

$$R_{i\bar{j}\alpha \bar{\beta }}=-{\partial }_{\alpha }{\partial }_{\bar{\beta }}{G}_{i\bar{j}}+G^{k\bar{l}}{\partial }_{\alpha }{G}_{i\bar{l}}{\partial }_{\bar{\beta }}{G}_{k\bar{j}}$$

$$(z;v)=(z^1,\cdots ,z^n;v^1,\cdots ,v^r),$$

$$v\in E\mapsto G(v)=G(v,\bar{v})=G_{i\bar{j}}{v}^i{\bar{v}}^j.$$

$$\partial \bar{\partial }G=-R_{i\bar{j}\alpha \bar{\beta }}{v}^i{\bar{v}}^{j}d{z}^{\alpha }\wedge d{\bar{z}}^{\beta }+G_{i\bar{j}}\delta v^i\wedge \delta {\bar{v}}^j.$$

$$-R_{i\bar{j}\alpha \bar{\beta }}{v}^i{\bar{v}}^{j}d{z}^{\alpha }\wedge d{\bar{z}}^{\beta }+G_{i\bar{j}}\delta v^i\wedge \delta {\bar{v}}^j$$

$$=-(-{\partial }_{\alpha }{\partial }_{\bar{\beta }}{G}_{i\bar{j}}+G^{\bar{l}k}{\partial }_{\alpha }{G}_{i\bar{l}}{\partial }_{\bar{\beta }}{G}_{k\bar{j}})v^i{\bar{v}}^{j}d{z}^{\alpha }\wedge d{\bar{z}}^{\beta }$$

$$+G_{i\bar{j}}(d{v}^i+G_{\alpha \bar{l}}{G}^{\bar{l}i}d{z}^{\alpha })\wedge (d{\bar{v}}^j+G_{\bar{\beta }k}{G}^{\bar{j}k}d{\bar{z}}^{\beta })$$

$$=G_{\alpha \bar{\beta }}d{z}^{\alpha }\wedge d{\bar{z}}^{\beta }+G_{\alpha \bar{j}}d{z}^{\alpha }\wedge d{\bar{v}}^j+G_{i\bar{\beta }}d{v}^i\wedge d{\bar{z}}^j+G_{i\bar{j}}d{v}^i\wedge d{\bar{v}}^j$$

$$=\partial \bar{\partial }G.$$

$$\Omega :={\pi }^{\ast }\omega +\sqrt{-1}\partial \bar{\partial }G.$$

$$\Omega =\sqrt{-1}{\Omega }_{\alpha \bar{\beta }}d{z}^{\alpha }\wedge d{\bar{z}}^{\beta }+\sqrt{-1}{G}_{i\bar{j}}\delta v^i\wedge \delta {\bar{v}}^j,$$

$${\Omega }_{\alpha \bar{\beta }}:=-R_{i\bar{j}\alpha \bar{\beta }}{v}^i{\bar{v}}^j+g_{\alpha \bar{\beta }}$$

$$\partial G=G_{\alpha }d{z}^{\alpha }+G_{i}d{v}^i=G_i(d{v}^i+G_{\alpha \bar{l}}{G}^{\bar{l}i}d{z}^{\alpha })=G_i\delta v^i.$$

$$\Vert \partial G{\Vert }^2=G_i{G}_{\bar{j}}{G}^{\bar{j}i}.$$

$$G_i{G}_{\bar{j}}{G}^{\bar{j}i}=G_{i\bar{l}}{\bar{v}}^l{G}_{k\bar{j}}{v}^k{G}^{\bar{j}i}=G_{k\bar{l}}{v}^k{\bar{v}}^l=G,$$

Full text

11footnotetext: 2000 Mathematics Subject Classification. 53C43, 53C21, 53C2522footnotetext: Key words and phrases. Griffiths negativity, Teichmüller space, Quasifuchsian space, Complex projective structure.33footnotetext: Research by Inkang Kim is partially supported by Grant NRF-2017R1A2A2A05001002 and research by Genkai Zhang is partially supported by Swedish Research Council (VR).

1. 1 Preliminaries

2. 1.1 Quasifuchsian space

3. 1.2 Space of complex projective structures on surface

4. 1.3 Embedding of quasifuchsian space into the space of complex projective structures

5. 1.4 Vector bundle isomorphism between quadratic differentials and Beltrami differentials

6. 2 Griffiths negativity and Kähler metric on the holomorphic vector bundles

7. 2.1 Griffiths negativity

8. 2.2 Kähler metrics on Griffiths negative vector bundles

9. 3 Curvature of the new Kähler metric

Introduction

Teichmüller space $\mathcal{T}(S)$ carries a natural mapping class group invariant Kähler metric, called a Weil-Petersson metric $g_{WP}$. There have been active studies on the properties of this metric since its birth. More recently, some new Kähler metrics with more desirable properties such as Kähler hyperbolicity have been found where the Kähler hyperbolicity means that the Kähler metric is complete with bounded curvatures and it has a bounded Kähler primitive. Such Kähler hyperbolic metrics are studied by McMullen [14] and Liu-Sun-Yau [13].

In Kleinian group theory, the quasifuchsian space $QF(S)$ is a quasiconformal deformation space of the Fuchsian space $F(S)$ which can be identified with $\mathcal{T}(S)$. By Bers’ simultaneous uniformization theorem, $QF(S)$ can be naturally identified with $\mathcal{T}(S)\times\mathcal{T}(\bar{S})$ where $\bar{S}$ is a surface with an orientation reversed. With this identification, the mapping class group acts diagonally on $QF(S)$ and $F(S)=\mathcal{T}(S)$ sits diagonally on $\mathcal{T}(S)\times\mathcal{T}(\bar{S})$. But this diagonal embedding is totally real. Hence if one gives a product Kähler metric on $QF(S)$, this metric is not an extension of a Kähler metric on $F(S)$. There have been several attempts to extend a Kähler metric of $\mathcal{T}(S)$ to $QF(S)$. Bridgeman and Taylor [4] described a quasi-metric which extends the Kähler metric of $\mathcal{T}(S)$ but it vanishes along the pure bending deformation vectors [5].

In this article, we give a completely new mapping class group invariant Kähler metric on $QF(S)$ which extends any Kähler metric on $\mathcal{T}(S)$. Indeed such a metric is already defined in the paper [11] a few years ago. The metric is defined by a Kähler potential which is a combination of $L^{2}$ norm of a fiber and a Kähler potential on the base $\mathcal{T}(S)$. We will see that $QF(S)$ can be embedded, via Bers embedding using the complex projective structures, in the holomorphic bundle over $\mathcal{T}(S)$ with fibers being quadratic holomorphic differentials as a bounded open neighborhood of the zero section. The Kähler metric we construct is the restriction to this open neighborhood. We choose then the Weil-Petersson metric on $\mathcal{T}(S)$ and show that the new Kähler metric on $QF(S)$ has similar properties such as its Kähler form has a bounded primitive and the curvature has non-positivity for some directions.

Theorem 0.1**.**

There exists a mapping class group invariant Kähler metric on $QF(S)$ which extends the Weil-Petersson metric on $\mathcal{T}(S)\subset QF(S)$. Furthermore the curvature of the metric is non-positive when evaluated on the tautological section (and vanishes along vertical directions), its Ricci curvature is bounded from above by $-\frac{1}{\pi(g-1)}$ when restricted to Teichmüller space, and its Kähler form has a bounded primitive.

Acknowledgment: The first author thanks C. McMullen for the communications on Kähler metrics on Teichmüller space and the suggestions.

1. Preliminaries

1.1. Quasifuchsian space

Recall that the isometry group of the hyperbolic 3-space $\mathbb{H}^{3}$ can be identified with $PSL(2,\mathbb{C})$. We use the unit ball in $\mathbb{R}^{3}$ as a realization of $\mathbb{H}^{3}$. The ideal boundary is then $S^{2}$ and is further identified with $\mathbb{CP}^{1}$ such that the action of $PSL(2,\mathbb{C})$ on $S^{2}$ is the natural extension of its isometric action on $\mathbb{H}^{3}$.

The Teichmüller space $\mathcal{T}(S)$ is realized as the space of Fuchsian representations, i.e., discrete and faithful representations $\rho:\pi_{1}(S)\rightarrow PSL(2,\mathbb{R})$ up to conjugacy. Let $\Gamma_{\rho}$ be the image of $\rho$, whence $\Gamma_{\rho}$ acts on $S^{2}$ by Möbius map preserving the equator. Then any quasiconformal map $f$ from $S^{2}$ into itself induces a quasiconformal deformation $\rho_{f}$ defined by

[TABLE]

If furthermore $\rho_{f}(\gamma)$ is an element of $PSL(2,\mathbb{C})$ for any $\gamma\in\pi_{1}(S)$ then it defines a representation of $\pi_{1}(S)$ in $PSL(2,\mathbb{C})$. Collection of such quasiconformal deformations of Fuchsian representations is denoted $QF(S)$ and is identified with an open set of a character variety $\chi(\pi_{1}(S),PSL(2,\mathbb{C}))$. Hence it has a natural induced complex structure from $\chi(\pi_{1}(S),PSL(2,\mathbb{C}))$.

If $\phi:\pi_{1}(S)\rightarrow PSL(2,\mathbb{C})$ is a quasifuchsian representation, then $M_{\phi}=\mathbb{H}^{3}/\phi(\pi_{1}(S))$ is a quasifuchsian hyperbolic 3-manifold which is homeomorphic to $S\times\mathbb{R}$. Then two ideal boundaries of $M_{\phi}$ define a pairs of points $(X,Y)\in\mathcal{T}(S)\times\mathcal{T}(\bar{S})$. This is known as Bers’ simultaneous uniformization of $QF(S)$ [2]. In this case, we denote $M_{\phi}$ by $Q(X,Y)$. In this identification, a Fuchsian representation $\rho:\pi_{1}(S)\rightarrow PSL(2,\mathbb{R})$ whose quotient $X=\mathbb{H}^{2}/\rho(\pi_{1}(S))$ is a point in $\mathcal{T}(S)$ gets identified with $(X,\bar{X})$.

The mapping class group $Mod(S)$ acts on the space of representations $\rho:\pi_{1}(S)\rightarrow PSL(2,\mathbb{C})$ by pre-composition $\phi\rho=\rho\circ\phi_{*}$ where $\phi\in Mod(S)$ and $\phi_{*}$ is the induced homomorphism on $\pi_{1}(S)$. Then $Mod(S)$ acts on $QF(S)=\mathcal{T}(S)\times\mathcal{T}(\bar{S})$ diagonally

[TABLE]

1.2. Space of complex projective structures on surface

A complex projective structure on $S$ is a maximal atlas $\{(\phi_{i},U_{i})|\phi_{i}:U_{i}\rightarrow S^{2}\}$ whose transition maps $\phi_{i}\circ\phi_{j}^{-1}$ are restrictions of Möbius maps. Then the developing map $dev:\widetilde{S}\rightarrow S^{2}$ gives rise to a holonomy representation $\rho:\pi_{1}(S)\rightarrow PSL(2,\mathbb{C})$. We denote the space of marked complex projective structures on $S$ by $\mathcal{P}(S)$. Since Möbius transformations are holomorphic, a projective structure determines a complex structure on $S$. In this way we obtain a forgetful map

[TABLE]

Obviously a Fuchsian representation $\rho:\pi_{1}(S)\rightarrow PSL(2,\mathbb{R})\subset PSL(2,\mathbb{C})$ preserving the equator of $S^{2}$ gives rise to an obvious projective structure by identifying $\mathbb{H}^{2}$ with the upper and lower hemisphere of $S^{2}$. This gives an embedding

[TABLE]

More generally, for $X\in\mathcal{T}(S)$ and $Z\in\pi^{-1}(X):=P(X)$, by conformally identifying $\widetilde{X}=\mathbb{H}^{2}$, we obtain a developing map $dev:\mathbb{H}^{2}\rightarrow S^{2}=\mathbb{CP}^{1}$ for $Z$. Hence the developing map can be regarded as a meromorphic function $f=dev$ on $\mathbb{H}^{2}$. Then the Schwarzian derivative

[TABLE]

descends to $X$ as a holomorphic quadratic differential. It is known that for any element in holomorphic quadratic differentials $Q(X)$ on $X$, one can show that there exists a complex projective structure over $X$ by solving Schwarzian linear ODE equation.

In this way, $\mathcal{P}(S)$ can be identified with a holomorphic vector bundle $\mathcal{Q}(S)$ over $\mathcal{T}(S)$ whose fiber over $X$ is $Q(X)$. In particular this identifies $P(X)$ with $Q(X)$ as affine spaces, [7], and the choice of a base point $Z_{0}$ in $P(X)$ gives an isomorphism $Z\rightarrow Z-Z_{0}$. Hence we will choose $Z_{0}=\sigma_{0}(X)$ and $\mathcal{T}(S)$ will be identified with zero section on $\mathcal{Q}(S)$.

1.3. Embedding of quasifuchsian space into the space of complex projective structures

Recall that given $X\in\mathcal{T}(S),Y\in\mathcal{T}(\bar{S})$ the Bers’ uniformization determines the quasifuchsian manifold $Q(X,Y)$. Then $Q(X,Y)$ has domain of discontinuity $\Omega_{+}\cup\Omega_{-}$ with $\Omega_{+}/Q(X,Y)=X,\Omega_{-}/Q(X,Y)=Y$ where $Q(X,Y)$ is viewed as a quasifuchsian representation into $PSL(2,\mathbb{C})$.

As a quotient of a domain in $\mathbb{CP}^{1}$ by a discrete group in $PSL(2,\mathbb{C})$, $\Omega_{-}/Q(X,Y)$ is a marked projective surface $\Sigma_{Y}(X)$. Then for a fixed $Y$, we obtain a quasifuchsian section, called a Bers’ embedding

[TABLE]

It is known that this map

[TABLE]

is a homeomorphism onto its image in $\mathcal{P}(\bar{S})$; see e.g. [7]. Under the identification of $\mathcal{P}(\bar{S})$ with $\mathcal{Q}(\bar{S})$ such that $\sigma_{0}(\mathcal{T}(\bar{S}))$ is a zero section,

[TABLE]

this embedding includes zero section which is the image of $\mathcal{T}(S)$.

The space $Q(Y)$ of quadratic differentials is also equipped with $L^{\infty}$-norm defined by

[TABLE]

where $\rho(z)|dz|$ is a hyperbolic metric on $Y$. Then by Nehari’s bound [14] we get

Theorem 1.1**.**

The above embedding of $Q(X,Y)$ into $Q(Y)$ is contained in a ball of radius $\frac{3}{2}$ in $Q(Y)$ where the norm is the $L^{\infty}$-norm on quadratic differentials.

Corollary 1.2**.**

The quasifuchsian space $QF(S)$ embeds into a neighborhood of a zero section in $\mathcal{Q}(\bar{S})$ which is contained in a ball of radius $9\pi(g-1)$ in $L^{2}$-norm on each fiber $Q(Y)$.

Proof.

The $L^{2}$-norm of a quadratic differential $\phi(z)dz^{2}$ is given by

[TABLE]

∎

1.4. Vector bundle isomorphism between quadratic differentials and Beltrami differentials

The holomorphic tangent bundle of Teichmüller space $\mathcal{T}(S)$ is a holomorphic vector bundle over Teichmüller space whose fiber over $X$ is the set of harmonic Beltrami differentials $B(X)$. For a harmonic Beltrami differential $\mu(z)\frac{d\bar{z}}{dz}$ over $X$ with a hyperbolic metric $g=\rho(z)|dz|$, the $L^{2}$-norm defines the Weil-Petterson metric

[TABLE]

on the tangent space of $\mathcal{T}(S)$. The set of harmonic Beltrami differentials $B(X)$ and $Q(X)$ are vector bundle isomorphic by the natural identification of differential forms with tangent vectors via the metric,

[TABLE]

The $L^{2}$-norms are by definition preserved,

[TABLE]

By Corollary 1.2, we get

Corollary 1.3**.**

Under this isomorphism between cotangent bundle and holomorphic tangent bundle of $\mathcal{T}(S)$, the quasifuchsian space $QF(S)$ embeds into a neighborhood of a zero section in the holomorphic tangent bundle of $\mathcal{T}(S)$ which is contained in a ball of radius $9\pi(g-1)$ in $L^{2}$-norm on each fiber $B(X)$.

2. Griffiths negativity and Kähler metric on the holomorphic vector bundles

2.1. Griffiths negativity

As elaborated above the space $QF(S)$ can be realized as an open set in the tangent bundle of $\mathcal{T}(S)$, and we shall construct metrics on $QF(S)$ using some general constructions. For that purpose we recall the notion of Griffiths positivity. Identifying $\mathcal{P}(S)$ with the holomorphic vector bundle $\mathcal{Q}(S)$ whose fiber over $Y\in\mathcal{T}(S)$ is $Q(Y)$, one can give a mapping class group invariant Kähler metric on $\mathcal{Q}(S)$ as follows. By a theorem of Berndtsson [3], one can show that $\mathcal{Q}(S)$ is Griffiths positive. See [11] for a proof. Hence its dual bundle $\mathcal{B}(S)=\mathcal{Q}^{*}(S)$, which is a tangent bundle of Teichmüller space whose fiber is the set of Beltrami differentials, is Griffiths negative. We fix in the rest of the paper this realization of $QF(S)$ as a subset in $\mathcal{Q}^{*}(S)$. The $L^{2}$-norm of a Beltrami differential $w=\mu(v)\frac{d\bar{v}}{dv}$ is given by

[TABLE]

where $v$ is a local holomorphic coordinate on $Y$ and $\rho(v)|dv|$ is a hyperbolic metric on $Y$. Here $(,)$ denotes the $L^{2}$ inner product over each fiber and $||\cdot||$ denotes its associated norm.

The Kähler metric depending on a constant $k>0$ and a Kähler metric on $\mathcal{T}(S)$, is constructed on $\mathcal{B}(S)=\mathcal{Q}^{*}(S)$ via Kähler potential

[TABLE]

where $w$ is an element in the fiber, $\psi$ is a Kähler potential on $\mathcal{T}(S)$ and $\pi:\mathcal{B}(S)\rightarrow\mathcal{T}(S)$ is a projection.

In local holomorphic coordinates $(z,x)$ around $w_{0}$, where $z=(z_{1},\cdots,z_{3g-3})$ is local holomorphic coordinates around $\pi(w_{0})=z_{0}$, and $w=\sum x^{\alpha}e_{\alpha}(z)$ with respect to local holomorphic sections $e_{\alpha}$, for a holomorphic tangent vector at $w$ $T=u+v$ with a canonical decomposition into $\mathcal{T}(S)$ direction $u$ and vertical fiber direction $v$, the norm of $T$ with respect to the Kähler metric $g$ defined by the Kähler potential $\Phi$ is given by

[TABLE]

where $R$ is a curvature of the Chern connection $\nabla$ on $\mathcal{B}(S)$ and $\nabla=\mathcal{D}+\bar{\partial}$ is a decomposition into $(1,0)$ and $(0,1)$ part of the connection. See [11] for details.

Since this construction is general, we treat this construction as general as possible in the following subsection.

2.2. Kähler metrics on Griffiths negative vector bundles

Let $\pi:E\to M$ be a holomorphic vector bundle of rank $r$ over a complex manifold $M$, $\dim M=n$. Let $\{e_{i}\}_{i=1}^{r}$ be a local holomorphic frame of $E$ and $\{z^{\alpha}\}_{\alpha=1}^{n}$ be local coordinates of $M$. Let $G$ be a Hermitian metric on $E$ with Griffiths negative curvature, that is

[TABLE]

for any non-zero vectors $v=v^{i}e_{i}\in E$ and $\xi=\xi^{\alpha}\frac{\partial}{\partial z^{\alpha}}\in TM$. Here

[TABLE]

denotes the Chern curvature tensor of the Hermitian metric $G$. With respect to the local holomorphic frame $\{e_{i}\}_{i=1}^{r}$, the complex manifold $E$ is equipped with the following holomorphic coordinates

[TABLE]

representating the point $v=v^{i}e_{i}\in E$. The Hermitian metric $G$ also gives the norm square function on $E$. By abuse of notation, we also denote it by $G$, i.e. the function

[TABLE]

Then $\partial\bar{\partial}G$ is a $(1,1)$-form on $E$.

Lemma 2.1**.**

Denote $\delta v^{i}:=dv^{i}+G_{\alpha\bar{l}}G^{\bar{l}i}dz^{\alpha}$. Then

[TABLE]

Proof.

This follows by a direct computation,

[TABLE]

∎

Now we assume $(M,\omega=\sqrt{-1}g_{\alpha\bar{\beta}}dz^{\alpha}\wedge d\bar{z}^{\beta})$ is a Kähler manifold and $(E,G)$ is Griffiths negative. Define the $(1,1)$-form

[TABLE]

Lemma 2.1 then implies that $\Omega$ is a Kähler metric on $E$. In terms of local coordinates $\Omega$ is

[TABLE]

where

[TABLE]

is a positive definite matrix. The differential $\partial G$ of $G$ is a globally defined one-form on $E$, and its norm square is $G$. Indeed,

[TABLE]

Its norm squre with respect to the metric $\Omega$ is

[TABLE]

Since $G=G_{i\bar{j}}v^{i}\bar{v}^{j}$, so $G_{i}=G_{i\bar{j}}\bar{v}^{j}$ and

[TABLE]

which yields that

[TABLE]

which is independent of the metric $\omega$.

Proposition 2.2**.**

The norm of the one-form $\partial G$ with respect to $\Omega$ is given by

[TABLE]

for any metric $\omega$ on $M$. In particular,

[TABLE]

on the disk bundle $S_{R}=\{(z,v)\in E|G(z,v)<R\}$.

As a corollary, we obtain

Corollary 2.3**.**

If $\omega$ is $d$-bounded, $\omega=d\beta$ for some (locally defined) bounded one-form $\beta$, then $\Omega$ is also $d$-bounded on any bounded domain of $E$ with $\Omega=d(\partial G+\pi^{*}\beta)$ and bounded one-form $\partial G+\pi^{*}\beta$.

Specifying to the space $QF(S)$ we find that new Kähler metric on $QF(S)$ has a bounded primitive if the Kähler form on $\mathcal{T}(S)$ has a bounded primitive.

3. Curvature of the new Kähler metric

In this section, we will calculate the curvature of the Kähler metric $\Omega$ (2.1). By [6, Section 2], the Hermitian metric $G$ gives a decomposition on the tangent bundle $TE$ of $E$, i.e.

[TABLE]

where the horizontal subbundle $\mathcal{H}$ and vertical subbundle $\mathcal{V}$ are given by

[TABLE]

By duality, the cotangent bundle $T^{*}E=\mathcal{H}^{*}\oplus\mathcal{V}^{*}$ with

[TABLE]

Let $\nabla=\nabla^{\prime}+\bar{\partial}$ denote the Chern connection of $\Omega$ and

[TABLE]

denote the Chern curvature of $\nabla$. Then

[TABLE]

where the last equality holds since $\bar{\partial}\left(\frac{\delta}{\delta z^{\beta}}\right)$ is vertical, and

[TABLE]

where the last equality follows from the fact $G_{i\bar{j}k}=0$ since $G_{i\bar{j}}$ is a metric along vertical directions. From (3.1) and (3.2), the curvature $R^{\Omega}$ is

[TABLE]

and

[TABLE]

Therefore, we obtain

Proposition 3.1**.**

The Chern curvature $R^{\Omega}$ satisfies

• (i)

$\langle R^{\Omega}\left(\frac{\partial}{\partial v^{i}}\right),\frac{\partial}{\partial v^{j}}\rangle=(R_{i\bar{l}\alpha\bar{\beta}}R_{k\bar{j}\gamma\bar{\sigma}}v^{k}\bar{v}^{l}\Omega^{\bar{\beta}\gamma}+R_{i\bar{j}\alpha\bar{\sigma}})dz^{\alpha}\wedge d\bar{z}^{\sigma}$;

• (ii)

$\langle R^{\Omega}(\frac{\delta}{\delta z^{\alpha}}),\frac{\delta}{\delta z^{\beta}}\rangle=\bar{\partial}(\partial\Omega_{\alpha\bar{\sigma}}\Omega^{\bar{\sigma}\gamma})\Omega_{\gamma\bar{\beta}}-R_{p\bar{l}\gamma\bar{\beta}}R_{k\bar{q}\alpha\bar{\sigma}}v^{k}\bar{v}^{l}G^{\bar{q}p}dz^{\gamma}\wedge d\bar{z}^{\sigma}$.

Proof.

(i) We compute the inner product according to (3.4),

[TABLE]

Note that $G_{i\bar{j}}G^{k\bar{j}}=\delta_{i}^{k}$, hence

[TABLE]

Then

[TABLE]

But $R_{i\bar{l}\alpha\bar{\beta}}=-\partial_{\alpha}\partial_{\bar{\beta}}G_{i\bar{l}}+G^{k\bar{j}}\partial_{\alpha}G_{i\bar{j}}\partial_{\bar{\beta}}G_{k\bar{l}}$ and $G=G(z,v)=G_{i\bar{j}}(z)v^{i}\bar{v}^{j}$, hence

[TABLE]

and

[TABLE]

Finally we get

[TABLE]

Similar calculations give

[TABLE]

(ii) Using (3.5) we compute

[TABLE]

∎

Remark 3.2**.**

From (i), when evaluated on a vertical vector, $\langle R^{\Omega}\left(\frac{\partial}{\partial v^{i}}\right),\frac{\partial}{\partial v^{j}}\rangle$ vanishes, that is $\langle R^{\Omega}(\frac{\partial}{\partial v^{k}},\bullet)\left(\frac{\partial}{\partial v^{i}}\right),\frac{\partial}{\partial v^{j}}\rangle=0$.

There exists a canonical holomorphic section of $\mathcal{V}$, that is

[TABLE]

which is called the tautological section (see e.g. [1, Section 3]). Denote

[TABLE]

From Proposition 3.1 the $(1,1)$-form $\langle R^{\Omega}(P),P\rangle$ is

[TABLE]

For any point $(z,v)$ outside the zero section, i.e in the set

[TABLE]

the vector $P(z,v)\neq 0$. So $(\Psi_{\alpha\bar{\beta}})$ is a positive definite matrix on $E^{o}$ by Griffiths negativity. Since

[TABLE]

is positive definite, so

[TABLE]

Thus, we obtain

Proposition 3.3**.**

$\langle\sqrt{-1}R^{\Omega}(P),P\rangle$* is a non-positive $(1,1)$-form on $E$.*

Remark 3.4**.**

Moreover, $\langle\sqrt{-1}R^{\Omega}(P),P\rangle$ is a strictly negative $(1,1)$-form on $E^{o}$ along the horizontal directions, that is

[TABLE]

for any nonzero vector $\xi=\xi^{\alpha}\frac{\delta}{\delta z^{\alpha}}\in\mathcal{H}_{(z,v)}$, $(z,v)\in E^{o}$. In fact, from (3.7), $\langle R^{\Omega}(\xi,\bar{\xi})(P),P\rangle=0$ if and only if

[TABLE]

Since $\left(\Psi^{\bar{\beta}\gamma}-\Omega^{\bar{\beta}\gamma}\right)$ is positive definite on $E^{o}$, so (3.8) is equivalent to

[TABLE]

On the other hand, $(\Psi_{\alpha\bar{\beta}})$ is positive definite on $E^{o}$ by Griffiths negativity of $(E,G)$, which implies that $\xi=0$.

The Ricci curvature of the Kähler metric is

[TABLE]

Denote by $\iota:M\to E$ the natural embedding (as the zero section of $E$), then

[TABLE]

is the $(1,1)$-form on $M$.

In particular, consider $E=TM$ and $(M,\omega)$ is Teichmüller space with Weil-Petersson metric, that is, $(G_{i\bar{j}})=(g_{\alpha\bar{\beta}})$ is Weil-Petersson metric. For any unit vector $\xi\in E=TM$, i.e. $\|\xi\|^{2}=1$, then

[TABLE]

where $Ric:=\bar{\partial}\partial\log\det(g_{\alpha\bar{\beta}})$ denotes the Ricci curvature of Weil-Petersson metric. From [15, Lemma 4.6 (i)], the Ricci curvature of Weil-Petersson metric satisfies

[TABLE]

where $g$ denotes the genus of Riemann surfaces. Substituting (3.11) into (3.10), we obtains

[TABLE]

Thus

Proposition 3.5**.**

Let $(M,\omega)$ be Teichmüller space with the Weil-Petersson metric, and let $E=TM$ be the holomorphic tangent bundle. When restricting to $M$, the Ricci curvature of $\Omega$ is bounded from above by $-\frac{1}{\pi(g-1)}$.

Now we begin to prove our main theorem:

Proof of Theorem 0.1.

From Corollary 1.3, the quasifuchsian space $QF(S)$ embeds into a neighborhood of a zero section in the holomorphic tangent bundle of $\mathcal{T}(S)$ which is contained in a ball of radius $9\pi(g-1)$ in $L^{2}$-norm on each fiber $B(X)$. Since the tangent bundle $\mathcal{B}(S)$ of $\mathcal{T}(S)$ with the Weil-Petersson metric $\omega_{WP}$ is Griffiths negative, so it defines a norm $G$ on $\mathcal{B}(S)$. Denote $\pi:\mathcal{B}(S)\to\mathcal{T}(S)$, then the following $(1,1)$-form

[TABLE]

defines a mapping class group invariant Kähler metric on $\mathcal{B}(S)$ (see [11]). From (2.2), one sees that $\Omega$ is an extension of the Weil-Petersson metric $\omega_{WP}$. From [14, Theorem 1.5], the Weil-Petersson metric $\omega_{WP}$ has a bounded primitive with respect to Weil-Petersson metric. By Corollary 2.3, the Kähler metric $\Omega$ also has a bounded primitive with respect to $\Omega$. And by Propositions 3.3, 3.5, the Chern curvature $R^{\Omega}$ of $\Omega$ is non-positive when evaluated on the tautological section $P$, and its Ricci curvature is bounded from above by $-\frac{1}{\pi(g-1)}$ when restricted to Teichmüller space. ∎

Remark 3.6**.**

Finally to put our results in perspective we remark that the space $\mathcal{P}(S)$ of marked complex projective structures is identified with the cotangent bundle of $\mathcal{T}(S)$ and the natural holonomy map $\mathcal{P}(S)\to\chi=\chi(\pi_{1}(S),PSL(2,\mathbb{C}))$ to the character variety is a local biholomorphic mapping by the results of Earle-Hejhal-Hubbard [8, 9, 10] (see also [7, Theorem 5.1]). Thus our constructions and results are also valid for $\mathcal{P}(S)$ and for its image in $\chi$. The space $QF(S)$ of quasifuchsian representations is also an open subset of $\chi$, $\mathcal{T}(S)\subset QF(S)\subset\chi$, and it might be interesting to understand the geometry of character variety $\chi$ using our metric on these open subsets.

The above remark applies also to the Hitchin component for any real split simple Lie group $G$ of real rank two, namely $G=SL(3,R),Sp(2,\mathbb{R}),G_{2}$. Indeed Labourie [12] generalized the construction in [11] of Kähler metric for $SL(3,\mathbb{R})$ to the above $G$. In this case the Hitchin component is proved to be a bundle over Teichmüller space with fiber being space of holomorphic differentials of degree $3,4,6$, respectively. Hence we obtain

Corollary 3.7**.**

The curvature of the Kähler metric on the Hitchin component for real split simple Lie groups of real rank 2 vanishes along vertical directions, and non-positive along tautological sections.