WEIL-PETERSSON Curvature operator

The Weil-Petersson curvature operator on the universal Teichmüller space

Zheng Huang Department of Mathematics, The City University of New York, Staten Island, NY 10314, USA The Graduate Center, The City University of New York, 365 Fifth Ave., New York, NY 10016, USA zheng.huang@csi.cuny.edu  and  Yunhui Wu Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing 100084, China yunhui_wu@mail.tsinghua.edu.cn
Abstract.

The universal Teichmüller space is an infinitely dimensional generalization of the classical Teichmüller space of Riemann surfaces. It carries a natural Hilbert structure, on which one can define a natural Riemannian metric, the Weil-Petersson metric. In this paper we investigate the Weil-Petersson Riemannian curvature operator of the universal Teichmüller space with the Hilbert structure, and prove the following:

  1. is non-positive definite.

  2. is a bounded operator.

  3. is not compact; the set of the spectra of is not discrete.

As an application, we show that neither the Quaternionic hyperbolic space nor the Cayley plane can be totally geodesically immersed in the universal Teichmüller space endowed with the Weil-Petersson metric.

2010 Mathematics Subject Classification:
Primary 30F60, Secondary 32G15

1. Introduction

1.1. The Weil-Petersson geometry on classical Teichmüller space

Moduli theory of Riemann surfaces and their generalizations continue to be inspiration for ideas and questions for many different mathematical fields since the times of Gauss and Riemann. In this paper, we study the Weil-Petersson geometry of the universal Teichmüller space.

Let be a closed oriented surface of genus where , and be the Teichmüller space of (space of hyperbolic metrics on modulo orientation preserving diffeomorphisms isotopic to the identity). The Teichmüller space is a manifold of complex dimension , with its cotangent space at identified as the space of holomorphic quadratic differentials on the conformal structure of the hyperbolic metric . The Weil-Petersson metric on Teichmüller space is obtained by duality from the natural pairing of holomorphic quadratic differentials. The Weil-Petersson geometry of Teichmüller space has been extensively studied: it is a Kählerian metric [Ahl61], incomplete [Chu76, Wol75] yet geodesically convex [Wol87]. Many features of the curvature property were also studied in detail by many authors (see a comprehensive survey [Wol11] and the book [Wol10]). Since intuitively we consider the universal Teichmüller space contains Teichmüller spaces of all genera, among those Weil-Petersson curvature features; it is known that the Weil-Petersson metric has negative sectional curvature, with an explicit formula for the Riemannian curvature tensor due to Tromba-Wolpert [Tro86, Wol86], strongly negative curvature in the sense of Siu [Sch86], dual Nakano negative curvature [LSY08], various curvature bounds in terms of the genus [Hua07b, Teo09, Wu17], good behavior of the Riemannian curvature operator on Teichmüller space [Wu14, WW15]. One can also refer to [BF06, Hua05, Hua07a, LSY04, LSYY13, Wol11, Wol10, Wol12b] for other aspects of the curvatures of the Weil-Petersson metric.

1.2. Main results

There are several well-known models of universal Teichmüller spaces. We will adapt the approach in [TT06] and use the disk model to define the universal Teichmüller space as a quotient of the space of bounded Beltrami differentials on the unit disk . Unlike the case in the classical Teichmüller space, the Petersson pairing for the bounded Beltrami differentials on is not well-defined on the whole tangent space of the universal Teichmüller space . To ramify this, Takhtajan-Teo [TT06] defined a Hilbert structure on such that the Petersson pairing is now meaningful on the tangent space at any point in this Hilbert structure. We denote the universal Teichmüller space with this Hilbert structure by . The resulting metric is the Weil-Petersson metric on . All terms will be defined rigorously in §2.

The Riemannian geometry of this infinitely dimensional deformation space is very intriguing. Takhtajan-Teo showed the Weil-Petersson metric on has negative sectional curvature, and constant Ricci curvature [TT06], and Teo [Teo09] proved the holomorphic sectional curvature has no negative upper bound.

We are interested in the Weil-Petersson curvature operator on . In general there are some fundamental questions regarding linear operators on manifolds: whether the operator is signed, whether it is bounded, and the behavior of its eigenvalues. In this paper, we investigate the Weil-Petersson curvature operator along these question lines. In particular, we prove:

Theorem 1.1.

Let be the Weil-Petersson Riemannian curvature operator on the universal Teichmüller space , then

  1. is non-positive definite on .

  2. For , if and only if there is an element such that , where is defined above.

As a direct corollary, we have:

Corollary 1.2.

[TT06] The sectional curvature of the Weil-Petersson metric on is negative.

Our second result is:

Theorem 1.3.

The curvature operator is bounded. More precisely, for any with , we have , where is the Euclidean norm for the wedge product defined in (4.1).

A direct consequence of Theorem  1.3 is:

Corollary 1.4.

[GBR15] The Riemannian Weil-Petersson curvature tensor (defined in (2.18)) is bounded.

Being bounded and non-positively definite are properties for the Weil-Petersson curvature operator on certain part of the classical Teichmüller space as well [Wu14, WW15], but noncompactness of is a more distinctive feature for . Our next result is:

Theorem 1.5.

The curvature operator is not a compact operator, more specifically, the set of spectra of is not discrete on the interval .

As an important application, in the last part of this paper we will address some rigidity questions on harmonic maps from certain symmetric spaces into . For harmonic map from a domain, which is either the Quaternionic hyperbolic space or the Cayley plane, into a non-positive curved target space, many beautiful rigidity results were established in [DM15, GS92, JY97, MSY93] and others. We prove the following:

Theorem 1.6.

Let be a lattice in a semisimple Lie group which is either or , and let be the isometry group of with respect to the Weil-Petersson metric. Then, any twisted harmonic map from into must be a constant, with respect to each homomorphism . Here the twisted map with respect to means that , for all .

1.3. Methods in the proofs

An immediate difficulty we have to cope with is that is an infinite dimensional manifold. There is however a basis for tangent vectors for the Hilbert structure that we can work with. With this basis, the Weil-Petersson Riemannian curvature tensor takes an explicit form. To prove the first two results, we need to generalize techniques developed in [Wu14, WW15] carefully and rigorously to the case of infinite dimensional Hilbert spaces.

Proof of the Theorem  1.5 is different. We prove a key estimate for the operator on an -dimensional subspace (Proposition  5.4), then bound the spectra of the curvature operator by the corresponding spectra of its projection onto this subspace to derive a contradiction.

1.4. Plan of the paper

The organization of the paper is as follows: in §2, we set up notations and preliminaries, in particular, we restrict ourselves in the classical setting to define Teichmüller space of closed surfaces and the Weil-Petersson metric in §2.1, its curvature operator on Teichmüller space is set up in §2.2, then we define the universal Teichmüller space and its Hilbert structure in §2.3, and introduce the basis for tangent vectors for the , and describe the Weil-Petersson Riemannian curvature operator on the universal Teichmüller space in §2.4. Main theorems are proved in sections §3, §4 and §5. And in the last section §6 we prove Theorem  1.6.

1.5. Acknowledgment

We acknowledge supports from U.S. national science foundation grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric Structures and Representation varieties” (the GEAR Network). This work was supported by a grant from the Simons Foundation (#359635, Zheng Huang) and a research award from the PSC-CUNY. Part of the work is completed when the second named author was a G. C. Evans Instructor at Rice University, he would like to thanks to the mathematics department for their support. He would also like to acknowledge a start-up research fund from Tsinghua University to finish this work. The authors would like to thank an anonymous referee whose comments are very helpful to impove the paper.

2. Preliminaries

2.1. Teichmüller space and its Weil-Petersson metric

Let be the unit disk with the Poincaré metric, and be a closed oriented surface of genus . Then by the uniformization theorem we have a hyperbolic structure on , where is a Fuchsian group, and is the group of orientation preserving isometries of . Writing as the complex coordinate on , the Poincaré metric is explicitly given as

It descends to a hyperbolic metric on the Riemann surface , which we denote by . Spaces of Beltrami differentials and holomorphic quadratic differentials on Riemann surfaces play a fundamental role in Teichmüller theory, and let us describe these spaces.

  1. : the space of bounded Beltrami differentials on . A Beltrami differential on a Riemann surface is a form in the form of , where is a function on satisfying:

  2. : the unit ball of , namely,

  3. : the space of holomorphic quadratic differentials on . A holomorphic quadratic differential is a form taking the form , where is a holomorphic function on satisfying:

    It is a basic fact in Riemann surface theory that is a Banach space of real dimension .

  4. : the space of harmonic Beltrami differentials on . A Beltrami differential is harmonic if there is a holomorphic quadratic differential such that

    (2.1)

    where is the hyperbolic metric on . Seeing from , the space consists of functions

    (2.2)

The Teichmüller space is the space of hyperbolic metrics on the surface , modulo orientation preserving biholomorphisms. Real analytically is isomorphic to , where two Beltrami differentials are equivalent if the unique quasiconformal maps between the extended complex plane coincide on the unit circle. At each point , its tangent space is identified as the space , while the cotangent space at is identified as the space .

Given two tangent vectors and in , the Weil-Petersson metric is defined as the following (Petersson) pairing:

(2.3)

where is the hyperbolic area element on . Writing as a metric tensor, we have

This is a Riemannian metric with many nice properties. There is an explicit formula for its curvature tensor due to Tromba-Wolpert ([Tro86, Wol86]):

(2.4)

Here the operator is defined as

(2.5)

where is the Laplace operator on with respect to the hyperbolic metric . This operator is fundamental in Teichmüller theory, and the following is well-known (see for instance [Wol86]):

Proposition 2.1.

The operator is a positive, self-adjoint operator on . Furthermore, let be a Green’s function for , then is positive, and : , we have

(2.6)

To simplify our calculations, we introduce the following notation:

Definition 2.2.

For any element ’s in the tangent space , we set:

(2.7)

Using this notation, the Weil-Petersson curvature tensor formula on Teichmüller space becomes

(2.8)

2.2. The Weil-Petersson curvature operator on Teichmüller space

We now introduce the Riemannian curvature operator for the Weil-Petersson metric on Teichmüller space . Note that this is a matrix of the real order , whose diagonal entries are the sectional curvatures.

Let be a neighborhood of in Teichmüller space , and we have as a local holomorphic coordinate on , where . Then forms a real smooth coordinate in , and

Let be the real tangent bundle of and be the exterior wedge product of and itself. For any , we have

and

(2.9)
Definition 2.3.

The Weil-Petersson curvature operator on Teichmüller space is defined on by

where ’s are tangent vectors at , and is the curvature tensor.

If we take a real orthonormal basis for , and set , then

and the curvature operator , for real coefficients , can be expressed as follows:

(2.10)

In [Wu14] the second named author proved the curvature operator is non-positively definite on Teichmüller space. Further analysis on was studied in [WW15]. We will generalize this fundamental operator to the setting of the universal Teichmüller space and reveal some geometric features for the Weil-Petersson metric on the universal Teichmüller space.

2.3. The universal Teichmüller space and its Hilbert structure

Introduced by Bers ([Ber65]), the universal Teichmüller space is a central subject for the theory of univalent functions. It contains all Teichmüller spaces of closed surfaces which are complex submanifolds.

Recall that every Riemann surface (or hyperbolic structure) on a closed surface is quotient of the Poincaré disk with a Fuchsian group : . Previously in §2.1, we have Teichmüller space isomorphic to a quotient space , where is the space of bounded Beltrami differentials on with super-norm less than one, and two such Beltrami differentials are equivalent if the unique quasiconformal maps induced by them between the extended complex plane coincide on the unit circle.

Let us set up some notations before we proceed. Letting be the identity group, we work in the Poincaré disk , we have similarly with §2.1:

  1. : the space of bounded functions on .

  2. : the unit ball of , namely,

  3. We will need two spaces of holomorphic functions on , both are analog to the space , the space of holomorphic quadratic differentials on . Let us define

    (2.11)

    where is the hyperbolic metric on . This is the space of holomorphic functions on with finite super-norm defined within (2.11).

    We also define

    (2.12)

    This is the space of holomorphic functions on with finite -norm defined within (2.12).

  4. For the notion of generalized “harmonic Beltrami differentials” on , we also have two spaces to introduce:

    (2.13)

    and

    (2.14)
Definition 2.4.

The universal Teichmüller space , where if and only if on the unit circle, and is the unique quasiconformal map between extended complex planes which fixes the points , and solves the Beltrami equation .

At any point in the universal Teichmüller space , the cotangent space is naturally identified with the Banach space defined in (2.11), and the tangent space is identified with the space defined in (2.13). It is then clear the Petersson pairing of functions in the space is not well-defined. However, for any , we write the Petersson pairing as the following inner product:

(2.15)

Then this defines a Hilbert structure on the universal Teichmüller space , introduced in ([TT06]), namely, endowed with this inner product, becomes an infinite dimensional complex manifold and Hilbert space. We denote this Hilbert manifold , which consists of all the points of the universal Teichmüller space , with tangent space identified as , a sub-Hilbert space of the Banach space . We call the resulting metric from (2.15) the Weil-Petersson metric on . The space we are dealing with is still very complicated: in the corresponding topology induced from the inner product above, the Hilbert manifold is a disjoint union of uncountably many components ([TT06]).

One of the most important tools for us is the Green’s function for the operator on the disk. We abuse our notation to denote the operator and its Green’s function, where is the Laplace operator on the Poincaré disk . Let us organize some properties we will use later into the following proposition.

Proposition 2.5.

[Hej76] The Green’s function satisfies the following properties:

  1. Positivity: for all ;

  2. Symmetry: for all ;

  3. Unit hyperbolic area: for all ;

  4. We denote the space of bounded smooth functions on , then for ,

    (2.16)

    Moreover, .

2.4. The curvature operator on the universal Teichmüller space

We have defined the Hilbert manifold and its Riemannian metric (2.15) for its tangent space which we will work with for the rest of the paper, let us now generalize the concept of the curvature operator (Definition  2.3) for Teichmüller space to . This has been done in more abstract settings, see for instance [pages 238-239, [Lan99]] or [Duc13].

We work in the Poincaré disk . On one hand, without Fuchsian group action, we are forced to deal with an infinite dimensional space of certain functions, on the other hand, the hyperbolic metric is explicit. This leads to some explicit calculations that one can take advantage of. First we note that the tangent space has an explicit orthonormal basis: we set, ,

(2.17)
Lemma 2.6.

[TT06, Teo09] The set forms an orthonormal basis with respect to the Weil-Petersson metric on .

Moreover, Takhtajan-Teo established the curvature tensor formula for the Weil-Petersson metric on , which takes the same form as Tromba-Wolpert’s formula for Teichmüller space of closed surfaces:

Theorem 2.7.

[TT06] For ’s in , the Riemannian curvature tensor for the Weil-Petersson metric (2.15) is given by:

(2.18)

Here we abuse our notation to use , where is the Laplace operator on the Poincaré disk .

Let be a neighborhood of and be a local holomorphic coordinate system on such that is orthonormal at , where ’s are explicitly defined in (2.17), we write , then is a real smooth coordinate system in , and we have:

Let be the real tangent bundle of and be the exterior wedge product of and itself. For any , we have

and

Following Lang [Chapter 9, [Lan99]], we define:

Definition 2.8.

The Weil-Petersson curvature operator is given as

and extended linearly, where are real tangent vectors, and is the curvature tensor for the Weil-Petersson metric.

It is easy to see that is a bilinear symmetric form.

3. Non-Positive Definiteness and Zero Level Set

In this section, we prove the first part of Theorem  1.1:

Theorem 3.1.

The operator is non-positive definite.

The strategy of our proof is the most direct approach, namely, lengthy but careful calculations using explicit nature of both the hyperbolic metric on , and the orthonormal basis on given by (2.17). We will verify the theorem by calculating with various combinations of bases elements, then extend bilinearly. We follow closely the argument in the proof of Theorem 1.1 in [Wu14], which was inspired by calculations in [LSY08].

3.1. Preparation

Note that the version of the operator on a closed surface is positive and self-adjoint on , and it plays a fundamental role in Teichmüller theory, but in the case of , the operator is noncompact, therefore we have to justify several properties carefully.

Proposition 3.2.

We have the following:

  1. The operator is self-adjoint on ;

  2. For any , we have also ;

  3. The operator is positive on .

Proof.

(i). For all , we have

(ii). From Proposition  2.5, we know . Using the positivity of the Green’s function , and , we estimate with the Cauchy-Schwarz inequality:

The last step we used self-adjointness of and the fact that . This proves .

(iii) Given any real function , let us denote and by (ii) above, it also lies in , then , and

Here we used that is negative definite on ([Hej76]).

The case when is complex valued can be proved similarly after working on real and imaginary parts separately. ∎

Recall that forms an orthonormal basis for , where ’s are given explicitly in (2.17). Using the coordinate system described in §2.4, we have

Naturally we will work with these three combinations. Let us define a few terms to simplify our calculations:

  1. Consider , where are real. We denote

    (3.1)
  2. The Green’s function of the operator : .

  3. Consider , where are real. We denote

    (3.2)

There are three types of basis elements in , however, in terms of the curvature operator, we only have to work with the first two types because of the next lemma:

Lemma 3.3.

We have the following:


Proof.

The Weil-Petersson metric on the Hilbert manifold is Kähler-Einstein ([TT06]), therefore its associated complex structure is an isometry on the tangent space such that and . Now it is easy to verify:

The other equality is proved similarly. ∎

3.2. Proof of Theorem  3.1

The curvature tensor in equation (2.18) has two terms. Set

Thus, the curvature tensor satisfies

(3.3)
Proof of Theorem  3.1.

We write

(3.4)

By Lemma  3.3, we only have to show . We pause to give an expression for . Since , we have

Now we work with these terms.

Lemma 3.4.

Using above notations, we have

Proof of Lemma 3.4.

First we recall the Weil-Petersson curvature tensor formula (2.18), notation in (2.7), and , then we take advantage of the Green’s function for and the fact that is self-adjoint on to calculate as follows:

For the second term in the equation above,