Weil-Petersson Metric Research Articles

Eta invariant and Selberg zeta function of odd type over convex co-compact hyperbolic manifolds

We show meromorphic extension and give a complete description of the divisors of a Selberg zeta function of odd type Z Γ , Σ o ( λ ) associated to the spinor bundle Σ on an odd dimensional convex co-compact hyperbolic manifold Γ \\ H 2 n + 1 . As a byproduct we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We show that there is a natural eta invariant η ( D ) associated to the Dirac operator D over a convex co-compact hyperbolic manifold Γ \\ H 2 n + 1 and that exp ( π i η ( D ) ) = Z Γ , Σ o ( 0 ) , thus extending Millson's formula to this setting. Under some assumption on the exponent of convergence of Poincaré series for the group Γ, we also define an eta invariant for the odd signature operator, and we show that for Schottky 3-dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the Zograf factorization formula relating two natural Kähler potentials for Weil–Petersson metric on Schottky space.

Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space

We consider a natural non-negative two-form G on quasifuchsian space that extends the Weil-Petersson metric on Teichmuller space. We describe completely the positive definite locus of G, showing that it is a positive definite metric off the fuchsian diagonal of quasifuchsian space and is only zero on the "pure-bending'' tangent vectors to the fuchsian diagonal . We show that G is equal to the pullback of the pressure metric from dynamics. We use the properties of G to prove that at any critical point of the Hausdorff dimension function on quasifuchsian space the Hessian of the Hausdorff dimension function must be positive definite on at least a half-dimensional subspace of the tangent space. In particular this implies that Hausdorff dimension has no local maxima on quasifuchsian space.

Quasi-Fuchsian 3-Manifolds and Metrics on Teichmüller Space

An almost Fuchsian 3-manifold is a quasi-Fuchsian manifold which contains an incompressible closed minimal surface with principal curvatures in the range of (−1, 1). By the work of Uhlenbeck, such a 3-manifold M admits a foliation of parallel surfaces, whose locus in Teichmuller space is represented as a path γ, we show that γ joins the conformal structures of the two components of the conformal boundary of M . Moreover, we obtain an upper bound for the Teichmuller distance between any two points on γ, in particular, the Teichmuller distance between the two components of the conformal boundary of M , in terms of the principal curvatures of the minimal surface in M . We also establish a new potential for the Weil-Petersson metric on Teichmuller space.

Weil-Petersson geometry of Teichmüller–Coxeter complex and its finite rank property

On a Teichmüller space, the Weil-Petersson metric is known to be incomplete. Taking metric and geodesic completions result in two distinct spaces, where the Hopf-Rinow theorem is no longer relevant due to the singular behavior of the Weil-Petersson metric. We construct a geodesic completion of the Teichmüller space through the formalism of Coxeter complex with the Teichmüller space as its non-linear non-homogeneous fundamental domain. We then show that the metric and geodesic completions both satisfy a finite rank property, demonstrating a similarity with the non-compact symmetric spaces of semi-simple Lie groups.

Ergodicity of the Geodesic Flow on Non-complete Negatively Curved Surfaces

We study the dynamics of the geodesic flow for a class of non-complete Riemannian metrics on a negatively curved surface M . These finite area surfaces are composed of finitely many “singular” surfaces of revolutions of the form y = x, r > 1 for 0 ≤ x ≤ 1 (thin pieces), together with connecting surfaces of bounded negative curvature (thick pieces). The curvature at every point is negative and bounded away from zero, but is unbounded in the “cusps.” The geodesic flow is non-complete because geodesics corresponding to singular unit tangent vectors pointing into the origin hit the cusps in finite time and then cease to be defined. Such singular unit tangent vectors are dense in the unit tangent bundle. Our main result is ergodicity of the geodesic flow (Theorem 5.1). It immediately follows that these metrics have dense geodesics in the unit tangent bundle SM . We also prove that the closed geodesics are dense in the unit tangent bundle (Theorem 6.1) The motivation for considering these metrics arises from studying the geodesic flow for the Weil-Petersson (WP) metric on two dimensional moduli spaces of Riemann surfaces., e.g., the moduli space for the once punctured torus. In several fundamental ways the geodesic flow we consider provides a good model for the WP geodesic flow. For example, Wolpert showed that the WP metric is also non-complete, has finite area, has negative curvature bounded away from zero, and is unbounded in the ”cusps.” Also, various authors have shown that the cusps can be approximated by singular surface of revolutions. At present, there is only a single technical obstruction (additional estimates on the derivatives of the GF in the cusp) that prevent us from applying this general method to prove ergodicity of the WP geodesic flow. See Section 7. The geometric properties of both types of flows imply that the geodesic flow is a uniformly hyperbolic dynamical system with singularities. The non-completeness causes pathologies in the stable and unstable manifolds, as for many billiard flows. For example, stable and unstable manifolds at a point, if they exist, may intersect a cusp point, and thus have only finite length. One needs extensions of “Pesin theory” to systems with singularities to study these geodesic flows. Another challenging aspect is that these surfaces may be simply connected, making the use of any of the traditional arguments involving boundaries of the covering spaces inapplicable. Studying the dynamics requires the development of new approches. Our strategy to prove ergodicity of the geodesic flow is to first establish non-uniform hyperbolicity, i.e., the Lyapunov exponents are non-zero at almost every point. We then prove that there

Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics and relative hyperbolicity

We analyze the coarse geometry of the Weil-Petersson metric on Teichm\"uller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, the "pants graph." We show that in dimension~3 the pants graph is strongly relatively hyperbolic with respect to naturally defined product regions and show any quasi-flat lies a bounded distance from a single product. For all higher dimensions there is no non-trivial collection of subsets with respect to which it strongly relatively hyperbolic; this extends a theorem of [BDM] in dimension 6 and higher into the intermediate range (it is hyperbolic if and only if the dimension is 1 or 2 [BF]). Stability and relative stability of quasi-geodesics in dimensions up through 3 provide for a strong understanding of the behavior of geodesics and a complete description of the CAT(0)-boundary of the Weil-Petersson metric via curve-hierarchies and their associated "boundary laminations."

The Weil–Petersson geometry of the moduli space of Riemann surfaces

In 2007, Z. Huang showed that in the thick part of the moduli space M g of compact Riemann surfaces of genus g, the sectional curvature of the Weil-Petersson metric is bounded below by a constant depending on the injectivity radius, but independent of the genus g. In this article, we prove this result by a different method. We also show that the same result holds for Ricci curvature. For the universal Teichmuller space equipped with a Hilbert structure induced by the Weil-Petersson metric, we prove that its sectional curvature is bounded below by a universal constant.

Behavior of geodesic-length functions on Teichmüller space

Let T be the Teichmuller space of marked genus g, n punctured Riemann surfaces with its bordification T the augmented Teichmuller space of marked Riemann surfaces with nodes, (Abi77, Ber74). Pro- vided with the WP metric T is a complete CAT(0) metric space, (DW03, Wol03, Yam04). An invariant of a marked hyperbolic struc- ture is the lengthof the geodesic in a free homotopy class. A basic feature of Teichmuller theory is the interplay of two-dimensional hyperbolic geometry, Weil-Petersson (WP) geometry and the behavior of geodesic-length functions. Our goal is to develop the understanding of the intrinsic local WP geometry through a study of the gradient and Hessian of geodesic-length functions. Considerations include ex- pansions for the WP pairing of gradients, expansions for the Hessian and covariant derivative, comparability models for the WP metric, as well as the behavior of WP geodesics including a description of the Alexandrov tangent cone at the augmentation. Approximations and applications for geodesics close to the augmentation are developed. An application for fixed points of group actions is described. Bound- ing configurations and functions on the hyperbolic plane is basic to our approach. Considerations include analyzing the orbit of a dis- crete group of isometries and bounding sums of the inverse square exponential-distance.

Dimension and rank for mapping class groups

We study the large scale geometry of the mapping class group, MCG(S). Our main result is that for any asymptotic cone of MCG(S), the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG(S). An application is a proof of Brock-Farb’s Rank Conjecture which asserts that MCG(S) has quasi-flats of dimension N if and only if it has a rank N free abelian subgroup. (Hamenstadt has also given a proof of this conjecture, using different methods.) We also compute the maximum dimension of quasi-flats in Teichmuller space with the Weil-Petersson metric.

New results on the geometry of the moduli space of Riemann surfaces

We briefly survey our recent results about the Mumford goodness of several canonical metrics on the moduli spaces of Riemann surfaces, including the Weil-Petersson metric, the Ricci metric, the Perturbed Ricci metric and the Kahler-Einstein metric. We prove the dual Nakano negativity of the Weil-Petersson metric. As applications of these results we deduce certain important results about the L 2-cohomology groups of the logarithmic tangent bundle over the compactified moduli spaces.

An extension of the Weil–Petersson metric to quasi-Fuchsian space

We define a natural semi-definite metric on quasi-fuchsian space, derived from geodesic current length functions and Hausdorff dimension, that extends the Weil–Petersson metric on Teichmuller space. We use this to describe a metric on Teichmuller space obtained by taking the second derivative of Hausdorff dimension and show that this metric is bounded below by the Weil–Petersson metric. We relate the change in Hausdorff dimension under bending along a measured lamination to the length in the Weil–Petersson metric of the associated earthquake vector of the lamination.

The Weil-Petersson geometry on the thick part of the moduli space of Riemann surfaces

In the thick part of the moduli space of Riemann surface, we show the sectional curvature of the Weil-Petersson metric is bounded independent of the genus of the surface.

Weil-Petersson volumes of the moduli spaces of CY manifolds

In this paper it is proved that the volumes of the moduli spaces of polarized CY manifolds with respect to the Weil-Petersson metrics are finite and they are rational numbers.

On Asymptotic Weil-Petersson Geometry of Teichmuller Space of Riemann Surfaces

We investigate the asymptotic behavior of curvatures of the Weil-Petersson metric in Teichmuller space. We use a pointwise curvature estimate to study directions, in the tangent space, of extremely negative curvature and directions of asymptotically zero curvatures.

Asymptotic geometry of the mapping class group and Teichmüller space

In this work, we study the asymptotic geometry of the mapping class group and Teichmueller space. We introduce tools for analyzing the geometry of `projection' maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several applications of this analysis. One of which is that the asymptotic cone of the mapping class group of any surface is tree-graded in the sense of Drutu and Sapir; this tree-grading has several consequences including answering a question of Drutu and Sapir concerning relatively hyperbolic groups. Another application is a generalization of the result of Brock and Farb that for low complexity surfaces Teichmueller space, with the Weil-Petersson metric, is delta-hyperbolic. Although for higher complexity surfaces these spaces are not delta-hyperbolic, we establish the presence of previously unknown negative curvature phenomena in the mapping class group and Teichmueller space for arbitrary surfaces.

Weil–Petersson isometries via the pants complex

We extend a theorem of Masur and Wolf which states that given a finite area hyperbolic surface S, every isometry of the Teichmuller space for S with the Weil-Petersson metric is induced by an element of the mapping class group for S. Our argument handles the previously untreated cases of the four times-punctured sphere, the once-punctured torus, and the twice-punctured torus.

Curvature and rank of Teichmüller space

Let S be a surface with genus g and n boundary components, and let d(S) = 3 g - 3 + n denote the number of curves in any pants decomposition of S . We employ metric properties of the graph of pants decompositions P ( S ) to prove that the Weil-Petersson metric on Teichmüller space Teich ( S ) is Gromov-hyperbolic if and only if d(S) ≤ 2. When d(S) ≥ 3, the Weil-Petersson metric has higher rank in the sense of Gromov (it admits a quasi-isometric embedding of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] k , k ≥ 2); when d(S) ≤ 2, we combine the hyperbolicity of the complex of curves and the relative hyperbolicity of P ( S ) to prove Gromov-hyperbolicity. We prove moreover that Teich ( S ) admits no geodesically complete, Mod ( S )-invariant, Gromov-hyperbolic metric of finite covolume when d(S) ≥ 3, and that no complete Riemannian metric of pinched negative curvature exists on Moduli space M(S) when d(S) ≥ 2.

Generalized Hodge metrics and BCOV torsion on Calabi-Yau moduli

We establish an unexpected relation among the Weil-Petersson metric, the generalized Hodge metrics and the BCOV torsion. Using this relation, we prove that certain kind of moduli spaces of polarized Calabi-Yau manifolds do not admit complete subvarieties. That is, there is no complete family for certain class of polarized Calabi-Yau manifolds. We also give an estimate of the complex Hessian of the BCOV torsion using the relation. After establishing a degenerate version of the Schwarz Lemma of Yau, we prove that the complex Hessian of the BCOV torsion is bounded by the Poincare metric.

Asymptotic Flatness of the Weil-Petersson Metric on Teichmüller Space

In this paper, we show that there is no negative upper bound for the sectional curvature of Teichmuller space of Riemann surfaces with the Weil–Petersson metric. Mathematics Subject Classifications (2000). 32G15, 53C21, 30F60.

The Velling-Kirillov metric on the universal Teichmüller curve

We extend Velling's approach and prove that the second variation of the spherical areas of a family of domains defines a Hermitian metric on the universal Teichmuller curve, whose pull-back to Diff +(S1)/S1 coincides with the Kirillov metric. We call this Hermitian metric the Velling-Kirillov metric. We show that the vertical integration of the square of the symplectic form of the Velling-Kirillov metric on the universal Teichmuller curve is the symplectic form that defines the Weil-Petersson metric on the universal Teichmuller space. Restricted to a finite dimensional Teichmuller space, the vertical integration of the corresponding form on the Teichmuller curve is also the symplectic form that defines the Weil-Petersson metric on the Teichmuller space.