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 Teichmüller curve, whose pull-back to Diff +(S ¹)/S ¹ 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 Teichmüller curve is the symplectic form that defines the Weil-Petersson metric on the universal Teichmüller space. Restricted to a finite dimensional Teichmüller space, the vertical integration of the corresponding form on the Teichmüller curve is also the symplectic form that defines the Weil-Petersson metric on the Teichmüller space.     

Liouville Action and Weil-Petersson Metric on Deformation Spaces, Global Kleinian Reciprocity and Holography

We rigorously define the Liouville action functional for the finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that classical action – the critical value of the Liouville action functional, considered as a function on the quasi-Fuchsian deformation space, is an antiderivative of a 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections. This result can be considered as global quasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity. We prove that the classical action is a Kähler potential of the Weil-Petersson metric. We also prove that the Liouville action functional satisfies holography principle, i.e., it is a regularized limit of the hyperbolic volume of a 3-manifold associated with a quasi-Fuchsian group. We generalize these results to a large class of Kleinian groups including finitely generated, purely loxodromic Schottky and quasi-Fuchsian groups, and their free combinations.     

Analytic Functions and Integrable Hierarchies–Characterization of Tau Functions

We prove the dispersionless Hirota equations for the dispersionless Toda, dispersionless coupled modified KP and dispersionless KP hierarchies using an idea from classical complex analysis. We also prove that the Hirota equations characterize the tau functions for each of these hierarchies. As a result, we establish the links between the hierarchies.     

Holomorphic Factorization of Determinants of Laplacians using Quasi-Fuchsian Uniformization

For a quasi-Fuchsian group Γ with ordinary set Ω, and Δ _n the Laplacian on n-differentials on Γ\Ω, we define a notion of a Bers dual basis for ker Δ _n . We prove that det , is, up to an anomaly computed by Takhtajan and the second author in (Commun. Math Phys 239(1-2):183–240, 2003), the modulus squared of a holomorphic function F(n), where F(n) is a quasi-Fuchsian analogue of the Selberg zeta function Z(n). This generalizes the D’Hoker–Phong formula det , and is a quasi-Fuchsian counterpart of the result for Schottky groups proved by Takhtajan and the first author in Analysis 16, 1291–1323, 2006.      

Dolbeault cohomology of G /( P,P )

Let G be a complex connected semi-simple Lie group, with parabolic subgroup P. Let (P,P) be its commutator subgroup. The generalized Borel-Weil theorem on flag manifolds has an analogous result on the Dolbeault cohomology . Consequently, the dimension of is either 0 or . In this paper, we show that the Dolbeault operator has closed image, and apply the Peter-Weyl theorem to show how q determines the value 0 or . For the case when P is maximal, we apply our result to compute the Dolbeault cohomology of certain examples, such as the punctured determinant bundle over the Grassmannian. Received: September 2, 1997; in final form February 9, 1998     

Bers isomorphism on the universal Teichmüller curve

We study the Bers isomorphism between the Teichmüller space of the parabolic cyclic group and the universal Teichmüller curve. We prove that this is a group isomorphism and its derivative map gives a remarkable relation between Fourier coefficients of cusp forms and Fourier coefficients of vector fields on the unit circle. We generalize the Takhtajan–Zograf metric to the Teichmüller space of the parabolic cyclic group, and prove that up to a constant, it coincides with the pull back of the Velling–Kirillov metric defined on the universal Teichmüller curve via the Bers isomorphism.      

Conformal Mappings and Dispersionless Toda Hierarchy II: General String Equations

Random current representation (RCR) for transverse field Ising models (TFIM) has been introduced in [14]. This representation is a space-time version of the classical RCR exploited by Aizenman et. al. [1,3,4]. In this paper we formulate and prove corresponding space-time versions of the classical switching lemma and show how they generate various correlation inequalities. In particular we prove exponential decay of truncated two-point functions at positive magnetic fields in the z-direction and address the issue of the sharpness of the phase transition.     

Ruelle zeta function for cofinite hyperbolic Riemann surfaces with ramification points

Letters in Mathematical Physics - We consider the Ruelle zeta function R(s) of a genus g hyperbolic Riemann surface with n punctures and v ramification points. R(s) is equal to $$Z(s)/Z(s+1)$$,...     

A Different Expression of the Weil–Petersson Potential on the Quasi-Fuchsian Deformation Space

We extend a definition of the Weil–Petersson potential on the universal Teichmüller space to the quasi-Fuchsian deformation space. We prove that up to a constant, this function coincides with the Weil–Petersson potential on the quasi-Fuchsian deformation space. As a result, we prove a lower bound for the potential on the quasi-Fuchsian deformation space 32G15, 30F60, 30F10