Browsing by Author "Teo, LP"

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Analytic functions and integrable hierarchies characterization of tau functions
(2003-04-01)
Teo, LP
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應用數學系
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Department of Applied Mathematics
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.
Dolbeault cohomology of G/(P,P)
(1999-03-01)
Chuah, MK
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Teo, LP
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應用數學系
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Department of Applied Mathematics
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 H-0,H-q(G/(P, P)). Consequently, the dimension of H-0,H-q(G/(P, P)) is either 0 or infinity. In this paper, we show that the Dolbeault operator <(partial derivative)over bar> has closed image, and apply the Peter-Weyl theorem to show how q determines the value 0 or infinity. 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.
Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography
(2003-08-01)
Takhtajan, LA
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Teo, LP
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應用數學系
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Department of Applied Mathematics
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 Kahler 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.
The Velling-Kirillov metric on the universal Teichmuller curve
(2004)
Teo, LP
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應用數學系
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Department of Applied Mathematics
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 +(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 Veiling-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.