Abstract

We characterize hyperbolic metrics on compact triangulated surfaces with boundary using a variational principle. As a consequence, a new parameterization of the Teichmüller space of compact surface with boundary is produced. In the new parameterization, the Teichmüller space becomes an open convex polytope. It is conjectured that the Weil-Petersson symplectic form can be expressed explicitly in terms of the new coordinate. 1.1. The purpose of this paper is to produce a new parameterization of the Teichmüller space of compact surface with non-empty boundary so that the lengths of the boundary components are fixed. In this new parameterization, the Teichmüller space becomes an explicit open convex polytope. Our result can be considered as the counter-part of

Citations

316	Intersection theory on the moduli space of curves and the matrix Airy function - Kontsevich - 1992
143	Geometry and Spectra of Compact Riemann Surfaces (Birkhäuser - Buser - 1992
100	The geometry and topology of 3–manifolds, lecture notes - Thurston - 1980
98	The decorated Teichmüller space of punctured surfaces - Penner - 1987
96	The second homology of the mapping class group of an orientable surface, Inventiones 72 - Harer - 1983
87	Convex Analysis and Nonlinear Optimization - Borwein, Lewis - 2006
57	Euclidean structures on simplicial surfaces and hyperbolic volume - Rivin - 1994
42	Variational principles for circle patterns and Koebe’s theorem - Bobenko, Springborn
32	Weil-Petersson volumes and intersection theory on the moduli spaces of curves - Mirzakhani - 2005
28	Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic - Bonahon - 1996
17	Characterizing the Delaunay decompositions of compact hyperbolic surfaces. math.GT/0103174 - Leibon - 2001
16	Decorated Teichmuller Theory of Bordered Surfaces - Penner
15	Kreispackungen und Triangulierungen - Brägger - 1992
12	Tiling the projective foliation space of a punctured surface - Mosher - 1988
9	Rigidity of Polyhedral Surfaces - Luo
7	Un principe variationnel pour les empilements de cercles - Verdiere, Yves - 1991
6	Volume and angle structures on 3-manifolds - Luo
5	Masahiko: An introduction to Teichmüller spaces. Translated and revised from the Japanese by the authors - Imayoshi, Taniguchi - 1992
4	Hyperideal polyhedra in hyperbolic manifolds - Schlenker - 2002
4	A variational principle for weighted Delaunay triangulations and hyperideal polyhedra, preprint - Springborn - 2006
2	private communication - Verdiere, Yves
2	A canonical cellular decomposition of the Teichmüller space of compact surfaces with boundary - Ushijima - 1999
1	A Characterization of spherical polyhedral surfaces - Luo
1	A Characterization of - Luo
1	A canonical cellular decomposition of the Teichmller space of compact surfaces with boundary - Ushijima - 1999