Abstract. Given a Riemann surface with boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at ∂S perpendicularly are coordinates on the Teichmüller space T (S). We compute the Weil-Petersson Poisson structure on T (S) in this system of coordinates and we prove that it limits pointwise to the piecewise-linear Poisson structure defined by Kontsevich on the arc complex of S. As a byproduct of the proof, we obtain a formula for the first-order variation of the distance between two closed geodesic under Fenchel-Nielsen deformation.

Abstract not found

Abstract. In this paper we give two different proofs of Bobenko and Springborn’s theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle pattern with proscribed intersection angles and cone angles on a cellular decomposed surface up to isometry (or similarity). 1.

Abstract: We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law and the Lengendre transformation of them. These include energies used by Colin de Verdiere, Braegger, Rivin, Cohen-Kenyon-Propp, Leibon and Bobenko-Springborn for variational principles on triangulated surfaces. Our study is based on a set of identities satisfied by the derivative of the cosine law. These identities which exhibit similarity in all spaces of constant curvature are probably a discrete analogous of the Bianchi