On Teichmüller spaces of surfaces with boundary

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 a compact surface with boundary is produced. In the new parameterization, the Teichmüller space becomes an explicit open convex polytope. Our results can be considered as a generalization of the simplicial coordinate of Penner [P1], [P2] for hyperbolic metrics with cusp ends to the case of surfaces with geodesic boundary. It is conjectured that the Weil-Petersson symplectic form can be expressed explicitly in terms of the new coordinate