On Teichmuller Space of Surface with Boundary

TL;DR

This paper introduces a new parametrization of the Teichmuller space for surfaces with boundary, representing it as an open convex polytope, and explores its geometric and symplectic properties.

Contribution

It provides a variational characterization of hyperbolic metrics and a novel convex polytope parametrization of the Teichmuller space for surfaces with boundary.

Findings

Teichmuller space is an open convex polytope under the new parametrization.

Hyperbolic metrics characterized via a variational principle.

Conjecture on explicit expression of Weil-Petersson form in new coordinates.

Abstract

We characterization hyperbolic metrics on compact surfaces with boundary using a variational principle. As a consequence, a new parametrization of the Teichmuller space of compact surface with boundary is produced. In the new parametrization, the Teichmuller 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.