Triangulated Riemann surfaces with boundary and the Weil-Petersson Poisson structure

TL;DR

This paper introduces a coordinate system for Teichmüller space of bordered Riemann surfaces using boundary geodesic arcs, computes the Weil-Petersson Poisson structure in these coordinates, and shows its convergence to a piecewise-linear structure, also deriving a variation formula for geodesic distances.

Contribution

It provides a new coordinate system for Teichmüller space and explicitly computes the Weil-Petersson Poisson structure in this setting, connecting it to Kontsevich's structure.

Findings

Computed Weil-Petersson Poisson structure in boundary arc coordinates

Proved the structure converges to Kontsevich's piecewise-linear Poisson structure

Derived a formula for geodesic distance variation under Fenchel-Nielsen deformation

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 the boundary of S perpendicularly are coordinates on the Teichmueller 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 geodesics under Fenchel-Nielsen deformation.