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

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

2006-10-23

arxiv.org/abs/math/0610698v4

J. Differential Geometry 81 (2009), pp. 391-436

Mathematics

Differential Geometry

44 pages, 12 figures (LaTeX 2e). Published version

Scientific paper

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.

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