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Triangulated riemann surfaces with boundary and the weil-petersson poisson structure

Gabriele Mondello

Differential Geometry mathscidoc:1609.10103

Journal of Differential Geometry, 81, (2), 391-436, 2009
Given a hyperbolic surface with geodesic 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 Teichm¨uller space T (S). We express the Weil-Petersson Poisson structure of 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. At the same time, we obtain a formula for the first-order variation of the distance between two closed geodesics under Fenchel-Nielsen deformation.
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@inproceedings{gabriele2009triangulated,
  title={TRIANGULATED RIEMANN SURFACES WITH BOUNDARY AND THE WEIL-PETERSSON POISSON STRUCTURE},
  author={Gabriele Mondello},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911175844280192760},
  booktitle={Journal of Differential Geometry},
  volume={81},
  number={2},
  pages={391-436},
  year={2009},
}
Gabriele Mondello. TRIANGULATED RIEMANN SURFACES WITH BOUNDARY AND THE WEIL-PETERSSON POISSON STRUCTURE. 2009. Vol. 81. In Journal of Differential Geometry. pp.391-436. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911175844280192760.
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