Geometric Approach to the Weil-Petersson Symplectic Form
Differential Geometry
2008-08-28 v2 Complex Variables
Abstract
In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic forms on the Teichmueller spaces of compact Riemann surfaces in a purely geometric way. The method can also be applied to situations like moduli spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available.
Keywords
Cite
@article{arxiv.0808.3741,
title = {Geometric Approach to the Weil-Petersson Symplectic Form},
author = {Reynir Axelsson and Georg Schumacher},
journal= {arXiv preprint arXiv:0808.3741},
year = {2008}
}