Geometric Approach to the Weil-Petersson Symplectic Form

TL;DR

This paper presents a geometric method to compare the Fenchel-Nielsen and Weil-Petersson symplectic forms on Teichmüller spaces, extending to moduli spaces where Kleinian group techniques are inapplicable.

Contribution

It introduces a geometric approach to relate symplectic forms on Teichmüller spaces, providing new insights beyond traditional methods.

Findings

Coincidence of Fenchel-Nielsen and Weil-Petersson forms established geometrically.

Method applicable to moduli spaces of weighted punctured Riemann surfaces.

First variation of geodesic lengths computed in families of complex manifolds.

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.