On the Symplectic Geometry of Deformations of a Hyperbolic Surface

Abstract

Let R be a Riemann surface. In this manuscript we consider a geometry on the moduli space X(R) for R, which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of R. Classically the space of flat metrics for a torus is the locally symmetric space 0(2) SL(2; R)/SL(2; Z). We shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative Euler characteristic. The Teichmiiller space T(R), a covering of the moduli space, is a complex Kihler manifold. A KUhler metric for T(R), defined in terms of the Petersson product for automorphic forms, was introduced by Weil, [1]. The Weil-Petersson metric is invariant under the covering transformations and so projects to the moduli space X(R). The metric provides a link between the function theory of R and the geometry of X(R). In the Fenchel-Nielsen manuscript [8] a deformation, based on an amalgamation construction for Fuchsian groups, is introduced. The deformation is defined geometrically by cutting the surface along a simple closed geodesic, rotating one side of the cut relative to the other, and attaching the sides in their new position. The hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface. Choose a free homotopy class [a] on the surface R; then for each marked surface R realize [a] by the closed geodesic aR. The Fenchel-Nielsen deformations for the athen define a 1-parameter group of diffeomorphisms of T(R), whose infinitesimal generator by definition is the Fenchel-Nielsen vector field ta. In [21] the Fenchel-Nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t * was begun. The Fenchel-Nielsen vector fields were found to be related to the geodesic length functions 1*, introduced by Fricke-Klein to provide coordinates for T(R).