Projective structures on Riemann surfaces and Kleinian groups

Abstract

Let S be a compact Riemann surface of genus 2, and let iv: U--->S be a holomorphic universal covering of S with the covering transformation group F , where U is the upper half plane {z e C : 1m z>0}. T h en , is a finitely generated Fuchsian group of the first kind on U and consists of hyperbolic Mobius transformations. We denote by B,(L, r) the Banach space of all holomorphic quadratic differentials for F defined on the lower half plane L . Namely, B A , F ) is the set of all holomorphic functions 0 on L satisfying