Differentials and Metrics on Riemann Surfaces

Abstract

Introduction. In this paper we show how to finite-dimensional spaces of differentials on a Riemann surface can be associated a finite collection of Riemannian metrics whose curvatures are closely related to the zeros of the differentials on the surface. In particular, we show that the Weierstrass points of a compact Riemann surface can be characterized as points of zero curvature of certain naturally defined metrics on the surface. The main theorem and application are in Part II; in Part I we have collected all the preliminary details. In the conclusion we point out the origin of this study and a possible direction for further research.