The hyperbolic metric and the geometry of the universal curve

Introduction 0.1. A basic deformation of a compact Riemann surface is pinching a nontrivial loop. Understanding the limiting case, where the loop degenerates to a point, is actually tantamount to understanding the Deligne-Mumford stable curve compactification Jt g of the classical moduli space of Riemann surfaces. Degenerating families of Riemann surfaces are readily given by cut and paste constructions in the complex analytic category or by hyperbolic geometry, following Fenchel and Nielsen. A basic question is to relate the two approaches; to find the expansion for the hyperbolic metric in terms of the complex parametrization. The motivation is twofold, we would like to be able to analyze the degeneration of an invariant of hyperbolic metrics by simply writing out its expansion in complex coordinates, and we would like to use the hyperbolic metric in the study of the analytic geometry of Jt g .