Hyperbolic Metrics on Surfaces with Boundary

Abstract

We discuss an alternative approach to the uniformisation problem on surfaces with boundary by representing conformal structures on surfaces M of general type by hyperbolic metrics with boundary curves of constant positive geodesic curvature. In contrast to existing approaches to this problem, the boundary curves of our surfaces (M, g) cannot collapse as the conformal structure degenerates which is important in applications in which (M, g) serves as domain of a PDE with boundary conditions.