# Metrics of constant curvature on a Riemann surface with two corners on the boundary - Mathematics > Differential Geometry

Metrics of constant curvature on a Riemann surface with two corners on the boundary - Mathematics > Differential Geometry - Download this document for free, or read online. Document in PDF available to download.

Abstract: We use PDE methods as developed for the Liouville equation to study theexistence of conformal metrics with prescribed singularities on surfaces withboundary, the boundary condition being constant geodesic curvature. Our firstresult shows that a disk with two corners admits a conformal metric withconstant Gauss curvature and constant geodesic curvature on its boundary if andonly if the two corners have the same angle. In fact, we can classify all thesolutions in a more general situation, that of the 2-sphere cut by two planes.

Author: ** Juergen Jost, Guofang Wang, Chunqin Zhou**

Source: https://arxiv.org/