Metrics of constant curvature on a Riemann surface with two corners on the boundary

Details Metrics of constant curvature on a Riemann surface with two corners on the boundary Metrics of constant curvature on a Riemann surface with two corners on the boundary

2007-12-19

arxiv.org/abs/0712.3162v1

Mathematics

Differential Geometry

to appear in Annales de l'IHP-ANL

Scientific paper

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

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