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

TL;DR

This paper investigates conformal metrics with constant curvature and boundary conditions on surfaces with corners, establishing existence criteria and classifying solutions for specific geometric configurations.

Contribution

It provides a classification of conformal metrics with constant curvature on surfaces with boundary and corners, extending PDE methods to new geometric settings.

Findings

A disk with two corners admits such a metric only if the corners have equal angles.

Classifies solutions for the 2-sphere cut by two planes.

Establishes existence and uniqueness conditions for these metrics.

Abstract

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.