Hyperbolic metrics on surfaces with boundary

TL;DR

This paper presents a new hyperbolic metric approach to the uniformisation problem on surfaces with boundary, ensuring boundary curves do not collapse during degeneration, which benefits PDE boundary condition applications.

Contribution

It introduces an alternative hyperbolic metric method for surfaces with boundary that maintains boundary curve integrity during conformal structure degeneration.

Findings

Boundary curves of the hyperbolic surfaces do not collapse during degeneration.

The approach preserves boundary conditions in PDE applications.

Provides a new perspective on uniformisation with boundary constraints.

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.