Prescribing Gaussian curvature on surfaces with conical singularities and geodesic boundary

TL;DR

This paper establishes a new existence result for conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary, including cases with both positive and negative cone angles.

Contribution

It provides the first known existence theorem in this setting, allowing for conical singularities with both positive and negative orders and multiple boundary components.

Findings

Existence of conformal metrics with prescribed Gaussian curvature on complex surfaces.

Inclusion of conical singularities with both positive and negative cone angles.

Application of variational methods to prove the main result.

Abstract

We study conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive a general existence result for surfaces with at least two boundary components. This seems to be the first result in this setting. Moreover, we allow to have conical singularities with both positive and negative orders, that is cone angles both less and grater than $2\pi$.