Prescribing curvatures on surfaces with conical singularities and corners

TL;DR

This paper introduces a novel variational approach to prescribe Gaussian and geodesic curvatures on surfaces with boundary, conical singularities, and corners, marking the first such result in this singular setting.

Contribution

It extends a recent variational formulation to surfaces with singularities, providing the first solutions for prescribed curvatures in such complex geometries.

Findings

Established existence of solutions for prescribed curvatures on singular surfaces

Extended variational methods to handle conical singularities and corners

First known results in this specific geometric setting

Abstract

This paper is concerned with the problem of prescribing Gaussian curvature and geodesic curvature in a compact surface with boundary with conical singularities and corners. Solutions are obtained using a new variational formulation, recently introduced for the regular counterpart of the problem and extended here to the singular case. As far as we know, this is the first result for the problem of prescribed curvatures in surfaces with singularities.