Prescribing Gaussian and geodesic curvature on disks

TL;DR

This paper studies how to prescribe Gaussian and geodesic curvature on a disk through conformal metric changes, introducing a new variational approach to establish existence of solutions under symmetry conditions.

Contribution

It develops a novel variational framework for prescribing curvatures on disks, addressing a Liouville-type equation with nonlinear boundary conditions.

Findings

Existence of solutions under symmetry assumptions

Introduction of a new variational method for the problem

Identification of minimizers in the prescribed curvature problem

Abstract

In this paper we consider the problem of prescribing the Gaussian and geodesic curvature on a disk and its boundary, respectively, via a conformal change of the metric. This leads us to a Liouville-type equation with a nonlinear Neumann boundary condition. We address the question of existence by setting the problem in a variational framework which seems to be completely new in the literature. We are able to find minimizers under symmetry assumptions.