Conformal metrics with prescribed Gaussian and geodesic curvatures

TL;DR

This paper addresses the problem of prescribing Gaussian and geodesic curvatures on compact surfaces with boundary through conformal metric deformations, introducing novel blow-up analysis techniques for diverging volume scenarios.

Contribution

It provides new existence results for prescribed curvatures using variational methods and introduces a pioneering blow-up analysis for solutions with diverging volume.

Findings

Established existence results via minimization and min-max methods.

Developed a blow-up analysis for solutions with diverging volume.

Utilized holomorphic domain-variations and Morse-index estimates.

Abstract

We consider the problem of prescribing the Gaussian and the geodesic curvatures of a compact surface with boundary by a conformal deformation of the metric. We derive some existence results using a variational approach, either by minimization of the Euler-Lagrange energy or via min-max methods. One of the main tools in our approach is a blow-up analysis of solutions, which in the present setting can have diverging volume. To our knowledge, this is the first time in which such an aspect is treated. Key ingredients in our arguments are: a blow-up analysis around a sequence of points different from local maxima; the use of holomorphic domain-variations; and Morse-index estimates.