Calculus of Variations and Geometric Measure Theory

R. Lopez-Soriano - A. Malchiodi - D. Ruiz

Conformal metrics with prescribed gaussian and geodesic curvatures

created by malchiodi on 29 Jun 2018



Inserted: 29 jun 2018
Last Updated: 29 jun 2018

Year: 2018


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.