A mean field problem approach for the double curvature prescription problem

TL;DR

This paper introduces a mean field approach to solve the problem of prescribing Gaussian and geodesic curvatures on compact surfaces with boundary, using variational methods for different Euler characteristic cases.

Contribution

It develops a new mean field-type formulation for the curvature prescription problem on surfaces with boundary, providing existence results via variational techniques.

Findings

Established existence results for positive, zero, and negative Euler characteristics.

Formulated the problem as a Liouville-type PDE with nonlinear Neumann boundary conditions.

Applied variational methods to prove solutions under various geometric conditions.

Abstract

In this paper we establish a new mean field-type formulation to study the problem of prescribing Gaussian and geodesic curvatures on compact surfaces with boundary, which is equivalent to the following Liouville-type PDE with nonlinear Neumann conditions: $$\left\{\begin{array}{ll} -\Delta u+2K_g=2Ke^u&\text{in }\Sigma\\ \partial_\nu u+2h_g=2he^\frac u2&\text{on }\partial\Sigma. \end{array}\right.$$ We provide three different existence results in the cases of positive, zero and negative Euler characteristics by means of variational techniques.