A Variational approach to Liouville equations

Abstract:  Liouville equations govern the change of Gaussian curvature under conformal deformations of the metric of a surface, and also arise in studying energies that depend on the spectrum of the Laplace operator. Uniformization or extremization properties are subtle in general, due to non-compactness phenomena, but can be attacked with different tools. We will describe a variational approach to the problem, considering cases when Gauss-Bonnet integrals are "large", as it might happen in presence of conical singularities. We prove existence of solutions combining geometric inequalities based on a micro/macroscopic description of the conformal volume distribution.

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