Colloquium : A Variational approach to Liouville equations

TIFR CAM Colloquium

Title : A Variational approach to Liouville equations

Speaker : Andrea Malchiodi

Date : March 7, 2023

Venue : TIFR CAM, Bangalore

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.