Equivariant mean field flow

TL;DR

This paper studies a gradient flow related to the mean field equation on compact Riemannian surfaces, proving its global existence and convergence under symmetry conditions.

Contribution

It introduces a gradient flow for the mean field equation on surfaces and establishes conditions for its convergence to solutions, extending understanding of mean field dynamics.

Findings

Flow exists for all time on compact surfaces.

Flow converges to solutions under symmetry conditions.

Provides conditions related to isometry group actions.

Abstract

We consider a gradient flow associated to the mean field equation on $(M,g)$ a compact riemanniann surface without boundary. We prove that this flow exists for all time. Moreover, letting $G$ be a group of isometry acting on $(M,g)$, we obtain the convergence of the flow to a solution of the mean field equation under suitable hypothesis on the orbits of points of $M$ under the action of $G$.