Equilibria of point-vortices on closed surfaces

TL;DR

This paper investigates equilibrium configurations of point vortices on closed surfaces, revealing how surface topology influences vortex equilibria and providing new results for related nonlinear equations.

Contribution

It introduces new existence results for vortex equilibria on various closed surfaces and links these to solutions of a nonlinear mean-field equation.

Findings

Existence of vortex equilibria depends on surface topology.

New existence results for singular mean-field equations with exponential nonlinearity.

Topological properties of surfaces determine vortex configuration possibilities.

Abstract

We discuss the existence of equilibrium configurations for the Hamiltonian point-vortex model on a closed surface $\Sigma$. The topological properties of $\Sigma$ determine the occurrence of three distinct situations, corresponding to $\mathbb{S}^2$, to $\mathbb{RP}^2$ and to $\Sigma \not=\mathbb{S}^2,\mathbb{RP}^2$. As a by-product, we also obtain new existence results for the singular mean-field equation with exponential nonlinearity.