Equilibria of point-vortices on closed surfaces

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 $\Sigma \ne {\mathbb{S}}^2,{\mathbb{RP}}^2$. As a by-product, we also obtain new existence results for the singular mean-field equation with exponential nonlinearity.

27 pages