Equilibria of vortex type Hamiltonians on closed surfaces

TL;DR

This paper proves the existence of critical points for vortex Hamiltonians on closed surfaces, extending results beyond the sphere and projective plane, with applications to fluid dynamics.

Contribution

It establishes the existence of critical points for vortex Hamiltonians on a broad class of closed surfaces, generalizing previous results to more complex topologies.

Findings

Critical points exist for arbitrary number of vortices and vorticities.

Results apply to surfaces not homeomorphic to sphere or projective plane.

Includes specific cases like Kirchhoff-Routh Hamiltonian.

Abstract

We prove the existence of critical points of vortex type Hamiltonians \[ H(p_1,\ldots, p_N) = \sum_{{i,j=1},{i\ne j}}^N \Gamma_i\Gamma_jG(p_i,p_j)+\psi(p_1,\dots,p_N) \] on a closed Riemannian surface $(\Sigma,g)$ which is not homeomorphic to the sphere or the projective plane. Here $G$ denotes the Green function of the Laplace-Beltrami operator in $\Sigma$, $\psi:\Sigma^N\to\mathbb{R}$ may be any function of class $C^1$, and $\Gamma_1,\dots,\Gamma_N\in\mathbb{R}\setminus\{0\}$ are the vorticities. The Kirchhoff-Routh Hamiltonian from fluid dynamics corresponds to $\psi = -\sum_{i=1}^N \Gamma_i^2h(p_i,p_i)$ where $h:\Sigma\times\Sigma\to\mathbb{R}$ is the regular part of the Laplace-Beltrami operator. We obtain critical points $p=(p_1,\dots,p_N)$ for arbitrary $N$ and vorticities $(\Gamma_1,\dots,\Gamma_N)$ in $\mathbb{R}^N\setminus V$ where $V$ is an explicitly given algebraic variety of codimension 1.