Equilibria of vortex type Hamiltonians on closed surfaces

Abstract

We prove the existence of critical points of vortex type Hamiltonians \[ H(p_1,\ldots, p_N) = \sum_{{i,j=1},{i\ne j}}^N Γ_iΓ_jG(p_i,p_j)+ψ(p_1,\dots,p_N) \] 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 $\psi =-{\sum }_{i=1}^N{\Gamma }_i^{2}h(p_i,p_i)$ 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.

18 pages, to appear in Topol. Methods Nonlinear Anal