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 $(Σ,g)$ which is not homeomorphic to the sphere or the projective plane. Here $G$ denotes the Green function of the Laplace-Beltrami operator in $Σ$, $ψ:Σ^N\to\mathbb{R}$ may be any function of class $C^1$, and $Γ_1,\dots,Γ_N\in\mathbb{R}\setminus\{0\}$ are the vorticities. The Kirchhoff-Routh Hamiltonian from fluid dynamics corresponds to $ψ= -\sum_{i=1}^N Γ_i^2h(p_i,p_i)$ where $h:Σ\timesΣ\to\mathbb{R}$ is the regular part of the Laplace-Beltrami operator. We obtain critical points $p=(p_1,\dots,p_N)$ for arbitrary $N$ and vorticities $(Γ_1,\dots,Γ_N)$ in $\mathbb{R}^N\setminus V$ where $V$ is an explicitly given algebraic variety of codimension 1.