Equilibria of vortex type Hamiltonians on closed surfaces - Ahmedou, Mohameden; Bartsch, Thomas; Fiernkranz, Tim (Topological Methods in Nonlinear Analysis, 2023)
We prove the existence of critical points of vortex type Hamiltonians \[ H(p_1,…, p_N) = _i,j=1i e j ^N Γ_iΓ_jG(p_i,p_j)+Ψ(p_1,…,p_N) \] 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, ΨΣ^N→R may be any function of class C^1, and Γ_1,…,Γ_NR{0} are the vorticities. The Kirchhoff-Routh Hamiltonian from fluid dynamics corresponds to Ψ(p) = -_i=1^N Γ_i^2h(p_i,p_i) where hΣ×Σ→R is the regular part of the Laplace-Beltrami operator. We obtain critical points p=(p_1,…,p_N) for arbitrary N and vorticities (Γ_1,…,Γ_N) in R^NV where V is an explicitly given algebraic variety of codimension 1.