The N-Vortex Problem on a Riemann Sphere

Abstract

We investigate the dynamical behaviours of the n-vortex problem with circulation vector \(\varvec{\Gamma }\) on a Riemann sphere \({\mathbb {S}}^2\), equipped with an arbitrary metric g. By mixing perspectives from Riemannian geometry and symplectic geometry, we prove that for any given \(\varvec{\Gamma }\), the Hamiltonian is a Morse function for \({\mathcal {C}}^2\) generic g. If some constraints are put on \(\varvec{\Gamma }\), then for such g the n-vortex problem possesses finitely many fixed points and infinitely many periodic orbits. Moreover, we exclude the existence of perverse symmetric orbits.