The N-vortex Problem on a Riemann Sphere

Abstract

This article investigates the dynamical behaviours of the $n$-vortex problem with vorticity $\mathbf{\Gamma }$ on a Riemann sphere ${\mathbb{S}}^2$ equipped with an arbitrary metric $g$. From perspectives of Riemannian geometry and symplectic geometry, we study the invariant orbits and prove that with some constraints on vorticity $\mathbf{\Gamma }$, the $n$-vortex problem possesses finitely many fixed points and infinitely many periodic orbits for generic $g$. Moreover, we verify the contact structure on hyper-surfaces of the vortex dipole, and exclude the existence of perverse symmetric orbits.

32 pages, 3 figures