Vortices on closed surfaces

Key words: point vortices, Riemann surfaces, Green functions. We consider N point vortices s j of strengths κ j moving on a closed (compact, boundaryless, orientable) surface S with riemannian metric g. As far as we know, only the sphere or surfaces of revolution, the latter qualitatively, have been treated in the available literature. The aim of this note is to present an intrinsic geometric formulation for the general case. Since the pioneer works of Bogomolov [8] and Kimura/Okamoto [36] on the sphere S 2 , it is known that stream function produced by a unit point vortex at s o ∈ S on a background uniform counter vorticity field is given by Green's function G g (s, s o) of the Laplace-Beltrami operator ∆ g = divg • grad g. It behaves as log d(s, s o)/2π near s o. Desingularizing the stream function