Dynamics Of Ginzburg-Landau And Gross-Pitaevskii Vortices On Manifolds

We consider the dissipative heat flow and conservative Gross-Pitaevskii dynamics associated with the Ginzburg-Landau energy posed on a Riemannian 2-manifold M. We show the limiting vortices of the solutions to these two problems evolve according to the gradient flow and Hamiltonian point-vortex flow respectively, associated with the renormalized energy on M. For the heat flow, we then specialize to the case where M is a sphere and study the limiting system of ODE's and establish an annihilation result. Finally, for the Ginzburg-Landau heat flow on a sphere, we derive some weighted energy identities.