  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.
