Instability of Ginzburg-Landau Vortices on Manifolds

We investigate two settings of Ginzburg-Landau posed on a manifold where vortices are unstable. The first is an instability result for critical points with vortices of the Ginzburg-Landau energy posed on a simply connected, compact, closed 2-manifold. The second is a vortex annihilation result for the Ginzburg-Landau heat flow posed on certain surfaces of revolution with boundary.