  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.
