Convergence of the Yang-Mills-Higgs flow on Gauged Holomorphic maps and applications.

The symplectic vortex equations admit a variational description as global minimum of the Yang-Mills-Higgs functional. We study its negative gradient flow on holomorphic pairs where is a connection on a principal -bundle over a closed Riemann surface and is an equivariant map into a Kähler Hamiltonian -manifold. The connection induces a holomorphic structure on the Kähler fibration and we require that descends to a holomorphic section of this fibration. We prove a Łojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the -topology when is equivariantly convex at infinity with proper moment map, is holomorphically aspherical and its Kähler metric is analytic. As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang-Mills-Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet's Kobayashi-Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment-weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem. [ABSTRACT FROM AUTHOR]