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

作者：Trautwein Samuel^*

来源：International Journal of Mathematics, 2018, 29(4): 1850024.

DOI：10.1142/S0129167X18500246

• Gauge theory
• vortex equation
• heat flow
• moment map
• symplectic reduction
• geometric invariant theory

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 (A, u) where A is a connection on a principal G-bundle P over a closed Riemann surface Sigma and u : P -> X is an equivariant map into a Kahler Hamiltonian G-manifold. The connection A induces a holomorphic structure on the Kahler fibration P x (G) X and we require that u descends to a holomorphic section of this fibration.
We prove a Lojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the W-1,W-2 x W-2,W-2-topology when X is equivariantly convex at infinity with proper moment map, X is holomorphically aspherical and its Kahler 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.

• 出版日期2018-4