Yang-Mills heat flow on gauged holomorphic maps (Q344366)

A gauged holomorphic map is a pair consisting of a connection in a principal \(K\)-bundle and a section of an associated fiber bundle that is holomorphic with respect to the connection. If such a gauged holomorphic map is a zero of a suitable Yang-Mills functional, it is called a vortex. The base manifold is assumed here to be a Riemannian surface, possibly with boundary. The fiber of the fiber bundle is a compact Kähler Hamiltonian \(K\)-manifold or a vector space with linear \(K\)-action.NEWLINENEWLINEAn obvious attempt to find vortices would be to follow the gradient flow lines of the Yang-Mills functional on the space of gauged holomorphic maps. They can converge to a zero of the functional, but more generally would be expected to approach a critical point. One of the main results is that the flow exists for all times and indeed converges to a critical point, modulo ``bubbling off'' of spheres which results from energy concentrating in isolated points. If the base manifold has a nonempty boundary, the flow limit is a vortex. The same holds for a closed base manifold if the flow is started with initial data of sufficiently small energy.NEWLINENEWLINEA pleasing aspect for the analyst is the great care that is invested in describing the Sobolev spaces adapted to the geometric setting of the paper.