Yang–Mills heat flow on gauged holomorphic maps

We study the gradient flow lines of a Yang–Mills-type functional on a space of gauged holomorphic maps. These maps are defined on a principal $K$-bundle on a Riemann surface, possibly with boundary, where $K$ is a compact connected Lie group. The target space of the gauged holomorphic maps is a compact Kähler Hamiltonian $K$-manifold or a symplectic vector space with linear $K$-action and a proper moment map. We prove long time existence of the gradient flow. The flow lines converge to critical points of the functional, modulo sphere bubbling in $X$. Symplectic vortices are the zeros of the functional we study. When the base Riemann surface has nonempty boundary, similar to Donaldson’s result in [10], we show that there is only a single stratum; that is, any element of $\mathcal{H}(P,X)$ can be complex gauge transformed to a symplectic vortex. This is a version of Mundet’s Hitchin–Kobayashi result [30] on a surface with boundary.

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Published 12 September 2016