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 Kahler 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 non empty boundary, similar to Donaldson's result in [10], we show that there is only a single stratum; that is, any element of 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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