Yang-Mills heat flow on gauged holomorphic maps

Abstract: We study the gradient flow lines of a Yang-Mills-type
functional on the space of gauged holomorphic maps $\mathcal{H}(P,X)$,
where $P$ is a principal bundle on a Riemann surface $\Sigma$ and $X$
is a Kahler Hamiltonian $G$-manifold. For compact $\Sigma$, possibly
with boundary, we prove long time existence of the gradient flow. The
flow lines converge to critical points of the functional. So, there is
a stratification on $\mathcal{H}(P,X)$ that is invariant under the
action of the complexified gauge group.
Symplectic vortices are the zeros of the functional we study. When
$\Sigma$ has boundary, similar to Donaldson's result for the Hermitian
Yang-Mills equations, we show that there is only a single stratum -
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 on a surface with boundary.