Yang-Mills heat flow on gauged holomorphic maps

TL;DR

This paper investigates the gradient flow of a Yang-Mills-type functional on gauged holomorphic maps over Riemann surfaces, proving long-term existence, convergence to critical points, and establishing a stratification invariant under gauge transformations.

Contribution

It extends the analysis of Yang-Mills flow to gauged holomorphic maps on surfaces with boundary, proving long-time existence and a unique stratum structure similar to known results for Hermitian Yang-Mills equations.

Findings

Flow lines converge to critical points

Existence of a gauge-invariant stratification

Unique gauge transformation to symplectic vortex on surfaces with boundary

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 K\"ahler 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.