Convergence of Yang-Mills-Higgs flow for twist Higgs pairs on Riemann surfaces

TL;DR

This paper proves that the gradient flow of the Yang-Mills-Higgs functional on twist Higgs pairs over Riemann surfaces converges to a critical point characterized by the Harder-Narasimhan-Seshadri filtration, extending previous results to twisted cases.

Contribution

It generalizes Wilkin's results by establishing convergence and describing the limiting bundle for twist Higgs pairs using a modified Chern-Weil inequality.

Findings

Convergence of the Yang-Mills-Higgs flow to a critical point.

Limiting bundle described by the Harder-Narasimhan-Seshadri filtration.

Extension of known results to twist Higgs bundles.

Abstract

We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle $(E,H_0)$ over a Riemann surface $X$. It is already known the gradient flow with initial data $(A_0,\phi_0)$ converges to a critical point $(A_\infty, \phi_\infty)$ of this functional. Using a modified Chern-Weil type inequality, we prove that the limiting twist Higgs bundle $(E, d_{A_\infty}", \phi_\infty)$ is given by the graded twist Higgs bundle defined by the Harder-Narasimhan-Seshadri filtration of the initial twist Higgs bundle $(E,d_{A_0}",\phi_0)$, generalizing Wilkin's results for untwist Higgs bundle.