Remarks on the Yang-Mills flow on a compact Kahler manifold

TL;DR

This paper investigates the behavior of the Yang-Mills flow on holomorphic vector bundles over compact Kähler manifolds, introducing new techniques to analyze curvature blow-up and stability conditions.

Contribution

It develops a barrier function and analytical methods to control curvature blow-up, linking flow behavior to the Harder-Narasimhan-Seshadri filtration under certain assumptions.

Findings

Curvature remains bounded away from a specific subvariety.

Techniques relate flow properties to stability conditions.

Assumptions connect to stability in simple cases.

Abstract

We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X. We construct a natural barrier function along the flow, and introduce some techniques to study the blow-up of the curvature along the flow. Making some technical assumptions, we show how our techniques can be used to prove that the curvature of the evolved connection is uniformly bounded away from an analytic subvariety determined by the Harder-Narasimhan-Seshadri filtration of E. We also discuss how our assumptions are related to stability in some simple cases.