A note on curvature estimate of the Hermitian-Yang-Mills flow

TL;DR

This paper investigates the curvature behavior of the Hermitian-Yang-Mills flow on holomorphic vector bundles, establishing uniform bounds away from certain subvarieties in specific cases.

Contribution

It provides a new curvature estimate for the Hermitian-Yang-Mills flow in a simplified setting, linking it to the Harder-Narasimhan-Seshadri filtration.

Findings

Curvature remains uniformly bounded away from the subvariety

Establishes a link between curvature bounds and bundle filtrations

Provides insights into the geometric behavior of the flow

Abstract

In this paper, we study the curvature estimate of the Hermitian-Yang-Mills flow on holomorphic vector bundles. In one simple case, we show that the curvature of the evolved Hermitian metric is uniformly bounded away from the analytic subvariety determined by the Harder-Narasimhan-Seshadri filtration of the holomorphic vector bundle.