The finite time blow-up of the Yang-Mills flow

TL;DR

This paper proves that solutions to the Yang-Mills flow on certain manifolds blow up in finite time under specific smallness conditions on initial energy or curvature, revealing critical thresholds for flow singularities.

Contribution

It establishes finite time blow-up results for the Yang-Mills flow on non-flat bundles and links small curvature conditions to the existence of projective flat structures.

Findings

Yang-Mills flow blows up in finite time with small initial energy.

Finite time blow-up occurs when initial curvature is near harmonic form.

Holomorphic bundles with small trace-free Chern curvature admit projective flat structures.

Abstract

In this paper, we shall prove that, on a non-flat Riemannian vector bundle over a compact Riemannian manifold, the smooth solution of the Yang-Mills flow will blow up in finite time if the energy of the initial connection is small enough. We also consider the finite time blow up for the Yang-Mills flow with the initial curvature near the harmonic form. Furthermore, when $E$ is a holomorphic vector bundle over a compact K\"ahler manifold, then $E$ will admit a projective flat structure if the trace free part of Chern curvature is small enough.