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

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.