The Yang-Mills flow in vector bundles over higher dimensional Riemannian manifolds
We develop a parabolic formulation of the Uhlenbeck gauge fixing theorem and use it to prove the maximal existence of the Yang-Mills flow in vector bundles over compact $n$-dimensional Riemannian manifolds for $n \geq 4$. We apply Moser's iteration technique \cite{Mo1} to give a new proof of the bound of the curvature. Using an idea of Hamilton on the Ricci flow \cite{Ha}, we also provide a new proof on uniform estimates of higher order derivatives. Furthermore, we investigate the blow-up of the Yang-Mills flow at the maximal existence time and improve an asymptotical result of Hong-Tian \cite{HT1}.
math.DG↗