The Yang–Mills $\alpha$-flow in vector bundles over four manifolds and its applications

Abstract

In this paper we introduce an $\alpha$-flow for the Yang-Mills functional in vector bundles over four dimensional Riemannian manifolds, and establish global existence of a unique smooth solution to the $\alpha$-flow with smooth initial value. We prove that the limit of the solutions of the $\alpha$-flow as $\alpha \to 1$ is a weak solution to the Yang-Mills flow. By an application of the $\alpha$-flow, we then follow the idea of Sacks and Uhlenbeck [22] to prove some existence results for Yang-Mills connections and improve the minimizing result of the Yang-Mills functional of Sedlacek [26]