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

TL;DR

This paper introduces an low for the Yang-Mills functional on four-dimensional manifolds, establishing global existence and connecting it to the Yang-Mills flow, with applications to existence and minimization of Yang-Mills connections.

Contribution

It develops a new low for Yang-Mills functional, proves its global existence, and uses it to obtain existence results for Yang-Mills connections and improve minimization results.

Findings

Established global existence of the low solution.

Showed the low limit as pproaches 1 yields a weak Yang-Mills flow solution.

Improved the minimizing results for the Yang-Mills functional.

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 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 to prove some existence results for Yang-Mills connections and improve the minimizing result of the Yang-Mills functional of Sedlacek.