The gauge fixing theorem with applications to the Yang-Mills flow over Riemannian manifolds

TL;DR

This paper extends Uhlenbeck's gauge fixing theorem to the Yang-Mills flow, establishing existence and uniqueness of weak solutions on compact manifolds, with significant improvements in four dimensions.

Contribution

It introduces a parabolic gauge fixing theorem for Yang-Mills flow and proves existence and uniqueness of weak solutions, enhancing understanding of Yang-Mills dynamics on manifolds.

Findings

Established a parabolic gauge fixing theorem for Yang-Mills flow.

Proved existence of weak solutions on compact manifolds.

Improved key lemma for uniqueness in four dimensions.

Abstract

In 1982, Uhlenbeck \cite {U2} established the well-known gauge fixing theorem, which has played a fundamental role for Yang-Mills theory. In this paper, we apply the idea of Uhlenbeck to establish a parabolic type of gauge fixing theorems for the Yang-Mills flow and prove existence of a weak solution of the Yang-Mills flow on a compact $n$-dimensional manifold with initial value $A_0$ in $W^{1,n/2}(M)$. When $n=4$, we improve a key lemma of Uhlenbeck (Lemma 2.7 of \cite {U2}) to prove uniqueness of weak solutions of the Yang-Mills flow on a four dimensional manifold.