Sharp estimate of global Coulomb gauge

TL;DR

This paper proves the existence of a global Coulomb gauge for $W^{1,2}$-connections on SU(2)-bundles over 4-manifolds, providing a sharp estimate in a new function space that refines $L^4$ bounds.

Contribution

It introduces a novel function space $ ext{L}^{4, ext{infty}}$ and establishes a sharp gauge estimate for connections with bounded curvature.

Findings

Existence of a global Coulomb gauge with finite singularities.

Introduction of the $ ext{L}^{4, ext{infty}}$ space with refined embedding properties.

Sharp estimate of the gauge form in the new function space.

Abstract

Let $A$ be a $W^{1,2}$-connection on a principle $\text{SU}(2)$-bundle $P$ over a compact $4$-manifold $M$ whose curvature $F_A$ satisfies $\|F_A\|_{L^2(M)}\le \Lambda$. Our main result is the existence of a global section $\sigma: M\to P$ with finite singularities on $M$ such that the connection form $\sigma^*A$ satisfies the Coulomb equation $d^*(\sigma^*A)=0$ and admits a sharp estimate $\|\sigma^*A\|_{\mathcal{L}^{4,\infty}(M)}\le C(M,\Lambda)$. Here $\mathcal{L}^{4,\infty}$ is a new function space we introduce in this paper that satisfies $L^4(M)\subsetneq \mathcal{L}^{4,\infty}(M)\subsetneq L^{4-\epsilon}(M)$ for all $\epsilon>0$.