Global Coulomb gauge in dimension 4

Wang, Yu

doi:https://doi.org/10.21985/n2-d2ch-yz08

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Global Coulomb gauge in dimension 4

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In this dissertation we study the connections over principal bundles in dimension four with bounded Yang-Mills energy, and present a new result on the existence a global Coulomb gauge with estimate in optimal space. To be precise, let $A$ be a $W^{1,2}$-connection on a principal $\text{SU}(2)$-bundle $P$ over a smooth 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 introduced in \cite{me} that satisfies $L^4(M)\subsetneq \mathcal{L}^{4,\infty}(M)\subsetneq L^{4-\epsilon}(M)$ for all $\epsilon>0$. More precisely, $\mathcal{L}^{4,\infty}(M)$ is the collection of measurable function $u$ such that $\|u\|_{\mathcal{L}^{4,\infty}(M)}\eqqcolon\|1/s_u\|_{L^{4,\infty}(M)}<\infty$, where $L^{4,\infty}$ is the classical Lorentz space and $s_u$ is the $L^4$ integrability radius function associated to $u$ defined by $s_{u}(x)=\sup\Big\{r:\sup_{y\in B_{r}(x)}\int_{B_r(y)}|u|^4 dV_g\le 1\Big\}.$ Briefly speaking, we achieve the estimate of $\|\sigma^*A\|_{\mathcal{L}^{4,\infty}(M)}$ by showing that $\sigma^*A$ is effectively $L^4$-integrable away from controllably many points on $M$. This extends a celebrated result of Uhlenbeck to the global case, and settles an open problem raised by Tristan Rivi\`ere.

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