Quantitative stratification of stationary connections

Abstract

Let $A$ be a connection of a principal bundle $P$ over a Riemannian manifold $M$, such that its curvature $F_A\in L_{\text{loc}}^2(M)$ satisfies the stationarity equation. It is a consequence of the stationarity that $\theta_A(x,r)=e^{cr^2}r^{4-n}\int_{B_r(x)}|F_A|^2$ is monotonically increasing in $r$, for some $c$ depending only on the local geometry of $M$. We are interested in the singular set defined by $S(A)=\{x: \lim_{r\to 0}\theta_A(x,r)\neq 0\}$, and its stratification $S^k(A)=\{x: \text{no tangent measure at $x$ is $(k+1)$-symmetric}\}$. We then introduce and study the quantitative stratification $S^k_{\epsilon}(A)$. Roughly speaking, $S^k_{\epsilon}(A)$ consists of points at which no tangent measure of $A$ is $\epsilon$-close to being $(k+1)$-symmetric. In the main Theorem, we show that $S^k_{\epsilon}$ is $k$-rectifiable and satisfies the Minkowski volume estimate $\text{Vol}(B_r(S^k_{\epsilon})\cap B_1)\le Cr^{n-k}$. Lastly, we apply the main theorems to the stationary Yang-Mills connections to obtain a rectifiability theorem that extends some previously known results by G. Tian.