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 ${\theta }_A(x,r)=e^{c{r}^2}{r}^{4-n}{\int }_{B_r(x)}|F_A{|}^2$ 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 $S(A)=\{x:{\lim }_{r\to 0}{\theta }_A(x,r)\ne 0\}$, 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 $S_{\epsilon }^k(A)$. Roughly speaking, $S_{\epsilon }^k(A)$ 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 $S_{\epsilon }^k$ is $k$-rectifiable and satisfies the Minkowski volume estimate $\text{Vol}(B_r(S_{\epsilon }^k)\cap B_1)\le C{r}^{n-k}$. 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.

27 pages

References in corpus (2)

• Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps
• The rectifiability of singular sets for geometric flows (I)--Yang-Mills flow