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 FA∈Lloc2⁢(M){F_{A}\in L_{\mathrm{loc}}^{2}(M)} satisfies the stationarity equation. It is a consequence of the stationarity that θA⁢(x,r)=ec⁢r2⁢r4-n⁢∫Br⁢(x)|FA|2⁢𝑑Vg{\theta_{A}(x,r)=e^{cr^{2}}r^{4-n}\int_{B_{r}(x)}|F_{A}|^{2}\,dV_{g}} 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:limr→0⁡θA⁢(x,r)≠0}{S(A)=\{x:\lim_{r\to 0}\theta_{A}(x,r)\neq 0\}}, and its stratification Sk⁢(A)={x:no tangent measure of A at x is (k+1)-symmetric}{S^{k}(A)=\{x:\text{no tangent measure of $A$ at $x$ is $(k+1)$-symmetric}\}}. We then introduce the quantitative stratification Sϵk⁢(A){S^{k}_{\epsilon}(A)}; roughly speaking Sϵk⁢(A){S^{k}_{\epsilon}(A)} is the set of points at which no ball Br⁢(x){B_{r}(x)} is ϵ-close to being (k+1){(k+1)}-symmetric. In the main theorems, we show that Sϵk{S^{k}_{\epsilon}} is k -rectifiable and satisfies the Minkowski volume estimate Vol⁡(Br⁢(Sϵk)∩B1)≤C⁢rn-k{\operatorname{Vol}(B_{r}(S^{k}_{\epsilon})\cap B_{1})\leq 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 in [G. Tian, Gauge theory and calibrated geometry. I, Ann. of Math. (2) 151 2000, 1, 193–268].