Quantitative stratification of stationary connections

Abstract

Abstract Let A be a connection of a principal bundle P over a Riemannian manifold M, such that its curvature F A ∈ L loc 2 ⁢ ( M ) {F_{A}\in L_{\mathrm{loc}}^{2}(M)} satisfies the stationarity equation. It is a consequence of the stationarity that θ A ⁢ ( x , r ) = e c ⁢ r 2 ⁢ r 4 - n ⁢ ∫ B r ⁢ ( x ) | F A | 2 ⁢ 𝑑 V g {\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 : lim r → 0 ⁡ θ A ⁢ ( x , r ) ≠ 0 } {S(A)=\{x:\lim_{r\to 0}\theta_{A}(x,r)\neq 0\}} , and its stratification S k ⁢ ( 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 S ϵ k ⁢ ( A ) {S^{k}_{\epsilon}(A)} ; roughly speaking S ϵ k ⁢ ( A ) {S^{k}_{\epsilon}(A)} is the set of points at which no ball B r ⁢ ( x ) {B_{r}(x)} is ϵ-close to being ( k + 1 ) {(k+1)} -symmetric. In the main theorems, we show that S ϵ k {S^{k}_{\epsilon}} is k-rectifiable and satisfies the Minkowski volume estimate Vol ⁡ ( B r ⁢ ( S ϵ k ) ∩ B 1 ) ≤ C ⁢ r n - 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].