Quantitative stratification of stationary connections

Yu Wang

doi:10.1515/crelle-2021-0005

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Quantitative stratification of stationary connections

Yu Wang

Veröffentlicht/Copyright: 24. Februar 2021

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2021 Heft 775

Abstract

Let A be a connection of a principal bundle P over a Riemannian manifold M, such that its curvature FA∈Lloc2⁢(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

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}, and its stratification Sk⁢(A)={x:no tangent measure of A at x is (k+1)-symmetric}. We then introduce the quantitative stratification Sϵk⁢(A); roughly speaking Sϵk⁢(A) is the set of points at which no ball Br⁢(x) is ϵ-close to being (k+1)-symmetric. In the main theorems, we show that Sϵk is k-rectifiable and satisfies the Minkowski volume estimate Vol⁡(Br⁢(Sϵk)∩B1)≤C⁢rn-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].

A Appendix

We give the proof of Theorem 4.2. The proof is by contradiction. Suppose the contrary holds. Then there exist an ϵ0, a sequence of stationary connections Ai, positive real number sequences δi→0 and ri→0 such that Ai is (k,δi)-symmetric in B1⁢(xi) with respect to Vk(i), and yi∈B1⁢(xi)\Bϵ0⁢(Vk(i)), where no such r*>ri exists that Ai is (k+1,ϵ0)-symmetric in the ball Br*⁢(yi). Assume that Vk(i) is 12-effectively spanned by 0,ξ1i,…,ξki.

Up to passing to a subsequence we may assume xi=0∈ℝn, yi→y∞ and ξα(i)→ξα(∞) for each α=1,…,k; denote by Vk∞ the k-plane spanned by 0,ξ1∞,…,ξk∞. For convenience, let us denote 0 by ξ0i for all i. Moreover, we assume that |FAi|2⁢d⁢Vg⁢⇀∗⁢μ in B16⁢(0). Thus, the fact that the stationary connection Ai is (k,δi)-symmetric in B1⁢(0) with respect to Vk(i) implies that μ is k-symmetric in B1⁢(0) with respect to Vk∞.

Let us choose r0 sufficiently small to be determined later. On the one hand, by the monotonicity of r4-n⁢μ⁢(Br⁢(y∞)) and the pigeonhole principle, for any r0 there exists r* with r0≥r*≥r0⋅(ϵ010)10⁢Λ/ϵ0≡r0⋅r⁢(n,Λ,ϵ0) such that

(A.1)|θA⁢(y∞,10⁢r*)-θA⁢(y∞,ϵ0⁢r*10)|<ϵ010.

On the other hand, let Vk(y∞) be the k-plane that is parallel to Vk(∞) and passing through y∞. In this case we may find k points {y∞,l}l=1k⊆Br*⁢(y∞)∩Vk(∞) such that together with y∞ they τ⁢(n)⁢r*-effectively span Vk(∞) in Br*⁢(y∞). By the k-symmetry of μ in B1⁢(0), we have

(A.2)|θ⁢(y∞,l,10⁢r*)-θ⁢(y∞,l,ϵ0⁢r*10)|<ϵ010,l=0,…,k,

where we denoted y∞ by y∞,0. Let y∞proj be the projection image of y∞ onto Vk∞. Denote by yr* the intersecting point of ∂⁡Br*⁢(y∞) with the line passing through y∞proj and y∞, and set d*=dist⁢(y∞proj,y∞). Because μ∈𝒱⁢(Λ,16), we use the fact that r4-n⁢μ⁢(Br⁢(y∞proj)) being constant in r∈[0,12] and apply Lemma 2.11 (especially (2.17)) to see that

(A.3)t4-n⁢μ⁢(Bt⁢(yr*))=s4-n⁢μ⁢(Bs⁢(y∞)),where ⁢st=d*d*+r*.

By (A.1), (A.3), the fact that d*≥ϵ0, and choosing a sufficiently small r0≤r0⁢(ϵ0), we obtain

(A.4)|θ⁢(yr*,s)-θ⁢(yr*,ϵ0⁢s)|<ϵ010,ϵ0⁢r*5≤s≤5⁢r*.

From (A.2) and (A.4), we see that yr*,y∞,y∞,1,…,y∞,k are (ϵ02,r*)-cone points at y∞; moreover, they 2⁢τ⁢(n)⁢r*-effectively span a (k+1)-plane. By Definition 2.9, we have that μ is (k+1,ϵ02)-symmetric on Br*⁢(y∞). By the weak-* convergence, Ai is (k+1,ϵ0)-symmetric on Br*⁢(yi) for all i sufficiently large. This gives a contradiction since r*>ri for all i sufficiently large. Thus we complete the proof of Theorem 4.2.

Acknowledgements

The author gratefully thanks his advisor A. Naber for interesting him in the problem and giving him constant support and encouragements. The author would also like to thank the anonymous referee for pointing out an issue in an earlier draft.

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Received: 2018-04-13

Revised: 2021-01-09

Published Online: 2021-02-24

Published in Print: 2021-06-01

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