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Quantitative stratification of stationary connections

  • Yu Wang


Let A be a connection of a principal bundle P over a Riemannian manifold M, such that its curvature FALloc2(M) satisfies the stationarity equation. It is a consequence of the stationarity that


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:limr0θA(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)Crn-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 δi0 and ri0 such that Ai is (k,δi)-symmetric in B1(xi) with respect to Vk(i), and yiB1(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=0n, yiy 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|2dVgμ 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-nμ(Br(y)) and the pigeonhole principle, for any r0 there exists r* with r0r*r0(ϵ010)10Λ/ϵ0r0r(n,Λ,ϵ0) such that


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=1kBr*(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


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

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

By (A.1), (A.3), the fact that d*ϵ0, and choosing a sufficiently small r0r0(ϵ0), we obtain


From (A.2) and (A.4), we see that yr*,y,y,1,,y,k are (ϵ02,r*)-cone points at y; moreover, they 2τ(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.


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.


[1] C. Breiner and T. Lamm, Quantitative stratification and higher regularity for biharmonic maps, Manuscripta Math. 148 (2015), no. 3–4, 379–398. 10.1007/s00229-015-0750-xSearch in Google Scholar

[2] J. Cheeger, R. Haslhofer and A. Naber, Quantitative stratification and the regularity of mean curvature flow, Geom. Funct. Anal. 23 (2013), no. 3, 828–847. 10.1007/s00039-013-0224-9Search in Google Scholar

[3] J. Cheeger, R. Haslhofer and A. Naber, Quantitative stratification and the regularity of harmonic map flow, Calc. Var. Partial Differential Equations 53 (2015), no. 1–2, 365–381. 10.1007/s00526-014-0752-7Search in Google Scholar

[4] J. Cheeger and A. Naber, Lower bounds on Ricci curvature and quantitative behavior of singular sets, Invent. Math. 191 (2013), no. 2, 321–339. 10.1007/s00222-012-0394-3Search in Google Scholar

[5] J. Cheeger and A. Naber, Quantitative stratification and the regularity of harmonic maps and minimal currents, Comm. Pure Appl. Math. 66 (2013), no. 6, 965–990. 10.1002/cpa.21446Search in Google Scholar

[6] J. Cheeger, A. Naber and D. Valtorta, Critical sets of elliptic equations, Comm. Pure Appl. Math. 68 (2015), no. 2, 173–209. 10.1002/cpa.21518Search in Google Scholar

[7] M. Focardi, A. Marchese and E. Spadaro, Improved estimate of the singular set of Dir-minimizing Q-valued functions via an abstract regularity result, J. Funct. Anal. 268 (2015), no. 11, 3290–3325. 10.1016/j.jfa.2015.02.011Search in Google Scholar

[8] F.-H. Lin, Gradient estimates and blow-up analysis for stationary harmonic maps, Ann. of Math. (2) 149 (1999), no. 3, 785–829. 10.2307/121073Search in Google Scholar

[9] P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Stud. Adv. Math. 44, Cambridge University, Cambridge 1995. 10.1017/CBO9780511623813Search in Google Scholar

[10] A. Naber and D. Valtorta, Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps, Ann. of Math. (2) 185 (2017), no. 1, 131–227. 10.4007/annals.2017.185.1.3Search in Google Scholar

[11] M. Petrache and T. Rivière, The resolution of the Yang–Mills Plateau problem in super-critical dimensions, Adv. Math. 316 (2017), 469–540. 10.1016/j.aim.2017.06.012Search in Google Scholar

[12] P. Price, A monotonicity formula for Yang–Mills fields, Manuscripta Math. 43 (1983), no. 2–3, 131–166. 10.1007/BF01165828Search in Google Scholar

[13] R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geometry 17 (1982), no. 2, 307–335. 10.4310/jdg/1214436923Search in Google Scholar

[14] L. Simon, Theorems on regularity and singularity of energy minimizing maps, Lectures in Math. ETH Zürich, Birkhäuser, Basel 1996. 10.1007/978-3-0348-9193-6Search in Google Scholar

[15] T. Tao and G. Tian, A singularity removal theorem for Yang–Mills fields in higher dimensions, J. Amer. Math. Soc. 17 (2004), no. 3, 557–593. 10.1090/S0894-0347-04-00457-6Search in Google Scholar

[16] G. Tian, Gauge theory and calibrated geometry. I, Ann. of Math. (2) 151 (2000), no. 1, 193–268. 10.2307/121116Search in Google Scholar

[17] K. K. Uhlenbeck, Connections with Lp bounds on curvature, Comm. Math. Phys. 83 (1982), no. 1, 31–42. 10.1007/BF01947069Search in Google Scholar

[18] K. K. Uhlenbeck, Removable singularities in Yang–Mills fields, Comm. Math. Phys. 83 (1982), no. 1, 11–29. 10.1007/BF01947068Search in Google Scholar

[19] K. Wehrheim, Uhlenbeck compactness, EMS Ser. Lect. Math., European Mathematical Society, Zürich 2004. 10.4171/004Search in Google Scholar

Received: 2018-04-13
Revised: 2021-01-09
Published Online: 2021-02-24
Published in Print: 2021-06-01

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