Abstract
Let Y n denote the Gromov-Hausdorff limit \(M^{n}_{i}\stackrel{d_{\mathrm{GH}}}{\longrightarrow} Y^{n}\) of v-noncollapsed Riemannian manifolds with \({\mathrm{Ric}}_{M^{n}_{i}}\geq-(n-1)\). The singular set \(\mathcal {S}\subset Y\) has a stratification \(\mathcal {S}^{0}\subset \mathcal {S}^{1}\subset\cdots\subset \mathcal {S}\), where \(y\in \mathcal {S}^{k}\) if no tangent cone at y splits off a factor ℝk+1 isometrically. Here, we define for all η>0, 0<r≤1, the k-th effective singular stratum \(\mathcal {S}^{k}_{\eta,r}\) satisfying \(\bigcup_{\eta}\bigcap_{r} \,\mathcal {S}^{k}_{\eta,r}= \mathcal {S}^{k}\). Sharpening the known Hausdorff dimension bound \(\dim\, \mathcal {S}^{k}\leq k\), we prove that for all y, the volume of the r-tubular neighborhood of \(\mathcal {S}^{k}_{\eta,r}\) satisfies \({\mathrm {Vol}}(T_{r}(\mathcal {S}^{k}_{\eta,r})\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},\eta)r^{n-k-\eta}\). The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let \(\mathcal {B}_{r}\) denote the set of points at which the C 2-harmonic radius is ≤r. If also the \(M^{n}_{i}\) are Kähler-Einstein with L 2 curvature bound, \(\| Rm\|_{L_{2}}\leq C\), then \({\mathrm {Vol}}( \mathcal {B}_{r}\cap B_{\frac{1}{2}}(y))\leq c(n,{\mathrm {v}},C)r^{4}\) for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the \(M^{n}_{i}\), we obtain a slightly weaker volume bound on \(\mathcal {B}_{r}\) which yields an a priori L p curvature bound for all p<2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations.
Similar content being viewed by others
References
Anderson, M.T.: Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102(2), 429–445 (1990)
Cheeger, J., Colding, T.H.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. Math. 144(1), 189–237 (1996)
Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46(3), 406–480 (1997)
Cheeger, J., Colding, T.H., Tian, G.: On the singularities of spaces with bounded Ricci curvature. Geom. Funct. Anal. 12(5), 873–914 (2002)
Chen, X.-X., Donaldson, S.: Volume estimates for Kähler Einstein metrics and rigidity of complex structures (2011). arXiv:1104.4331v1 [math]
Chen, X.-X., Donaldson, S.: Volume estimates for Kähler Einstein metrics: the three dimensional case (2011). arXiv:1104.0270v2 [math]
Cheeger, J.: Integral bounds on curvature elliptic estimates and rectifiability of singular sets. Geom. Funct. Anal. 13(1), 20–72 (2003)
Cheeger, J., Kleiner, B., Naor, A.: Compression bounds for Lipschitz maps from the Heisenberg group to L 1. Acta Math. 207(2), 291–373 (2011)
Cheeger, J., Naber, A.: Quantitative stratification and the regularity of harmonic maps and minimal currents. Commun. Pure Appl. Math. (2011, to appear). arXiv:1107.1197 [math]
Chakrabarti, D., Shaw, M.-C.: The Cauchy-Riemann equations on product domains. Math. Ann. 349, 977–998 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cheeger, J., Naber, A. Lower bounds on Ricci curvature and quantitative behavior of singular sets. Invent. math. 191, 321–339 (2013). https://doi.org/10.1007/s00222-012-0394-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-012-0394-3