Regularity of Einstein manifolds and the codimension 4 conjecture. (English) Zbl 1335.53057

The authors consider the class of pointed Riemannian manifolds \((M^n,g,p)\) with bounded Ricci curvature \(|\mathrm{Ric}|\leq n-1\) which satisfies the noncollapsing assumption about the radius-one ball \(\mathrm{Vol}(B_1(p)) >v >0 \). They are interested in the properties of pointed Gromov-Hausdorff limits \[ (M^n_j,d_j,p_j) \to (X,d,p) \] of a sequence of such manifolds with respect to the Riemannian distance \(d_j\). There are many deep results. The most important is a proof of the following codimension-4 conjecture: Theorem. The singular set \(S\) of the limit space \(X\) has Hausdorff or Minkowski dimension \(\dim S \leq n-4\).
Combining this result with the ideas of quantitative stratification, the authors get an a priori \( L^q\) estimate on the full Riemannian curvature for all \( q < 2\). In the case of Einstein manifolds, they improve this to estimate the regularity scale. These results are applied for proving the following conjecture by Anderson about 4-dimensional Riemannian manifolds:
Theorem. There exists a constant \(C = C(v, D)\) such that if \( M^4\) satisfies \(|\mathrm{Ric}_{M^4}|\leq 3\), \(\mathrm{Vol}(B_1(p)) > v > 0\) and \(\operatorname{diam}(M^4) \leq D\), then \( M^4\) can have one of at most \( C\) diffeomorphism types.
The authors conjecture that this result holds in all dimensions.
It is shown also that noncollapse Riemannian 4-manifolds with bounded Ricci curvature admit an a priori \(L^2\) estimate on the Riemannian curvature tensor.


53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Full Text: DOI arXiv


[1] L. Ambrosio, N. Gigli, and G. Savaré, ”Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below,” Invent. Math., vol. 195, iss. 2, pp. 289-391, 2014. · Zbl 1312.53056 · doi:10.1007/s00222-013-0456-1
[2] L. Ambrosio, N. Gigli, and G. Savaré, ”Metric measure spaces with Riemannian Ricci curvature bounded from below,” Duke Math. J., vol. 163, iss. 7, pp. 1405-1490, 2014. · Zbl 1304.35310 · doi:10.1215/00127094-2681605
[3] M. T. Anderson, ”Ricci curvature bounds and Einstein metrics on compact manifolds,” J. Amer. Math. Soc., vol. 2, iss. 3, pp. 455-490, 1989. · Zbl 0694.53045 · doi:10.2307/1990939
[4] M. T. Anderson, ”Convergence and rigidity of manifolds under Ricci curvature bounds,” Invent. Math., vol. 102, iss. 2, pp. 429-445, 1990. · Zbl 0711.53038 · doi:10.1007/BF01233434
[5] M. T. Anderson, ”Hausdorff perturbations of Ricci-flat manifolds and the splitting theorem,” Duke Math. J., vol. 68, iss. 1, pp. 67-82, 1992. · Zbl 0767.53029 · doi:10.1215/S0012-7094-92-06803-7
[6] M. T. Anderson, ”Einstein metrics and metrics with bounds on Ricci curvature,” in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Basel, 1995, pp. 443-452. · Zbl 0840.53036
[7] M. T. Anderson and J. Cheeger, ”\(C^\alpha\)-compactness for manifolds with Ricci curvature and injectivity radius bounded below,” J. Differential Geom., vol. 35, iss. 2, pp. 265-281, 1992. · Zbl 0774.53021
[8] M. T. Anderson and J. Cheeger, ”Diffeomorphism finiteness for manifolds with Ricci curvature and \(L^{n/2}\)-norm of curvature bounded,” Geom. Funct. Anal., vol. 1, iss. 3, pp. 231-252, 1991. · Zbl 0764.53026 · doi:10.1007/BF01896203
[9] S. Bando, A. Kasue, and H. Nakajima, ”On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth,” Invent. Math., vol. 97, iss. 2, pp. 313-349, 1989. · Zbl 0682.53045 · doi:10.1007/BF01389045
[10] S. Bando, ”Bubbling out of Einstein manifolds,” Tohoku Math. J., vol. 42, iss. 2, pp. 205-216, 1990. · Zbl 0719.53025 · doi:10.2748/tmj/1178227654
[11] D. Bakry and M. Émery, ”Diffusions hypercontractives,” in Séminaire de Probabilités, XIX, 1983/84, New York: Springer-Verlag, 1985, vol. 1123, pp. 177-206. · Zbl 0561.60080 · doi:10.1007/BFb0075847
[12] J. Cheeger, ”Finiteness theorems for Riemannian manifolds,” Amer. J. Math., vol. 92, pp. 61-74, 1970. · Zbl 0194.52902 · doi:10.2307/2373498
[13] J. Cheeger, ”Integral bounds on curvature elliptic estimates and rectifiability of singular sets,” Geom. Funct. Anal., vol. 13, iss. 1, pp. 20-72, 2003. · Zbl 1086.53051 · doi:10.1007/s000390300001
[14] J. Cheeger and T. H. Colding, ”Lower bounds on Ricci curvature and the almost rigidity of warped products,” Ann. of Math., vol. 144, iss. 1, pp. 189-237, 1996. · Zbl 0865.53037 · doi:10.2307/2118589
[15] J. Cheeger and T. H. Colding, ”On the structure of spaces with Ricci curvature bounded below. I,” J. Differential Geom., vol. 46, iss. 3, pp. 406-480, 1997. · Zbl 0902.53034
[16] J. Cheeger, T. H. Colding, and G. Tian, ”On the singularities of spaces with bounded Ricci curvature,” Geom. Funct. Anal., vol. 12, iss. 5, pp. 873-914, 2002. · Zbl 1030.53046 · doi:10.1007/PL00012649
[17] J. Cheeger and A. Naber, ”Lower bounds on Ricci curvature and quantitative behavior of singular sets,” Invent. Math., vol. 191, iss. 2, pp. 321-339, 2013. · Zbl 1268.53053 · doi:10.1007/s00222-012-0394-3
[18] J. Cheeger, A. Naber, and D. Valtorta, ”Critical sets of elliptic equations,” Comm. Pure Appl. Math., vol. 68, iss. 2, pp. 173-209, 2015. · Zbl 1309.35012 · doi:10.1002/cpa.21518
[19] X. -X. Chen and S. K. Donaldson, ”Volume estimates for Kähler-Einstein metrics and rigidity of complex structures,” J. Differential Geom., vol. 93, iss. 2, pp. 191-201, 2013. · Zbl 1281.32019
[20] T. H. Colding, ”Ricci curvature and volume convergence,” Ann. of Math., vol. 145, iss. 3, pp. 477-501, 1997. · Zbl 0879.53030 · doi:10.2307/2951841
[21] T. H. Colding and A. Naber, ”Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications,” Geom. Funct. Anal., vol. 23, iss. 1, pp. 134-148, 2013. · Zbl 1271.53042 · doi:10.1007/s00039-012-0202-7
[22] R. Dong, ”Nodal sets of eigenfunctions on Riemann surfaces,” J. Differential Geom., vol. 36, iss. 2, pp. 493-506, 1992. · Zbl 0776.53024
[23] N. Gigli, A. Mondino, and G. Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, 2013. · Zbl 1398.53044
[24] G. H. Golub and C. F. Van Loan, Matrix Computations, Third ed., Johns Hopkins University Press, Baltimore, MD, 1996. · Zbl 0865.65009
[25] Q. Han and F. Lin, Nodal Sets of Solutions of Elliptic Differential Equations.
[26] H. Hein and A. Naber, Isolated Einstein singularities with singular tangent cones.
[27] A. Mondino and A. Naber, Structure theory of metric-measure spaces with lower Ricci curvature bounds I, 2014.
[28] A. Naber and D. Valtorta, ”Sharp estimates on the first eigenvalue of the \(p\)-Laplacian with negative Ricci lower bound,” Math. Z., vol. 277, iss. 3-4, pp. 867-891, 2014. · Zbl 1320.58018 · doi:10.1007/s00209-014-1282-x
[29] P. Petersen, Riemannian Geometry, New York: Springer-Verlag, 1998, vol. 171. · Zbl 0914.53001 · doi:10.1007/978-1-4757-6434-5
[30] R. Schoen and S. -T. Yau, Lectures on Differential Geometry, Cambridge, MA: International Press, 1994, vol. I. · Zbl 0830.53001
[31] G. Tian, ”On Calabi’s conjecture for complex surfaces with positive first Chern class,” Invent. Math., vol. 101, iss. 1, pp. 101-172, 1990. · Zbl 0716.32019 · doi:10.1007/BF01231499
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.