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Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. (English) Zbl 1260.53067
Authors’ abstract: We prove a new estimate on manifolds with a lower Ricci bound which asserts that the geometry of balls centered on a minimizing geodesic can change in at most a Hölder continuous way along the geodesic. We give examples that show that the Hölder exponent, along with essentially all the other consequences that follow from this estimate, are sharp.
Among the applications is that the regular set is convex for any noncollapsed limit of Einstein metrics. In the general case of a potentially collapsed limit of manifolds with just a lower Ricci curvature bound, we show that the regular set is weakly convex and a.e. convex. We also show two conjectures of Cheeger-Colding. One of these asserts that the isometry group of any, even collapsed, limit of manifolds with a uniform lower Ricci curvature bound is a Lie group. The other asserts that the dimension of any limit space is the same everywhere.

53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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[1] U. Abresch and D. Gromoll, ”On complete manifolds with nonnegative Ricci curvature,” J. Amer. Math. Soc., vol. 3, iss. 2, pp. 355-374, 1990. · Zbl 0704.53032
[2] 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
[3] 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
[4] Y. Burago, M. Gromov, and G. Perelman, ”A. D. Aleksandrov spaces with curvatures bounded below,” Russian Math. Surveys, vol. 47, iss. 2(284), pp. 1-58, 1992. · Zbl 0802.53018
[5] 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
[6] E. Calabi, ”On Ricci curvature and geodesics,” Duke Math. J., vol. 34, pp. 667-676, 1967. · Zbl 0153.51501
[7] J. Cheeger, Degeneration of Riemannian Metrics under Ricci Curvature Bounds, Pisa: Scuola Normale Superiore, 2001. · Zbl 1055.53024
[8] 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
[9] 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
[10] J. Cheeger and T. H. Colding, ”On the structure of spaces with Ricci curvature bounded below. II,” J. Differential Geom., vol. 54, iss. 1, pp. 13-35, 2000. · Zbl 1027.53042
[11] J. Cheeger and T. H. Colding, ”On the structure of spaces with Ricci curvature bounded below. III,” J. Differential Geom., vol. 54, iss. 1, pp. 37-74, 2000. · Zbl 1027.53043
[12] T. H. Colding, ”Shape of manifolds with positive Ricci curvature,” Invent. Math., vol. 124, iss. 1-3, pp. 175-191, 1996. · Zbl 0871.53027
[13] T. H. Colding, ”Large manifolds with positive Ricci curvature,” Invent. Math., vol. 124, iss. 1-3, pp. 193-214, 1996. · Zbl 0871.53028
[14] T. H. Colding, ”Ricci curvature and volume convergence,” Ann. of Math., vol. 145, iss. 3, pp. 477-501, 1997. · Zbl 0879.53030
[15] T. H. Colding, ”Spaces with Ricci curvature bounds,” in Proceedings of the International Congress of Mathematicians, Vol. II, 1998, pp. 299-308. · Zbl 0955.53020
[16] T. H. Colding and A. Naber, Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications. · Zbl 1271.53042
[17] T. H. Colding and A. Naber, Lower Ricci curvature, branching and the bi-Lipschitz structure of uniform Reifenberg spaces. · Zbl 1288.53031
[18] T. Eguchi and A. J. Hanson, ”Asymptotically flat self-dual solutions to Euclidean gravity,” Phys. Lett. B, vol. 74, iss. 1, pp. 82-106, 1978. · Zbl 0409.53020
[19] K. Fukaya, ”Collapsing of Riemannian manifolds and eigenvalues of Laplace operator,” Invent. Math., vol. 87, iss. 3, pp. 517-547, 1987. · Zbl 0589.58034
[20] K. Fukaya, ”Metric Riemannian geometry,” in Handbook of Differential Geometry. Vol. II, Elsevier/North-Holland, Amsterdam, 2006, pp. 189-313. · Zbl 1163.53021
[21] K. Fukaya and T. Yamaguchi, ”Isometry groups of singular spaces,” Math. Z., vol. 216, iss. 1, pp. 31-44, 1994. · Zbl 0797.53033
[22] S. Gallot, ”Volumes, courbure de Ricci et convergence des variétés (d’après T. H. Colding et Cheeger-Colding),” in Séminaire Bourbaki. Vol. 1997/98, , 1998, vol. 252, p. exp. no. 835, 3, 7-32. · Zbl 0976.53038
[23] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second ed., New York: Springer-Verlag, 1983, vol. 224. · Zbl 0562.35001
[24] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Boston, MA: Birkhäuser, 2007. · Zbl 1113.53001
[25] M. Gromov, J. Lafontaine, and P. Pansu, Structures Metriques pour les Varieties Riemanniennces, Paris: Cedid/Fernand Nathan, 1981.
[26] K. Grove and P. Petersen, ”Manifolds near the boundary of existence,” J. Differential Geom., vol. 33, iss. 2, pp. 379-394, 1991. · Zbl 0729.53045
[27] S. Honda, ”Bishop-Gromov type inequality on Ricci limit spaces,” J. Math. Soc. Japan, vol. 63, iss. 2, pp. 419-442, 2011. · Zbl 1252.53040
[28] P. Li and S. Yau, ”On the parabolic kernel of the Schrödinger operator,” Acta Math., vol. 156, iss. 3-4, pp. 153-201, 1986. · Zbl 0611.58045
[29] X. Menguy, ”Noncollapsing examples with positive Ricci curvature and infinite topological type,” Geom. Funct. Anal., vol. 10, iss. 3, pp. 600-627, 2000. · Zbl 0971.53030
[30] X. Menguy, ”Examples of strictly weakly regular points,” Geom. Funct. Anal., vol. 11, iss. 1, pp. 124-131, 2001. · Zbl 0990.53025
[31] X. C. Menguy, Examples of Manifolds and Spaces with Positive Ricci Curvature, ProQuest LLC, Ann Arbor, MI, 2000. · Zbl 0981.53019
[32] X. Menguy, ”Examples of nonpolar limit spaces,” Amer. J. Math., vol. 122, iss. 5, pp. 927-937, 2000. · Zbl 0981.53019
[33] G. Perelman, ”Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers,” in Comparison Geometry, Cambridge: Cambridge Univ. Press, 1997, vol. 30, pp. 157-163. · Zbl 0890.53038
[34] G. Perelman, ”A complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and nonunique asymptotic cone,” in Comparison Geometry, Cambridge: Cambridge Univ. Press, 1997, vol. 30, pp. 165-166. · Zbl 0887.53038
[35] A. Petrunin, ”Parallel transportation for Alexandrov space with curvature bounded below,” Geom. Funct. Anal., vol. 8, iss. 1, pp. 123-148, 1998. · Zbl 0903.53045
[36] 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
[37] G. Wei, ”Manifolds with a lower Ricci curvature bound,” in Surveys in Differential Geometry. Vol. XI, Somerville, MA: International Press, 2007, pp. 203-227. · Zbl 1151.53036
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