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On low-dimensional Ricci limit spaces

Published online by Cambridge University Press:  11 January 2016

Shouhei Honda
Shouhei Honda*
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan, honda@kurims.kyoto-u.ac.jp
*
Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan, honda@math.kyushu-u.ac.jp
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Abstract

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We call a Gromov–Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space. Furthermore, we prove that any Ricci limit space has integral Hausdorff dimension, provided that its Hausdorff dimension is not greater than 2. We also classify 1-dimensional Ricci limit spaces.

Keywords

53C2053C21
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 209 , March 2013 , pp. 1 - 22
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

References

[1] Burago, D., Burago, Y., and Ivanov, S., A Course in Metric Geometry, Grad. Stud. Math. 33, Amer. Math. Soc., Providence, 2001. MR 1835418.Google Scholar
[2] Cheeger, J. and Colding, T. H., Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1996), 189–237. MR 1405949. DOI 10.2307/2118589.Google Scholar
[3] Cheeger, J. and Colding, T. H., On the structure of spaces with Ricci curvature bounded below, I, J. Differential Geom. 46 (1997), 406–480. MR 1484888.Google Scholar
[4] Cheeger, J. and Colding, T. H., On the structure of spaces with Ricci curvature bounded below, II, J. Differential Geom. 54 (2000), 13–35. MR 1815410.Google Scholar
[5] Cheeger, J. and Colding, T. H., On the structure of spaces with Ricci curvature bounded below, III, J. Differential Geom. 54 (2000), 37–74. MR 1815411.Google Scholar
[6] Colding, T. H., Ricci curvature and volume convergence, Ann. of Math. (2) 145 (1997), 477–501. MR 1454700. DOI 10.2307/2951841.Google Scholar
[7] Fukaya, K., Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987), 517–547. MR 0874035. DOI 10.1007/BF01389241.CrossRefGoogle Scholar
[8] Fukaya, K., “Hausdorff convergence of Riemannian manifolds and its applications” in Recent Topics in Differential and Analytic Geometry, Adv. Stud. Pure Math. 18-I, Academic Press, Boston, 1990, 143–238. MR 1145256.Google Scholar
[9] Fukaya, K., “Metric Riemannian geometry” in Handbook of Differential Geometry, Vol. 2, Elsevier/North-Holland, Amsterdam, 2006. MR 2194670. DOI 10.1016/S1874-5741(06)80007-5.Google Scholar
[10] Gromov, M., Metric Structures for Riemannian and Non-Riemannian Spaces, with appendices by Katz, M., Pansu, P., and Semmes, S., Birkhäuser, Boston, 1999. MR 1699320.Google Scholar
[11] Honda, S., Ricci curvature and almost spherical multi-suspension, Tohoku Math. J. (2) 61 (2009), 499–522. MR 2598247. DOI 10.2748/tmj/1264084497.Google Scholar
[12] Honda, S., Bishop–Gromov type inequality on Ricci limit spaces, J. Math. Soc. Japan 63 (2011), 419–442. MR 2793106.Google Scholar
[13] Honda, S., A note on one-dimensional regular sets, in preparation.Google Scholar
[14] Menguy, X., Examples of nonpolar limit spaces, Amer. J. Math. 122 (2000), 927–937. MR 1781925.Google Scholar
[15] Menguy, X., Noncollapsing examples with positive Ricci curvature and infinite topological type, Geom. Funct. Anal. 10 (2000), 600–627. MR 1779615. DOI 10.1007/PL00001632.Google Scholar
[16] Menguy, X., Examples of strictly weakly regular points, Geom. Funct. Anal. 11 (2001), 124–131. MR 1829644. DOI 10.1007/PL00001667.Google Scholar
[17] Ohta, S., On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), 805–828. MR 2341840. DOI 10.4171/CMH/110.Google Scholar
[18] Simon, L., Lectures on Geometric Measure Theory, Proc. Centre Math. Anal. Austral. Nat. Univ. 3, Australian National University, Canberra, 1983. MR 0756417.Google Scholar
[19] Sormani, C., The almost rigidity of manifolds with lower bounds on Ricci curvature and minimal volume growth, Comm. Anal. Geom. 8 (2000), 159–212. MR 1730892.Google Scholar
[20] Watanabe, M., Local cut points and metric measure spaces with Ricci curvature bounded below, Pacific J. Math. 233 (2007), 229–256. MR 2366375. DOI 10.2140/pjm.2007.233.229.Google Scholar