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**On low-dimensional Ricci limit spaces.**
*(English)*
Zbl 1266.53044

There is a detailed theory of the possible Gromov-Hausdorff limits of sequences of complete pointed Riemannian manifolds with Ricci curvature bounded from below. Seminal results in this theory were obtained nearly 15 years ago in the papers [J. Cheeger and T. H. Colding, J. Differ. Geom. 46, No. 3, 406–480 (1997; Zbl 0902.53034); ibid. 54, No. 1, 13–35 (2000; Zbl 1027.53042); ibid. 54, No. 1, 37–74 (2000; Zbl 1027.53043)]. This paper considers low-dimensional Ricci limit spaces, where “low” means Hausdorff dimension less than two. The first result states that except for points, the Hausdorff dimension of the limit is greater than or equal to \(1\) and less than \(2\) if and only if there there are not regular sets of dimension greater than one, if and only if the volume of any regular set of dimension greater than one is zero, if and only if the space is isomorphic to a compact \(1\)-dimensional Riemannian manifold with boundary. The second result considers the Alexandrov points of a Ricci limit space. An Alexandrov point is one having a neighborhood in which all triangles are thin in the Alexandrov sense. The second result shows that any Alexandrov point in a Ricci limit with non-empty \(1\)-dimensional regular set has local Hausdorff dimension equal to one. Assuming that the Hausdorff dimension of the complement of the cut locus of a point in the tangent cone is the same as the Hausdorff limit of the tangent cone, Honda is able to show that the dimension of any Ricci limit space is an integer.

Reviewer: David Auckly (Manhattan)

### MSC:

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

53C20 | Global Riemannian geometry, including pinching |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

### References:

[1] | D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry , Grad. Stud. Math. 33 , Amer. Math. Soc., Providence, 2001. · Zbl 0981.51016 |

[2] | J. Cheeger and T. H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products , Ann. of Math. (2) 144 (1996), 189-237. · Zbl 0865.53037 · doi:10.2307/2118589 |

[3] | J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below, I , J. Differential Geom. 46 (1997), 406-480. · Zbl 0902.53034 |

[4] | J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below, II , J. Differential Geom. 54 (2000), 13-35. · Zbl 1027.53042 |

[5] | J. Cheeger and T. H. Colding, On the structure of spaces with Ricci curvature bounded below, III , J. Differential Geom. 54 (2000), 37-74. · Zbl 1027.53043 |

[6] | T. H. Colding, Ricci curvature and volume convergence , Ann. of Math. (2) 145 (1997), 477-501. · Zbl 0879.53030 · doi:10.2307/2951841 |

[7] | K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator , Invent. Math. 87 (1987), 517-547. · Zbl 0589.58034 · doi:10.1007/BF01389241 |

[8] | K. Fukaya, “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. · Zbl 0754.53004 |

[9] | K. Fukaya, “Metric Riemannian geometry” in Handbook of Differential Geometry, Vol. 2 , Elsevier/North-Holland, Amsterdam, 2006. · Zbl 1163.53021 · doi:10.1016/S1874-5741(06)80007-5 |

[10] | M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces , with appendices by M. Katz, P. Pansu, and S. Semmes, Birkhäuser, Boston, 1999. · Zbl 0953.53002 |

[11] | S. Honda, Ricci curvature and almost spherical multi-suspension , Tohoku Math. J. (2) 61 (2009), 499-522. · Zbl 1198.53043 · doi:10.2748/tmj/1264084497 |

[12] | S. Honda, Bishop-Gromov type inequality on Ricci limit spaces , J. Math. Soc. Japan 63 (2011), 419-442. · Zbl 1252.53040 · doi:10.2969/jmsj/06320419 |

[13] | S. Honda, A note on one-dimensional regular sets , in preparation. · JFM 61.1523.01 |

[14] | X. Menguy, Examples of nonpolar limit spaces , Amer. J. Math. 122 (2000), 927-937. · Zbl 0981.53019 · doi:10.1353/ajm.2000.0041 |

[15] | X. Menguy, Noncollapsing examples with positive Ricci curvature and infinite topological type , Geom. Funct. Anal. 10 (2000), 600-627. · Zbl 0971.53030 · doi:10.1007/PL00001632 |

[16] | X. Menguy, Examples of strictly weakly regular points , Geom. Funct. Anal. 11 (2001), 124-131. · Zbl 0990.53025 · doi:10.1007/PL00001667 |

[17] | S. Ohta, On the measure contraction property of metric measure spaces , Comment. Math. Helv. 82 (2007), 805-828. · Zbl 1176.28016 · doi:10.4171/CMH/110 |

[18] | L. Simon, Lectures on Geometric Measure Theory , Proc. Centre Math. Anal. Austral. Nat. Univ. 3 , Australian National University, Canberra, 1983. · Zbl 0546.49019 |

[19] | C. Sormani, The almost rigidity of manifolds with lower bounds on Ricci curvature and minimal volume growth , Comm. Anal. Geom. 8 (2000), 159-212. · Zbl 0970.53024 |

[20] | M. Watanabe, Local cut points and metric measure spaces with Ricci curvature bounded below , Pacific J. Math. 233 (2007), 229-256. · Zbl 1152.53030 · doi:10.2140/pjm.2007.233.229 |

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