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Examples of Ricci limit spaces with non-integer Hausdorff dimension

Abstract

We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of Cheeger-Colding [CC00a, Page 15] about collapsing Ricci limit spaces.

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References

  1. Lionel Bérard-Bergery. Quelques exemples de variétés riemanniennes complètes non compactes à courbure de Ricci positive. C. R. Acad. Sci. Paris Sér. I Math., 302(4) (1986), 159–161.

    MathSciNet  MATH  Google Scholar 

  2. Jeff Cheeger and Tobias H. Colding. Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. Math. (2), 144(1) (1996), 189–237.

    MathSciNet  Article  Google Scholar 

  3. Jeff Cheeger and Tobias H. Colding. On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom., 46(3) (1997), 406–480.

    MathSciNet  Article  Google Scholar 

  4. Jeff Cheeger and Tobias H. Colding. On the structure of spaces with Ricci curvature bounded below. II. J. Differ. Geom., 54(1) (2000), 13–35.

    MathSciNet  MATH  Google Scholar 

  5. Jeff Cheeger and Tobias H. Colding. On the structure of spaces with Ricci curvature bounded below. III. J. Differ. Geom., 54(1) (2000), 37–74.

    MathSciNet  MATH  Google Scholar 

  6. Tobias H. Colding and Aaron Naber. Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann. Math., 176(2) (2012), 1173–1229.

    MathSciNet  Article  Google Scholar 

  7. Qin Deng. Hölder continuity of tangent cones in RCD(K,N) spaces and applications to non-branching. arXiv:2009.07956, (2020).

  8. Kenji Fukaya. Theory of convergence for Riemannian orbifolds. Jpn. J. Math. (N.S.), 12 (1) (1986), 121–160.

    MathSciNet  Article  Google Scholar 

  9. Kenji Fukaya and Takao Yamaguchi. The fundamental groups of almost nonnegatively curved manifolds. Ann. Math., 136(2) (1992), 253–333.

    MathSciNet  Article  Google Scholar 

  10. Xavier Menguy. Examples of nonpolar limit spaces. Am. J. Math., 122(5) (2000), 927–937.

    MathSciNet  Article  Google Scholar 

  11. Aaron Naber. Conjectures and open questions on the structure and regularity of spaces with lower Ricci curvature bounds. SIGMA Symmetry Integrability Geom. Methods Appl., 16:Paper No. 104 (2020), 8.

  12. Philippe Nabonnand. Sur les variétés riemanniennes complètes à courbure de Ricci positive. C. R. Acad. Sci. Paris Sér. A-B, 291(10) (1980), A591–A593.

    MathSciNet  MATH  Google Scholar 

  13. Jiayin Pan. On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature. Geom. Topol., 25(2) (2021), 1059–1085.

    MathSciNet  Article  Google Scholar 

  14. Jiayin Pan. Nonnegative Ricci curvature and escape rate gap. J. Reine Angew. Math.https://doi.org/10.1515/crelle-2021-0065 (2021).

  15. Guofang Wei. Examples of complete manifolds of positive Ricci curvature with nilpotent isometry groups. Bull. Am. Math. Soc. (N.S.), 19(1) (1988), 311–313.

    MathSciNet  Article  Google Scholar 

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J. Pan was partially supported by AMS Simons travel grant during the preparation of this paper, and is currently supported by Fields Postdoc Fellowship.

G. Wei is partially supported by NSF DMS 1811558, 2104704.

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Pan, J., Wei, G. Examples of Ricci limit spaces with non-integer Hausdorff dimension. Geom. Funct. Anal. 32, 676–685 (2022). https://doi.org/10.1007/s00039-022-00598-4

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