Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 021, 10 pages      arXiv:2002.08612
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Collapsed Ricci Limit Spaces as Non-Collapsed RCD Spaces

Shouhei Honda
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Received February 20, 2020, in final form March 25, 2020; Published online April 01, 2020

In this short note we provide several conjectures on the regularity of measured Gromov-Hausdorff limit spaces of Riemannian manifolds with Ricci curvature bounded below, from the point of view of the synthetic treatment of lower bounds on Ricci curvature for metric measure spaces.

Key words: metric measure space; Ricci curvature; Laplacian; Hausdorff measure.

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  1. Ambrosio L., Calculus, heat flow and curvature-dimension bounds in metric measure spaces, in Proceedings of the International Congress of Mathematicians - Rio de Janeiro 2018, Vol. I, Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018, 301-340.
  2. Ambrosio L., Bruè E., Trevisan D., Lusin-type approximation of Sobolev by Lipschitz functions, in Gaussian and ${\rm RCD}(K,\infty)$ spaces, Adv. Math. 339 (2018), 426-452, arXiv:1712.06315.
  3. Ambrosio L., Gigli N., Mondino A., Rajala T., Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure, Trans. Amer. Math. Soc. 367 (2015), 4661-4701, arXiv:1207.4924.
  4. Ambrosio L., Gigli N., Savaré G., Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014), 1405-1490, arXiv:1109.0222.
  5. Ambrosio L., Honda S., New stability results for sequences of metric measure spaces with uniform Ricci bounds from below, in Measure Theory in Non-Smooth Spaces, Partial Differ. Equ. Meas. Theory, De Gruyter Open, Warsaw, 2017, 1-51, arXiv:1605.07908.
  6. Ambrosio L., Honda S., Local spectral convergence in ${\rm RCD}^*(K,N)$ spaces, Nonlinear Anal. 177 (2018), 1-23, arXiv:1703.04939.
  7. Ambrosio L., Honda S., Portegies J.W., Tewodrose D., Embedding of ${\rm RCD}^*(K, N)$-spaces in $L^2$ via eigenfunctions, arXiv:1812.03712.
  8. Ambrosio L., Honda S., Tewodrose D., Short-time behavior of the heat kernel and Weyl's law on ${\rm RCD}^*(K,N)$ spaces, Ann. Global Anal. Geom. 53 (2018), 97-119, arXiv:1701.03906.
  9. Ambrosio L., Mondino A., Savaré G., Nonlinear diffusion equations and curvature conditions in metric measure spaces, Mem. Amer. Math. Soc. 262 (2019), v+121 pages, arXiv:1509.07273.
  10. Anderson M.T., Hausdorff perturbations of Ricci-flat manifolds and the splitting theorem, Duke Math. J. 68 (1992), 67-82.
  11. Bérard P., Besson G., Gallot S., Embedding Riemannian manifolds by their heat kernel, Geom. Funct. Anal. 4 (1994), 373-398.
  12. Bruè E., Semola D., Constancy of dimension for ${\rm RCD}^*(K,N)$ spaces via regularity of Lagrangian flows, Comm. Pure Appl. Math., to appear, arXiv:1803.04387.
  13. Burago Y., Gromov M., Perelman G., A.D. Aleksandrov spaces with curvatures bounded below, Russian Math. Surveys 47 (1992), no. 2, 1-58.
  14. Cavalletti F., Milman E., The globalization theorem for the curvature dimension condition, arXiv:1612.07623.
  15. Cheeger J., Colding T.H., On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), 406-480.
  16. Cheeger J., Colding T.H., On the structure of spaces with Ricci curvature bounded below. II, J. Differential Geom. 54 (2000), 13-35.
  17. Cheeger J., Colding T.H., On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom. 54 (2000), 37-74.
  18. Cheeger J., Jiang W., Naber A., Rectifiability of singular sets in noncollapsed spaces with Ricci curvature bounded below, arXiv:1805.07988.
  19. Colding T.H., Naber A., Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications, Ann. of Math. 176 (2012), 1173-1229, arXiv:1102.5003.
  20. Colding T.H., Naber A., Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications, Geom. Funct. Anal. 23 (2013), 134-148, arXiv:1108.3244.
  21. De Philippis G., Gigli N., Non-collapsed spaces with Ricci curvature bounded from below, J. Éc. polytech. Math. 5 (2018), 613-650, arXiv:1708.02060.
  22. De Philippis G., Marchese A., Rindler F., On a conjecture of Cheeger, in Measure Theory in Non-Smooth Spaces, Partial Differ. Equ. Meas. Theory, De Gruyter Open, Warsaw, 2017, 145-155, arXiv:1607.02554.
  23. Erbar M., Kuwada K., Sturm K.-T., On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces, Invent. Math. 201 (2015), 993-1071, arXiv:1303.4382.
  24. Fukaya K., Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987), 517-547.
  25. Gigli N., The splitting theorem in non-smooth context, arXiv:1302.5555.
  26. Gigli N., Nonsmooth differential geometry - an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc. 251 (2018), v+161 pages, arXiv:1407.0809.
  27. Gigli N., Han B.-X., Independence on $p$ of weak upper gradients on ${\rm RCD}$ spaces, J. Funct. Anal. 271 (2016), 1-11, arXiv:1407.7350.
  28. Gigli N., Mondino A., Savaré G., Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc. 111 (2015), 1071-1129, arXiv:1311.4907.
  29. Gigli N., Pasqualetto E., Behaviour of the reference measure on ${\rm RCD}$ spaces under charts, Comm. Pure Appl. Math., to appear, arXiv:1607.05188.
  30. Hajłasz P., Koskela P., Sobolev meets Poincaré, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1211-1215.
  31. Han B.-X., Ricci tensor on ${\rm RCD}^*(K,N)$ spaces, J. Geom. Anal. 28 (2018), 1295-1314, arXiv:1412.0441.
  32. Hattori K., The nonuniqueness of the tangent cones at infinity of Ricci-flat manifolds, Geom. Topol. 21 (2017), 2683-2723, arXiv:1503.07278.
  33. Honda S., A weakly second-order differential structure on rectifiable metric measure spaces, Geom. Topol. 18 (2014), 633-668, arXiv:1112.0099.
  34. Honda S., New differential operator and non-collapsed ${\rm RCD}$ spaces, Geom. Topol., to appear, arXiv:1905.00123.
  35. Honda S., Sun S., Zhang R., A note on the collapsing geometry of hyperkähler four manifolds, Sci. China Math. 62 (2019), 2195-2210.
  36. Kapovitch V., Perelman's stability theorem, in Surveys in Differential Geometry, Vol. XI, Surv. Differ. Geom., Vol. 11, Int. Press, Somerville, MA, 2007, 103-136, arXiv:math.DG/0703002.
  37. Kapovitch V., Mondino A., On the topology and the boundary of $N$-dimensional ${\rm RCD}(K,N)$ spaces, arXiv:11907.02614.
  38. Kell M., Mondino A., On the volume measure of non-smooth spaces with Ricci curvature bounded below, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), 593-610, arXiv:1607.02036.
  39. Kitabeppu Y., A sufficient condition to a regular set being of positive measure on ${\rm RCD}$ spaces, Potential Anal. 51 (2019), 179-196, arXiv:1708.04309.
  40. Lott J., Villani C., Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), 903-991, arXiv:math.DG/0412127.
  41. Menguy X., Examples of nonpolar limit spaces, Amer. J. Math. 122 (2000), 927-937.
  42. Menguy X., Noncollapsing examples with positive Ricci curvature and infinite topological type, Geom. Funct. Anal. 10 (2000), 600-627.
  43. Menguy X., Examples of strictly weakly regular points, Geom. Funct. Anal. 11 (2001), 124-131.
  44. Mondino A., Naber A., Structure theory of metric measure spaces with lower Ricci curvature bounds, J. Eur. Math. Soc. 21 (2019), 1809-1854, arXiv:1405.2222.
  45. Naber A., The geometry of Ricci curvature, in Proceedings of the International Congress of Mathematicians - Seoul 2014, Vol. II, Kyung Moon Sa, Seoul, 2014, 911-937.
  46. Perelman G., Spaces with curvature bounded below II, Preprint.
  47. Petrunin A., Alexandrov meets Lott-Villani-Sturm, Müster J. Math. 4 (2011), 53-64, arXiv:1003.5948.
  48. Rajala T., Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations 44 (2012), 477-494, arXiv:1107.4842.
  49. Sturm K.-T., On the geometry of metric measure spaces. I, Acta Math. 196 (2006), 65-131.
  50. Sturm K.-T., On the geometry of metric measure spaces. II, Acta Math. 196 (2006), 133-177.
  51. Villani C., Optimal transport: old and new, Grundlehren der Mathematischen Wissenschaften, Vol. 338, Springer-Verlag, Berlin, 2009.bibitemYYamaguchi T., Collapsing and pinching under a lower curvature bound, Ann. of Math. 133 (1991), 317-357.
  52. Yamaguchi T., Collapsing and pinching under a lower curvature bound, Ann. of Math. 133 (1991), 317-357.
  53. Zhang H.-C., Zhu X.-P., Ricci curvature on Alexandrov spaces and rigidity theorems, Comm. Anal. Geom. 18 (2010), 503-553, arXiv:0912.3190.
  54. Zhang H.-C., Zhu X.-P., Weyl's law on ${\rm RCD}^*(K, N)$ metric measure spaces,Comm. Anal. Geom. 27 (2019), 1869-1914, arXiv:1701.01967.

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