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

SIGMA 16 (2020), 123, 25 pages      arXiv:2008.12419
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing

Shaosai Huang a, Xiaochun Rong b and Bing Wang c
a) Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA
b) Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA
c) Institute of Geometry and Physics, and School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui Province, 230026, China

Received August 30, 2020, in final form November 23, 2020; Published online November 30, 2020

We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kähler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.

Key words: almost flat manifold; collapsing geometry; locally bounded Ricci covering geometry; nilpotent Killing structure; Ricci flow.

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  1. Bamler R.H., Zhang Q.S., Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature, Adv. Math. 319 (2017), 396-450, arXiv:1501.01291.
  2. Bamler R.H., Zhang Q.S., Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature - Part II, Calc. Var. Partial Differential Equations 58 (2019), 49, 14 pages, arXiv:1506.03154.
  3. Bochner S., Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946), 776-797.
  4. Buser P., Karcher H., Gromov's almost flat manifolds, Astérisque 81 (1981), 148 pages.
  5. Buyalo S.V., Collapsing manifolds of nonpositive curvature. I, Leningrad Math. J. 1 (1990), 1135-1155.
  6. Buyalo S.V., Collapsing manifolds of nonpositive curvature. II, Leningrad Math. J. 1 (1990), 1371-1399.
  7. Cai Q., Rong X., Collapsing construction with nilpotent structures, Geom. Funct. Anal. 18 (2009), 1503-1524.
  8. Cao J., Cheeger J., Rong X., Splittings and Cr-structures for manifolds with nonpositive sectional curvature, Invent. Math. 144 (2001), 139-167.
  9. Cao J., Cheeger J., Rong X., Local splitting structures on nonpositively curved manifolds and semirigidity in dimension 3, Comm. Anal. Geom. 12 (2004), 389-415.
  10. Cavalletti F., Mondino A., Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds, Invent. Math. 208 (2017), 803-849, arXiv:1502.06465.
  11. Chau A., Tam L.-F., Yu C., Pseudolocality for the Ricci flow and applications, Canad. J. Math. 63 (2011), 55-85, arXiv:math.DG/0701153.
  12. Cheeger J., Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61-74.
  13. Cheeger J., Colding T.H., On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), 406-480.
  14. Cheeger J., Fukaya K., Gromov M., Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992), 327-372.
  15. Cheeger J., Gromoll D., The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971), 119-128.
  16. Cheeger J., Gromov M., Collapsing Riemannian manifolds while keeping their curvature bounded. I, J. Differential Geom. 23 (1986), 309-346.
  17. Cheeger J., Gromov M., Collapsing Riemannian manifolds while keeping their curvature bounded. II, J. Differential Geom. 32 (1990), 269-298.
  18. Cheeger J., Jiang W., Naber A., Rectifiability of singular sets in noncollapsed spaces with Ricci curvature bounded below, arXiv:1805.07988.
  19. Cheeger J., Rong X., Collapsed Riemannian manifolds with bounded diameter and bounded covering geometry, Geom. Funct. Anal. 5 (1995), 141-163.
  20. Cheeger J., Rong X., Existence of polarized $F$-structures on collapsed manifolds with bounded curvature and diameter, Geom. Funct. Anal. 6 (1996), 411-429.
  21. Chen L., Rong X., Xu S., Quantitative volume space form rigidity under lower Ricci curvature bound II, Trans. Amer. Math. Soc. 370 (2018), 4509-4523, arXiv:1606.05709.
  22. Chen L., Rong X., Xu S., Quantitative volume space form rigidity under lower Ricci curvature bound I, J. Differential Geom. 113 (2019), 227-272, arXiv:1604.06986.
  23. Chen X., Wang B., Space of Ricci flows I, Comm. Pure Appl. Math. 65 (2012), 1399-1457, arXiv:0902.1545.
  24. Chen X., Wang B., Remarks of weak-compactness along Kähler Ricci flow, in Proceedings of the Seventh International Congress of Chinese Mathematicians, Vol. II, Adv. Lect. Math. (ALM), Vol. 44, Int. Press, Somerville, MA, 2019, 203-233, arXiv:1605.01374.
  25. Chen X., Wang B., Space of Ricci flows (II) - Part B: Weak compactness of the flows, J. Differential Geom. 116 (2020), 1-123, arXiv:1405.6797.
  26. Colding T.H., Shape of manifolds with positive Ricci curvature, Invent. Math. 124 (1996), 175-191.
  27. Colding T.H., Ricci curvature and volume convergence, Ann. of Math. 145 (1997), 477-501.
  28. 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.
  29. Dai X., Wang X., Wei G., On the variational stability of Kähler-Einstein metrics, Comm. Anal. Geom. 15 (2007), 669-693.
  30. Dai X., Wei G., Ye R., Smoothing Riemannian metrics with Ricci curvature bounds, Manuscripta Math. 90 (1996), 49-61, arXiv:dg-ga/9411014.
  31. Fang F., Rong X., Positive pinching, volume and second Betti number, Geom. Funct. Anal. 9 (1999), 641-674.
  32. Fang F., Rong X., The second twisted Betti number and the convergence of collapsing Riemannian manifolds, Invent. Math. 150 (2002), 61-109.
  33. Fukaya K., Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987), 517-547.
  34. Fukaya K., Collapsing Riemannian manifolds to ones of lower dimensions, J. Differential Geom. 25 (1987), 139-156.
  35. Fukaya K., A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom. 28 (1988), 1-21.
  36. Fukaya K., Collapsing Riemannian manifolds to ones with lower dimension. II, J. Math. Soc. Japan 41 (1989), 333-356.
  37. Fukaya K., Metric Riemannian geometry, in Handbook of Differential Geometry, Vol. II, Elsevier/North-Holland, Amsterdam, 2006, 189-313.
  38. Greene R.E., Wu H., Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988), 119-141.
  39. Gromov M., Almost flat manifolds, J. Differential Geometry 13 (1978), 231-241.
  40. Gromov M., Structures métriques pour les variétés riemanniennes, Textes Mathématiques, Vol. 1, CEDIC, Paris, 1981.
  41. Gross M., Tosatti V., Zhang Y., Collapsing of abelian fibered Calabi-Yau manifolds, Duke Math. J. 162 (2013), 517-551.
  42. Gross M., Tosatti V., Zhang Y., Gromov-Hausdorff collapsing of Calabi-Yau manifolds, Comm. Anal. Geom. 24 (2016), 93-113, arXiv:1304.1820.
  43. Gross M., Wilson P.M.H., Large complex structure limits of $K3$ surfaces, J. Differential Geom. 55 (2000), 475-546, arXiv:math.DG/0008018.
  44. Hamilton R.S., Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), 255-306.
  45. Hamilton R.S., The formation of singularities in the Ricci flow, in Surveys in Differential Geometry, Vol. II (Cambridge, MA, 1993), Int. Press, Cambridge, MA, 1995, 7-136.
  46. Hein H.-J., Sun S., Viaclovsky J., Zhang R., Nilpotent structures and collapsing Ricci-flat metrics on $K3$ surfaces, arXiv:1807.09367.
  47. Hochard R., Short-time existence of the Ricci flow on complete, non-collapsed $3$-manifolds with Ricci curvature bounded from below, arXiv:1603.08726.
  48. Huang H., Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry, Front. Math. China 15 (2020), 69-89, arXiv:2002.07383.
  49. Huang H., Kong L., Rong X., Xu S., Collapsed manifolds with Ricci bounded covering geometry, Trans. Amer. Math. Soc. 373 (2020), 8039-8057, arXiv:1808.03774.
  50. Huang H., Rong X., Collapsed manifolds with Ricci curvature and local rewinding volume bounded below, in preparation.
  51. Huang S., On the long-time behavior of immortal Ricci flows, arXiv:1908.05410.
  52. Huang S., Notes on Ricci flows with collapsing initial data (I): Distance distortion, Trans. Amer. Math. Soc. 373 (2020), 4389-4414, arXiv:1808.07394.
  53. Huang S., Wang B., Rigidity of the first Betti number via Ricci flow smoothing, arXiv:2004.09762.
  54. Huang S., Wang B., Ricci flow smoothing for locally collapsing manifolds, arXiv:2008.09956.
  55. Kapovitch V., Mixed curvature almost flat manifolds, arXiv:1911.09212.
  56. Kapovitch V., Li N., On dimensions of tangent cones in limit spaces with lower Ricci curvature bounds, J. Reine Angew. Math. 742 (2018), 263-280, arXiv:1506.02949.
  57. Kapovitch V., Wilking B., Structure of fundamental groups of manifolds with Ricci curvature bounded below, arXiv:1105.5955.
  58. Kontsevich M., Soibelman Y., Homological mirror symmetry and torus fibrations, in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001, 203-263, arXiv:math.SG/0011041.
  59. Ledrappier F., Wang X., An integral formula for the volume entropy with applications to rigidity, J. Differential Geom. 85 (2010), 461-477, arXiv:0911.0370.
  60. Li Y., SYZ conjecture for Calabi-Yau hypersurfaces in the Fermat family, arXiv:1912.02360.
  61. Lott J., Some geometric properties of the Bakry-Émery-Ricci tensor, Comment. Math. Helv. 78 (2003), 865-883, arXiv:math.DG/0211065.
  62. Lott J., Dimensional reduction and the long-time behavior of Ricci flow, Comment. Math. Helv. 85 (2010), 485-534, arXiv:0711.4063.
  63. Lu P., A local curvature bound in Ricci flow, Geom. Topol. 14 (2010), 1095-1110, arXiv:0906.3784.
  64. Molino P., Riemannian foliations, Progress in Mathematics, Vol. 73, Birkhäuser Boston, Inc., Boston, MA, 1988.
  65. Naber A., Tian G., Geometric structures of collapsing Riemannian manifolds II, J. Reine Angew. Math. 744 (2018), 103-132, arXiv:0804.2275.
  66. Naber A., Zhang R., Topology and $\varepsilon$-regularity theorems on collapsed manifolds with Ricci curvature bounds, Geom. Topol. 20 (2016), 2575-2664, arXiv:1412.1326.
  67. O'Neill B., The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459-469.
  68. Perelman G., The entropy formula for the Ricci flow and its applications, arXiv:math.DG/0211159.
  69. Petersen P., Wei G., Ye R., Controlled geometry via smoothing, Comment. Math. Helv. 74 (1999), 345-363, arXiv:dg-ga/9508012.
  70. Petrunin A., Rong X., Tuschmann W., Collapsing vs. positive pinching, Geom. Funct. Anal. 9 (1999), 699-735.
  71. Petrunin A., Tuschmann W., Diffeomorphism finiteness, positive pinching, and second homotopy, Geom. Funct. Anal. 9 (1999), 736-774.
  72. Rong X., The existence of polarized $F$-structures on volume collapsed $4$-manifolds, Geom. Funct. Anal. 3 (1993), 474-501.
  73. Rong X., The limiting eta invariants of collapsed three-manifolds, J. Differential Geom. 37 (1993), 535-568.
  74. Rong X., Rationality of geometric signatures of complete $4$-manifolds, Invent. Math. 120 (1995), 513-554.
  75. Rong X., Collapsed manifolds with bounded sectional curvature and applications, in Surveys in Differential Geometry, Vol. XI, Surv. Differ. Geom., Vol. 11, Int. Press, Somerville, MA, 2007, 1-23.
  76. Rong X., Manifolds of Ricci curvature and local rewinding volume bounded below, Sci. Sin. Math. 48 (2018), 791-806.
  77. Rong X., A new proof of Gromov's theorem on almost, arXiv:1906.03377.
  78. Rong X., A generalized Gromov's theorem on almost flat manifolds and applications, in preparation.
  79. Ruh E.A., Almost flat manifolds, J. Differential Geometry 17 (1982), 1-14.
  80. Shi W.-X., Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), 223-301.
  81. Strominger A., Yau S.-T., Zaslow E., Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), 243-259, arXiv:hep-th/9606040.
  82. Tian G., Wang B., On the structure of almost Einstein manifolds, J. Amer. Math. Soc. 28 (2015), 1169-1209, arXiv:1202.2912.
  83. Tosatti V., Collapsing Calabi-Yau manifolds, Surv. Differ. Geom. 23 (2020), 305-337, arXiv:2003.00673.
  84. Wang B., The local entropy along Ricci flow Part A: the no-local-collapsing theorems, Camb. J. Math. 6 (2018), 267-346, arXiv:1706.08485.
  85. Yano K., Bochner S., Curvature and Betti numbers, Annals of Mathematics Studies, Vol. 32, Princeton University Press, Princeton, N.J., 1953.

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