Collapsed manifolds with Ricci bounded covering geometry

Huang, Hongzhi; Kong, Lingling; Rong, Xiaochun; Xu, Shicheng

doi:10.1090/tran/8177

Collapsed manifolds with Ricci bounded covering geometry
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by Hongzhi Huang, Lingling Kong, Xiaochun Rong and Shicheng Xu PDF

Trans. Amer. Math. Soc. 373 (2020), 8039-8057 Request permission

Abstract:

We study collapsed manifolds with Ricci bounded covering geometry, i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Applying the techniques in the Ricci flow, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on the collapsed manifolds with (sectional curvature) local bounded covering geometry, to the manifolds with (global) Ricci bounded covering geometry.

References

Michael T. Anderson, Hausdorff perturbations of Ricci-flat manifolds and the splitting theorem, Duke Math. J. 68 (1992), no. 1, 67–82. MR 1185818, DOI 10.1215/S0012-7094-92-06803-7
R. Bamler and B. Wilking, The Ricci flow on manifolds with almost non-negative curvature operators, Unpublished notes (2016).
Qingsong Cai and Xiaochun Rong, Collapsing construction with nilpotent structures, Geom. Funct. Anal. 18 (2009), no. 5, 1503–1524. MR 2481735, DOI 10.1007/s00039-009-0700-4
F. Cavalletti and A. Mondino, Almost Euclidean isoperimetric inequalities in spaces satisfying local Ricci curvature lower bounds, arxiv1703.02119v1. (2017).
Albert Chau, Luen-Fai Tam, and Chengjie Yu, Pseudolocality for the Ricci flow and applications, Canad. J. Math. 63 (2011), no. 1, 55–85. MR 2779131, DOI 10.4153/CJM-2010-076-2
Jeff Cheeger and Tobias H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1996), no. 1, 189–237. MR 1405949, DOI 10.2307/2118589
Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406–480. MR 1484888
Jeff Cheeger, Kenji Fukaya, and Mikhael Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992), no. 2, 327–372. MR 1126118, DOI 10.1090/S0894-0347-1992-1126118-X
Jeff Cheeger and Mikhael Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. I, J. Differential Geom. 23 (1986), no. 3, 309–346. MR 852159
Jeff Cheeger and Mikhael Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. II, J. Differential Geom. 32 (1990), no. 1, 269–298. MR 1064875
Bing-Long Chen and Xi-Ping Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geom. 74 (2006), no. 1, 119–154. MR 2260930
Lina Chen, Xiaochun Rong, and Shicheng Xu, Quantitative volume space form rigidity under lower Ricci curvature bound I, J. Differential Geom. 113 (2019), no. 2, 227–272. MR 4023292, DOI 10.4310/jdg/1571882427
Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects, Mathematical Surveys and Monographs, vol. 163, American Mathematical Society, Providence, RI, 2010. MR 2604955, DOI 10.1090/surv/163
Tobias H. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124 (1996), no. 1-3, 193–214. MR 1369415, DOI 10.1007/s002220050050
Tobias H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 (1997), no. 3, 477–501. MR 1454700, DOI 10.2307/2951841
Xianzhe Dai, Guofang Wei, and Rugang Ye, Smoothing Riemannian metrics with Ricci curvature bounds, Manuscripta Math. 90 (1996), no. 1, 49–61. MR 1387754, DOI 10.1007/BF02568293
Kenji Fukaya, Collapsing Riemannian manifolds to ones of lower dimensions, J. Differential Geom. 25 (1987), no. 1, 139–156. MR 873459
Kenji Fukaya, A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom. 28 (1988), no. 1, 1–21. MR 950552
Kenji Fukaya, Collapsing Riemannian manifolds to ones with lower dimension. II, J. Math. Soc. Japan 41 (1989), no. 2, 333–356. MR 984756, DOI 10.2969/jmsj/04120333
Kenji Fukaya and Takao Yamaguchi, The fundamental groups of almost non-negatively curved manifolds, Ann. of Math. (2) 136 (1992), no. 2, 253–333. MR 1185120, DOI 10.2307/2946606
M. Gromov, Almost flat manifolds, J. Differential Geometry 13 (1978), no. 2, 231–241. MR 540942, DOI 10.4310/jdg/1214434488
Mark Gross and P. M. H. Wilson, Large complex structure limits of $K3$ surfaces, J. Differential Geom. 55 (2000), no. 3, 475–546. MR 1863732
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
H. Hein, S. Sun, J. Viaclovsky, and R. Zhang, Nilpotent structures and collapsing Ricci-flat manifolds on K3 surfaces, arXiv 2341633 [math.DG].
Hongzhi Huang, Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry, Front. Math. China 15 (2020), no. 1, 69–89. MR 4074346, DOI 10.1007/s11464-020-0824-2
H. Huang and X. Rong, Collapsed manifolds with local bounded Ricci covering geometry (In preparation).
Y. Li, On collapsed Calabi-Yau fibrations, arXiv: 1706.10250v1 [math.DG] (June, 2017).
Aaron Naber and Ruobing Zhang, Topology and $\varepsilon$-regularity theorems on collapsed manifolds with Ricci curvature bounds, Geom. Topol. 20 (2016), no. 5, 2575–2664. MR 3556347, DOI 10.2140/gt.2016.20.2575
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math/0211159v1 (2002).
Peter Petersen, Guofang Wei, and Rugang Ye, Controlled geometry via smoothing, Comment. Math. Helv. 74 (1999), no. 3, 345–363. MR 1710067, DOI 10.1007/s000140050093
Xiaochun Rong, Convergence and collapsing theorems in Riemannian geometry, Handbook of geometric analysis, No. 2, Adv. Lect. Math. (ALM), vol. 13, Int. Press, Somerville, MA, 2010, pp. 193–299. MR 2743443
X. Rong, A new proof of the Gromov’s theorem on almost flat manifolds, ariXiv:1906.03377v2 (2020).
X. Rong, A generalization of Gromov’s theorem on almost flat manifolds and its application, To appear.
Ernst A. Ruh, Almost flat manifolds, J. Differential Geometry 17 (1982), no. 1, 1–14. MR 658470
Wan-Xiong Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), no. 1, 223–301. MR 1001277
Wan-Xiong Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989), no. 2, 303–394. MR 1010165
Takao Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. (2) 133 (1991), no. 2, 317–357. MR 1097241, DOI 10.2307/2944340