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Gromov–Hausdorff limits of aspherical manifolds

Xiaochun Rong

Geometry & Topology 30 (2026) 2685–2712
Bibliography
1 I Belegradek, Iterated circle bundles and infranilmanifolds, Osaka J. Math. 57 (2020) 165 MR4052634
2 E Bruè, A Naber, D Semola, Stability of tori under lower sectional curvature, Geom. Topol. 28 (2024) 3961 MR4843753
3 Q Cai, X Rong, Collapsing construction with nilpotent structures, Geom. Funct. Anal. 18 (2009) 1503 MR2481735
4 J Cheeger, T H Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. 144 (1996) 189 MR1405949
5 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, I, J. Differential Geom. 46 (1997) 406 MR1484888
6 J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, III, J. Differential Geom. 54 (2000) 37 MR1815411
7 J Cheeger, M Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded, I, J. Differential Geom. 23 (1986) 309 MR852159
8 J Cheeger, M Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded, II, J. Differential Geom. 32 (1990) 269 MR1064875
9 J Cheeger, A Naber, Regularity of Einstein manifolds and the codimension 4 conjecture, Ann. of Math. 182 (2015) 1093 MR3418535
10 J Cheeger, K Fukaya, M Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992) 327 MR1126118
11 J Cheeger, W Jiang, A Naber, Rectifiability of singular sets of noncollapsed limit spaces with Ricci curvature bounded below, Ann. of Math. 193 (2021) 407 MR4226910
12 T H Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124 (1996) 193 MR1369415
13 T H Colding, Ricci curvature and volume convergence, Ann. of Math. 145 (1997) 477 MR1454700
14 T H Colding, A Naber, Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications, Ann. of Math. 176 (2012) 1173 MR2950772
15 B Farb, S Weinberger, Isometries, rigidity and universal covers, Ann. of Math. 168 (2008) 915 MR2456886
16 F T Farrell, W C Hsiang, Topological characterization of flat and almost flat Riemannian manifolds Mn (n3, 4), Amer. J. Math. 105 (1983) 641 MR704219
17 K Fukaya, Collapsing Riemannian manifolds to ones of lower dimensions, J. Differential Geom. 25 (1987) 139 MR873459
18 K Fukaya, A compactness theorem of a set of aspherical Riemannian orbifolds, from: "A fête of topology" (editors Y Matsumoto, T Mizutani, S Morita), Academic (1988) 391 MR928409
19 K Fukaya, Collapsing Riemannian manifolds to ones with lower dimension, II, J. Math. Soc. Japan 41 (1989) 333 MR984756
20 K Fukaya, T Yamaguchi, The fundamental groups of almost non-negatively curved manifolds, Ann. of Math. 136 (1992) 253 MR1185120
21 M Gromov, Almost flat manifolds, J. Differential Geometry 13 (1978) 231 MR540942
22 M Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983) 1 MR697984
23 M Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser (2007) MR2307192
24 H Huang, Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry, Front. Math. China 15 (2020) 69 MR4074346
25 S Huang, B Wang, Ricci flow smoothing for locally collapsing manifolds, Calc. Var. Partial Differential Equations 61 (2022) 64 MR4376543
26 H Huang, L Kong, X Rong, S Xu, Collapsed manifolds with Ricci bounded covering geometry, Trans. Amer. Math. Soc. 373 (2020) 8039 MR4169681
27 S Huang, X Rong, B Wang, Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing, Symmetry Integrability Geom. Methods Appl. 16 (2020) 123 MR4180049
28 E Hupp, A Naber, K H Wang, Lower Ricci curvature and nonexistence of manifold structure, Geom. Topol. 29 (2025) 443 MR4855995
29 V Kapovitch, Regularity of limits of noncollapsing sequences of manifolds, Geom. Funct. Anal. 12 (2002) 121 MR1904560
30 V Kapovitch, Perelman’s stability theorem, from: "Surveys in differential geometry, XI" (editors J Cheeger, K Grove), Surv. Differ. Geom. 11, International (2007) 103 MR2408265
31 V Kapovitch, B Wilking, Structure of fundamental groups of manifolds with Ricci curvature bounded below, preprint (2011) arXiv:1105.5955v2
32 B Kloeckner, S Sabourau, Mixed sectional-Ricci curvature obstructions on tori, J. Topol. Anal. 12 (2020) 713 MR4139212
33 E Le Donne, Metric Lie groups : Carnot–Carathéodory spaces from the homogeneous viewpoint, 306, Springer (2025) MR4995404
34 M Nakayama, On the S1-fibred nilBott tower, Osaka J. Math. 51 (2014) 67 MR3192532
35 Y Otsu, On manifolds of positive Ricci curvature with large diameter, Math. Z. 206 (1991) 255 MR1091941
36 J Pan, The Grushin hemisphere as a Ricci limit space with curvature 1, Proc. Amer. Math. Soc. Ser. B 10 (2023) 71 MR4566173
37 J Pan, G Wei, Examples of Ricci limit spaces with non-integer Hausdorff dimension, Geom. Funct. Anal. 32 (2022) 676 MR4431126
38 G Perelman, Alexandrov spaces with curvatures bounded from below, II, preprint (1991)
39 G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002) arXiv:math/0211159
40 M S Raghunathan, Discrete subgroups of Lie groups, 68, Springer (1972) MR507234
41 X Rong, Convergence and collapsing theorems in Riemannian geometry, from: "Handbook of geometric analysis, II" (editors L Ji, P Li, R Schoen, L Simon), Adv. Lect. Math. 13, International (2010) 193 MR2743443
42 X Rong, Manifolds of Ricci curvature and local rewinding volume bounded below, Scientia Sinica Mathematica 48 (2018) 791
43 X Rong, Collapsed manifolds with local Ricci bounded covering geometry, Sci. China Math. 69 (2026) 1585 MR5030287
44 X Rong, X Yao, A generalized π2-diffeomorphism finiteness theorem, Front. Math. China 15 (2020) 399 MR4094521
45 E A Ruh, Almost flat manifolds, J. Differential Geometry 17 (1982) 1 MR658470
46 M Simon, Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below, J. Reine Angew. Math. 662 (2012) 59 MR2876261
47 C Sormani, G Wei, Hausdorff convergence and universal covers, Trans. Amer. Math. Soc. 353 (2001) 3585 MR1837249
48 J Wang, Ricci limit spaces are semi-locally simply connected, J. Differential Geom. 128 (2024) 1301 MR4810225
49 D G Wright, Contractible open manifolds which are not covering spaces, Topology 31 (1992) 281 MR1167170
50 S Xu, Precompactness of domains with lower Ricci curvature bound under Gromov–Hausdorff topology, J. Reine Angew. Math. 828 (2025) 149 MR4979238
51 S Zamora, Tori can’t collapse to an interval, Electron. Res. Arch. 29 (2021) 2637 MR4288132
52 S Zamora, Torus covers with controlled volume and diameter, Proc. Amer. Math. Soc. 154 (2026) 849 MR5016563
53 S Zhou, On the Gromov-Hausdorff limits of Tori with Ricci conditions, preprint (2023) arXiv:2309.10997
54 S Zhou, Examples of Ricci limit spaces with infinite holes, preprint (2024) arXiv:2404.00619