Pith. sign in

REVIEW 1 major objections 98 references

Topology of two-dimensional collapsed spaces with lower Ricci bounds

T0 review · 1 major / 0 minor · reviewed 2026-06-26 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Collapsed metric measure spaces with lower Ricci bounds and essential dimension two are topological surfaces, possibly with boundary.

desk verdict The paper proves collapsed metric measure spaces with lower Ricci bound and essential dimension two are topological surfaces, possibly with boundary. read the letter →

arxiv 2606.19189 v1 pith:3R5445Y3 submitted 2026-06-17 math.DG

Elia Bruè , Qin Deng This is my paper · ORCID
classification math.DG
keywords Riccicurvaturecollapsedspacesmetricmeasuretopologicalsurfacesessentialdimensionlowerbounddifferentialgeometry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper shows that metric measure spaces which collapse while keeping Ricci curvature bounded from below and having essential dimension two must be homeomorphic to surfaces that may have boundary. This matters because such spaces appear as limits of sequences of Riemannian manifolds with curvature controls, and knowing their topology rules out many singular behaviors that could otherwise occur in the collapsed regime. A sympathetic reader would see this as closing a gap between the non-collapsed case, where dimension matches the topological dimension, and the collapsed setting where the two can differ. The argument proceeds by showing that the lower Ricci bound prevents the formation of topologies incompatible with a surface structure once the essential dimension is fixed at two.

What carries the argument

The notion of essential dimension two together with the lower Ricci bound on a collapsed metric measure space, which together force the underlying topology to be that of a surface with boundary.

What would settle it

Construction of a metric measure space that collapses, satisfies a lower Ricci bound, has essential dimension two, yet fails to be homeomorphic to any surface with boundary.

Watch

Extended reading notes

Core claim

We prove that collapsed metric measure spaces with Ricci curvature bounded below and essential dimension two are topological surfaces, possibly with boundary.

Load-bearing premise

The spaces must have essential dimension exactly two while remaining collapsed under the lower Ricci bound.

Editorial extensions

If this is right

  • Any such space is homeomorphic to a 2-manifold with boundary and therefore admits a triangulation.
  • Local charts exist that make the space look like the plane or half-plane away from a controlled set.
  • The topology is stable under small perturbations that preserve the Ricci lower bound and essential dimension.
  • Limits of sequences of 2-dimensional Riemannian manifolds with uniform lower Ricci bound must be surfaces if they collapse.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The result may extend to questions of stability of topology under Gromov-Hausdorff convergence in the collapsed setting.
  • It suggests that essential dimension provides a useful way to stratify the possible topologies of Ricci limit spaces.
  • Similar arguments could be tested in higher essential dimensions to see where the surface conclusion stops holding.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove that collapsed metric measure spaces with Ricci curvature bounded below and essential dimension two are topological surfaces, possibly with boundary.

Significance. If correct, the result would classify the topology of a class of collapsed spaces under Ricci lower bounds in dimension two, extending known results from non-collapsed Riemannian manifolds to the metric measure space setting and potentially informing the study of limits and singularities.

major comments (1)
  1. [Abstract] The provided text consists only of the abstract stating the claim, with no definitions of 'collapsed', 'essential dimension two', the precise Ricci bound, or any proof structure, derivations, or technical arguments; this prevents assessment of whether the topological conclusion follows from the hypotheses.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The single major comment concerns the content provided for review. We address it directly below. The full manuscript is available on arXiv:2606.19189 and contains all definitions, statements, and arguments.

read point-by-point responses

  1. Referee: [Abstract] The provided text consists only of the abstract stating the claim, with no definitions of 'collapsed', 'essential dimension two', the precise Ricci bound, or any proof structure, derivations, or technical arguments; this prevents assessment of whether the topological conclusion follows from the hypotheses.

    Authors: The comment is accurate regarding the text initially provided. The complete manuscript on arXiv supplies the definitions of collapsed metric measure spaces (in the sense of Gromov-Hausdorff limits with dimension drop), the notion of essential dimension two, the precise lower Ricci bound (typically Ric ≥ -(n-1) in the appropriate normalized sense for the dimension), and the full proof that such spaces are topological surfaces, possibly with boundary. We will ensure the journal receives the complete manuscript rather than an abstract-only excerpt. No alterations to the mathematical content of the paper are required. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper states a direct topological classification theorem for collapsed metric measure spaces under a lower Ricci bound and essential dimension two. The abstract and available text supply no equations, fitted parameters, self-referential definitions, or load-bearing self-citations that reduce the claimed result to its inputs by construction. As a standard proof in differential geometry relying on external geometric definitions, the derivation is self-contained.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the standard axioms of metric measure spaces, the definition of lower Ricci curvature bounds in the synthetic sense, and the notion of essential dimension two. No free parameters or invented entities are visible in the abstract. Because the full text is unavailable, the ledger cannot be audited beyond these background items.

assumptions (1)
  • domain assumption Metric measure spaces admit a well-defined notion of essential dimension and collapsed behavior under lower Ricci bounds.
    Invoked by the statement of the theorem in the abstract.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Topology of two-dimensional collapsed spaces with lower Ricci bounds." pith.science (2026). https://pith.science/paper/3R5445Y3

@misc{pith2026260619189,
  author       = {Pith},
  title        = {Pith review of: Topology of two-dimensional collapsed spaces with lower Ricci bounds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3R5445Y3}},
  note         = {Machine review of arXiv:2606.19189}
}
read the original abstract

We prove that collapsed metric measure spaces with Ricci curvature bounded below and essential dimension two are topological surfaces, possibly with boundary.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

98 extracted references · 8 canonical work pages

  1. [1]

    Ambrosio

    L. Ambrosio. Heat flow and curvature-dimension bounds in metric measure spaces. Proceedings of the International Congress of Mathematicians, Rio de Janeiro 2018. Vol. I. Plenary lectures, 301--340. World Sci. Publ., Hackensack, NJ, 2018

    2018

  2. [2]

    Ambrosio, N

    L. Ambrosio, N. Gigli, and G. Savaré. Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163 (2014), no. 7, 1405-1490

    2014

  3. [3]

    Ambrosio, A

    L. Ambrosio, A. Mondino, and G. Savaré. Nonlinear diffusion equations and curvature conditions in metric measure spaces. Mem. Amer. Math. Soc. 262 (2019), no. 1270, v+121 pp

    2019

  4. [4]

    Anderson

    M.-T. Anderson. Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102 (1990), no. 2, 429–445

    1990

  5. [5]

    Bacher and K.-T

    K. Bacher and K.-T. Sturm. Localization and tensorization properties of the curvature-dimension condition for metric measure spaces. J. Funct. Anal. 259 (2010), no. 1, 28--56

    2010

  6. [6]

    Brena, N

    C. Brena, N. Gigli, S. Honda, and X. Zhu. Weakly non-collapsed RCD spaces are strongly non-collapsed. J. Reine Angew. Math. 794 (2023), 215--252

    2023

  7. [7]

    Breuillard, B

    E. Breuillard, B. Green, and T. Tao

  8. [8]

    E. Bruè, A. Naber, and D. Semola. Boundary regularity and stability for spaces with Ricci bounded below. Invent. Math. 228 (2022), no. 2, 777--891

    2022

Show all 98 references
  1. [9]

    E. Bruè, A. Pigati, and D. Semola. Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below. Preprint arXiv:2405.03839 (2024)

    2024

  2. [10]

    E. Bruè, E. Pasqualetto, and D. Semola

  3. [11]

    Brué and D

    E. Brué and D. Semola. Constancy of the dimension for RCD (K,N) spaces via regularity of Lagrangian flows . Comm. Pure Appl. Math. 73 (2020), no. 6, 1141--1204

    2020

  4. [12]

    Burago, Y

    D. Burago, Y. Burago, and S. Ivanov

  5. [13]

    Cavalletti

    F. Cavalletti. Monge problem in metric measure spaces with Riemannian curvature-dimension condition. Nonlinear Anal. 99 (2014), 136--151

    2014

  6. [14]

    Cavalletti and E

    F. Cavalletti and E. Milman. The globalization theorem for the curvature-dimension condition. Invent. Math. 226 (2021), no. 1, 1--137

    2021

  7. [15]

    Cheeger and T.-H

    J. Cheeger and T.-H. Colding. Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. of Math. (2) 144 (1996), no. 1, 189--237

    1996

  8. [16]

    Cheeger and T.-H

    J. Cheeger and T.-H. Colding. On the structure of spaces with Ricci curvature bounded below. I. J. Differential Geom. 46 (1997), no. 3, 406--480

    1997

  9. [17]

    Cheeger and T.-H

    J. Cheeger and T.-H. Colding. On the structure of spaces with Ricci curvature bounded below. II. J. Differential Geom. 54 (2000), no. 1, 13--35

    2000

  10. [18]

    Cheeger, T.-H

    J. Cheeger, T.-H. Colding, and G. Tian. On the singularities of spaces with bounded Ricci curvature. Geom. Funct. Anal. 12 (2002), no. 5, 873–914

    2002

  11. [19]

    Cheeger, K

    J. Cheeger, K. Fukaya, and M. Gromov. Nilpotent structures and invariant metrics on collapsed manifolds. J. Amer. Math. Soc. 5 (1992), no. 2, 327--372

    1992

  12. [20]

    Cheeger and D

    J. Cheeger and D. Gromoll. The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differential Geometry 6 (1971/72), 119--128

    1971

  13. [21]

    Cheeger and M

    J. Cheeger and M. Gromov. Collapsing Riemannian manifolds while keeping their curvature bounded. I. J. Differential Geom. 23 (1986), no. 3, 309--346

    1986

  14. [22]

    Cheeger and M

    J. Cheeger and M. Gromov. Collapsing Riemannian manifolds while keeping their curvature bounded. II. J. Differential Geom. 32 (1990), no. 1, 269--298

    1990

  15. [23]

    Cheeger, W

    J. Cheeger, W. Jiang, and A. Naber. Rectifiability of singular sets of noncollapsed limit spaces with Ricci curvature bounded below. Ann. of Math. (2) 193 (2021), no. 2, 407--538

    2021

  16. [24]

    Cheeger and A

    J. Cheeger and A. Naber. Regularity of Einstein manifolds and the codimension 4 conjecture. Ann. of Math. (2) 182 (2015), no. 3, 1093--1165

    2015

  17. [25]

    Cheeger and G

    J. Cheeger and G. Tian. Curvature and injectivity radius estimates for Einstein 4-manifolds. J. Amer. Math. Soc. 19 (2006), no. 2, 487--525

    2006

  18. [26]

    L. Chen. 2-regular points in Ricci limit spaces. Preprint arXiv:2406.08306 (2024)

    2024

  19. [27]

    J. S. Calcut, H. C. King, and L. C. Siebenmann. Connected sum at infinity and Cantrell--Stallings hyperplane unknotting. Rocky Mountain J. Math. 42 (2012), no. 6, 1803--1862

    2012

  20. [28]

    urzeste Wege und Totalkr\

    S. Cohn-Vossen, K\"urzeste Wege und Totalkr\"ummung auf Fl\"achen. Compositio Math. 2 (1935), 69--133

    1935

  21. [29]

    T.-H. Colding. Shape of manifolds with positive Ricci curvature. Invent. Math. 124 (1996), no. 1-3, 175--191

    1996

  22. [30]

    T.-H. Colding. Ricci curvature and volume convergence. Ann. of Math. (2) 145 (1997), no. 3, 477--501

    1997

  23. [31]

    Colding and A

    T.-H. Colding and A. Naber. Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann. of Math. (2) 176 (2012), no. 2, 1173--1229

    2012

  24. [32]

    Colding and A

    T.-H. Colding and A. Naber. Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications. Geom. Funct. Anal. 23 (2013), no. 1, 134--148

    2013

  25. [33]

    Cucinotta, M

    A. Cucinotta, M. Magnabosco, and D. Semola. New topological restrictions for spaces with nonnegative Ricci curvature. Preprint arXiv:2601.14231 (2026)

    2026

  26. [34]

    De Philippis and N

    G. De Philippis and N. Gilgi. From volume cone to metric cone in the nonsmooth setting. Geom. Funct. Anal. 26 (2016), no. 6, 1526--1587

    2016

  27. [35]

    De Philippis and N

    G. De Philippis and N. Gilgi. Non-collapsed spaces with Ricci curvature bounded from below. J. Éc. polytech. Math. 5 (2018), 613--650

    2018

  28. [36]

    Q. Deng. Hölder continuity of tangent cones and non-branching in RCD (K, N) spaces. University of Toronto (Canada), 2021

    2021

  29. [37]

    Q. Deng. Hölder continuity of tangent cones in RCD (K, N) spaces and applications to nonbranching . Geom. Topol. 29 (2025), no. 2, 1037--1114

    2025

  30. [38]

    Deng and A

    Q. Deng and A. Pigati. Topology of non-collapsed three-dimensional RCD spaces. Preprint arXiv:2512.21961 (2025)

    2025

  31. [39]

    Q. Deng, J. Santos-Rod\'iguez, S. Zamora, and X. Zhao

  32. [40]

    Eguchi and A.-J

    T. Eguchi and A.-J. Hanson. Self-dual solutions to Euclidean gravity. Ann. Physics 120 (1979), no. 1, 82–106

    1979

  33. [41]

    Erbar, K

    M. Erbar, K. Kuwada, and K.-T. Sturm. On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces. Invent. Math. 201 (2015), no. 3, 993--1071

    2015

  34. [42]

    K. Fukaya. Collapsing of Riemannian manifolds and eigenvalues of Laplace operator. Invent. Math. 87 (1987), no. 3, 517–547

    1987

  35. [43]

    K. Fukaya. Collapsing Riemannian manifolds to ones of lower dimensions. J. Differential Geom. 25 (1987), no. 1, 139--156

    1987

  36. [44]

    K. Fukaya. A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters. J. Differential Geom. 28 (1988), no. 1, 1--21

    1988

  37. [45]

    K. Fukaya. Collapsing Riemannian manifolds to ones of lower dimensions. II. J. Math. Soc. Japan 41 (1989), no. 2, 333--356

    1989

  38. [46]

    N. Gigli. The splitting theorem in non-smooth context. Preprint arXiv:1302.5555 (2013)

    2013 arXiv

  39. [47]

    N. Gigli. On the differential structure of metric measure spaces and applications. Mem. Amer. Math. Soc., 236.1113 (2015)

    2015

  40. [48]

    N. Gigli. De Giorgi and Gromov working together. Preprint arXiv:2306.14604 (2023)

    2023

  41. [49]

    Gigli, A

    N. Gigli, A. Mondino, and G. Savaré. Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows. Proc. Lond. Math. Soc. (3) 111 (2015), no. 5 1071--1129

    2015

  42. [50]

    M. Gromov. Almost flat manifolds. J. Differential Geom. 13 (1978), no. 2, 231--241

    1978

  43. [51]

    M. Gromov. Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. No. 53 (1981), 53--73

    1981

  44. [52]

    S. Honda. On low-dimensional Ricci limit spaces. Nagoya Math. J. 209 (2013), 1--22

    2013

  45. [53]

    S. Honda. New differential operator and noncollapsed RCD spaces. Geom. Topol. 24 (2020), no. 4, 2127--2148

    2020

  46. [54]

    E. Hupp, A. Naber, and K.-H. Wang. Lower Ricci curvature and nonexistence of manifold structure. Geom. Topol. 29 (2025), no. 1, 443--477

    2025

  47. [55]

    Jiang and A

    W. Jiang and A. Naber. L^2 curvature bounds on manifolds with bounded Ricci curvature. Ann. of Math. (2) 193 (2021), no. 1, 107--222

    2021

  48. [56]

    Johansson

    I. Johansson. Topologische Untersuchungen über unverzweigte Überlagerungsflächen. Skriften norske Vidensk.-Akad. Oslo Math.-natur. Kl. 1 (1931), 1-69

    1931

  49. [57]

    Kell and A

    M. Kell and A. Mondino. On the volume measure of non-smooth spaces with Ricci curvature bounded below. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 2, 593--610

    2018

  50. [58]

    Kapovitch and A

    V. Kapovitch and A. Mondino. On the topology and the boundary of N -dimensional RCD (K,N) spaces . Geom. Topol. 25 (2021), no. 1, 445--495

    2021

  51. [59]

    Kapovitch and B

    V. Kapovitch and B. Wilking

  52. [60]

    Ketterer

    C. Ketterer. Cones over metric measure spaces and the maximal diameter theorem. J. Math. Pures Appl. 103 (2015), no. 5, 1228--1275

    2015

  53. [61]

    Kitabeppu

    Y. Kitabeppu. A sufficient condition to a regular set being of positive measure on RCD spaces . Potential Anal. 51 (2019), no. 2, 179--196

    2019

  54. [62]

    Kitabeppu and S

    Y. Kitabeppu and S. Lakzian. Characterization of low dimensional RCD^*(K,N) spaces. Anal. Geom. Metr. Spaces 4 (2016), no. 1, 187--215

    2016

  55. [63]

    Z. Li. The globalization theorem for CD (K,N) on locally finite spaces . Ann. Mat. Pura Appl. (4) 203 (2024), no. 1, 49--70

    2024

  56. [64]

    G. Liu. 3-manifolds with nonnegative Ricci curvature. Invent. Math. 193 (2013), no. 2, 367--375

    2013

  57. [65]

    Lytchak and S

    A. Lytchak and S. Stadler. Ricci curvature in dimension 2. J. Eur. Math. Soc. (JEMS) 25 (2023), no. 3, 845--867

    2023

  58. [66]

    Lott and C

    J. Lott and C. Villani. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169 (2009), no. 3, 903--991

    2009

  59. [67]

    X. Menguy. Noncollapsing examples with positive Ricci curvature and infinite topological type. Geom. Funct. Anal. 10 (2000), no. 3, 600--627

    2000

  60. [68]

    J. Milnor. A note on curvature and fundamental group. J. Differential Geometry 2 (1968), 1--7

    1968

  61. [69]

    Mondino and A

    A. Mondino and A. Naber. Structure theory of metric measure spaces with lower Ricci curvature bounds. J. Eur. Math. Soc. (JEMS) 21 (2019), no. 6, 1809--1854

    2019

  62. [70]

    Mondino and G

    A. Mondino and G. Wei. On the universal cover and the fundamental group of an RCD ^*(K,N) -space . J. Reine Angew. Math. 753 (2019), 211--237

    2019

  63. [71]

    R. L. Moore. On the foundations of plane analysis situs. Trans. Amer. Math. Soc. 17 (1916), no. 2, 131--164

    1916

  64. [72]

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

    2020

  65. [73]

    Naber and R

    A. Naber and R. Zhang. Topology and -regularity theorems on collapsed manifolds with Ricci curvature bounds. Geom. Topol. 20 (2016), no. 5, 2575--2664

    2016

  66. [74]

    S. Nadler. Continuum theory. An introduction. Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker Inc., New York, 1992

    1992

  67. [75]

    S.-I. Ohta. On the measure contraction property of metric measure spaces. Comment. Math. Helv. 82 (2007), no. 4, 805--828

    2007

  68. [76]

    Y. Otsu. On manifolds of positive Ricci curvature with large diameter. Math. Z. 206 (1991), no. 2, 255--264

    1991

  69. [77]

    Pan and G

    J. Pan and G. Wei. Examples of Ricci limit spaces with non-integer Hausdorff dimension. Geom. Funct. Anal. 32 (2022), no. 3, 676--685

    2022

  70. [78]

    Pan and G

    J. Pan and G. Wei. Semi-local simple connectedness of non-collapsing Ricci limit spaces. J. Eur. Math. Soc. (JEMS) 24 (2022), no. 12, 4027--4062

    2022

  71. [79]

    Perel'man

    G. Perel'man. Alexandrov spaces with curvatures bounded from below. II. Preprint (1991)

    1991

  72. [80]

    Perel'man

    G. Perel'man. Elements of Morse theory on Aleksandrov spaces. (Russian) ; translated from Algebra i Analiz 5 (1993), no. 1, 232--241 St. Petersburg Math. J. 5 (1994), no. 1, 205--213

    1993

  73. [81]

    Petrunin

    A. Petrunin. Parallel transportation for Alexandrov space with curvature bounded below. Geom. Funct. Anal. 8 (1998), no. 1, 123--148

    1998

  74. [82]

    Schoen and S

    R. Schoen and S. T. Yau. Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature. Seminar on Differential Geometry, pp. 209--228, Ann. of Math. Stud., No. 102, Princeton Univ. Press, Princeton, NJ, 1982

    1982

  75. [83]

    M. Simon. Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below. J. Reine Angew. Math. 662 (2012), 59--94

    2012

  76. [84]

    Simon and P

    M. Simon and P. M. Topping. Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces. Geom. Topol. 25 (2021), no. 2, 913--948

    2021

  77. [85]

    Simon and P

    M. Simon and P. M. Topping. Local control on the geometry in 3D Ricci flow. J. Differential Geom. 122 (2022), no. 3, 467--518

    2022

  78. [86]

    Sormani and G

    C. Sormani and G. Wei. Hausdorff convergence and universal covers. Trans. Amer. Math. Soc. 353 (2001), no. 9, 3585--3602

    2001

  79. [87]

    Sormani and G

    C. Sormani and G. Wei. Universal covers for Hausdorff limits of noncompact spaces. Trans. Amer. Math. Soc. 356 (2004), no. 3, 1233--1270

    2004

  80. [88]

    Stillwell

    J. Stillwell. Classical topology and combinatorial group theory. Grad. Texts in Math., 72 Springer-Verlag, New York, 1993, xii+334 pp

    1993

  81. [89]

    K.-T. Sturm. On the geometry of metric measure spaces. I. Acta Math. 196 (2006), no. 1, 65--131

    2006

  82. [90]

    K.-T. Sturm. On the geometry of metric measure spaces. II. Acta Math. 196 (2006), no. 1, 133--177

    2006

  83. [91]

    K.-T. Sturm. Metric measure spaces and synthetic Ricci bounds: fundamental concepts and recent developments. European Congress of Mathematics. Proceedings of the 8th congress, 125--159. EMS Press, Berlin, 2023

    2023

  84. [92]

    V. Timorin. Moore's theorem. Preprint arXiv:1001.5140 (2010)

    2010 arXiv

  85. [93]

    J. Wang. RCD ^*(K,N) spaces are semi-locally simply connected . J. Reine Angew. Math. 806 (2024), 1--7

    2024

  86. [94]

    J. Wang. Ricci limit spaces are semi-locally simply connected. J. Differential Geom. 128 (2024), no. 3, 1301--1314

    2024

  87. [95]

    Yamaguchi

    T. Yamaguchi. Manifolds of almost nonnegative Ricci curvature. J. Differential Geom. 28 (1988), no. 1, 157--167

    1988

  88. [96]

    Yamaguchi

    T. Yamaguchi. Collapsing and pinching under a lower curvature bound. Ann. of Math. 133 (1991), 317--357

    1991

  89. [97]

    S. Zhou. Examples of Ricci limit spaces with infinite holes. Preprint arXiv:2404.00619 (2024)

    2024

  90. [98]

    S.-H. Zhu. A finiteness theorem for Ricci curvature in dimension three. J. Differential Geom. 37 (1993), no. 3, 711--727

    1993

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.