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Manifolds with minimal radial curvature bounded from below and big volume


Author: Valery Marenich
Journal: Trans. Amer. Math. Soc. 352 (2000), 4451-4468
MSC (1991): Primary 53C20, 53C21
DOI: https://doi.org/10.1090/S0002-9947-00-02634-9
Published electronically: June 14, 2000
MathSciNet review: 1779483
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Abstract:

We prove that a convergence in the Gromov-Hausdorff distance of manifolds with minimal radial curvature bounded from below by 1 to the standard sphere is equivalent to a volume convergence.


References [Enhancements On Off] (What's this?)

  • [C] J. Cheeger, Critical points of distance functions and applications to geometry, Lecture Notes in Math. 1504 (1991), 1-38. MR 94a:53075
  • [Cl1] T. H. Colding, Shape of manifolds with positive Ricci curvature, Invent. Math. 124 (1996), 175-191. MR 96k:53027
  • [Cl2] T. H. Colding, Large manifolds with positive Ricci curvature, Invent. Math. 124 (1996), 193-214. MR 96k:53028
  • [Cl3] T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. 145 (1997), 477-501. MR 98d:53050
  • [GP] K. Grove, P. Petersen, A radius sphere theorem, Invent. Mathematicae 112 (1993), 577 - 583. MR 94e:53034
  • [GS] K. Grove, K. Shiohama, A generalized sphere theorem, Ann. of Math. 126 (1977), 201-211. MR 58:18268
  • [M1] Y. Machigashira, Manifolds with pinched radial curvature, Proc AMS 118 (1993), 979-985. MR 93i:53038
  • [M2] Y. Machigashira, Complete open manifolds of non-negative radial curvature, Pacific J. Math. 165 (1994), 153-160. MR 95h:53054
  • [MM] V. Marenich, S. J. X. Mendonça, Manifolds with minimal radial curvature bounded from below and big radius, Indiana Univ. Math. J. 48 (1999). CMP 2000:04
  • [MS] Y. Machigashira and K. Shiohama, Riemannian manifolds with positive radial curvature, Japan J. Math. 19 (1994), 419-430. MR 95f:53080
  • [OSY] Y. Otsu, K. Shiohama, T. Yamaguchi, A new version of differentiable sphere theorem, Invent. Math. 98 (1989), 219-228. MR 91i:53049
  • [Y1] T. Yamaguchi, Lipschitz convergence of manifolds of positive Ricci curvature with large volume, Math. Ann. 284 (1989), 423-436. MR 90c:53144
  • [Y2] T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. 133 (1991), 317-357. MR 92b:53067

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Additional Information

Valery Marenich
Affiliation: IMECC - UNICAMP, Campinas, Brazil
Email: marenich@ime.unicamp.br

DOI: https://doi.org/10.1090/S0002-9947-00-02634-9
Keywords: Sphere theorems, radial minimal curvature
Received by editor(s): February 3, 1999
Published electronically: June 14, 2000
Additional Notes: Supported by FAPERJ and CNPq
Article copyright: © Copyright 2000 American Mathematical Society

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