Wikipedia

Gromov's compactness theorem (geometry)

This article is about Gromov's compactness theorem in Riemannian geometry. For Gromov's compactness theorem in symplectic topology, see Gromov's compactness theorem (topology).

In Riemannian geometry, Gromov's (pre)compactness theorem states that the set of compact Riemannian manifolds of a given dimension, with Ricci curvature ≥ c and diameter ≤ D is relatively compact in the Gromov–Hausdorff metric.[1][2] It was proved by Mikhail Gromov in 1981.[2][3]

This theorem is a generalization of Myers's theorem.[4]

References

  1. ^ Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isenberg, James; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei (2010), The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects, Mathematical Surveys and Monographs, 163, Providence, Rhode Island: American Mathematical Society, p. 396, doi:10.1090/surv/163, ISBN 978-0-8218-4661-2, MR 2604955
  2. ^ a b Bär, Christian; Lohkamp, Joachim; Schwarz, Matthias (2011), Global Differential Geometry, Springer Proceedings in Mathematics, 17, Springer, p. 94, ISBN 9783642228421.
  3. ^ Gromov, Mikhael (1981), Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], 1, Paris: CEDIC, ISBN 2-7124-0714-8, MR 0682063. As cited by Bär, Lohkamp & Schwarz (2011).
  4. ^ Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004), Riemannian Geometry, Universitext, Springer, p. 179, ISBN 9783540204930.


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