Gromov's compactness theorem (geometry)

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

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. 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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