Asian Journal of Mathematics

Volume 19 (2015)

Number 3

Reeb stability and the Gromov-Hausdorff limits of leaves in compact foliations

Pages: 433 – 464

DOI: http://dx.doi.org/10.4310/AJM.2015.v19.n3.a3

Author

Pablo Lessa (Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Paris, France; and Centro de Matemática, Facultad de Ciencias, Universidad de la República, Montevideo, Uruguay)

Abstract

We show that the Gromov–Hausdorff limit of a sequence of leaves in a compact foliation is a covering space of the limiting leaf which is no larger than this leaf’s holonomy cover. We also show that convergence to such a limit is smooth instead of merely Gromov–Hausdorff. Corollaries include Reeb’s local stability theorem, part of Epstein’s local structure theorem for foliations by compact leaves, and a continuity theorem of Álvarez and Candel. Several examples are discussed.

Keywords

foliations, Riemannian geometry, convergence of Riemannian manifolds

2010 Mathematics Subject Classification

53C12, 57R30

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