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: https://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

Full Text (PDF format)

Published 19 June 2015