Asian Journal of Mathematics
Volume 19 (2015)
Reeb stability and the Gromov-Hausdorff limits of leaves in compact foliations
Pages: 433 – 464
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.
foliations, Riemannian geometry, convergence of Riemannian manifolds
2010 Mathematics Subject Classification