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

TL;DR

This paper investigates the limits of leaves in compact foliations, establishing conditions for Gromov-Hausdorff convergence, smooth convergence, and deriving several classical stability and structure theorems as corollaries.

Contribution

It provides a unified framework connecting Gromov-Hausdorff limits with classical foliation stability theorems, offering new insights into leaf convergence and holonomy covers.

Findings

Gromov-Hausdorff limit of leaves is a covering space of the limiting leaf

Convergence to the limit is smooth rather than just Gromov-Hausdorff

Reeb's local stability theorem is derived as a corollary

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 Alvarez and Candel. Several examples are discussed.