Transverse minimal foliations on unit tangent bundles and applications

Sérgio R. Fenley and Rafael Potrie

Vol. 343 (2026), No. 1, 39–118

DOI: 10.2140/pjm.2026.343.39

Abstract

We show that given two transverse minimal foliations on the unit tangent bundle of a surface of genus $\ge 2$, their intersection either is an Anosov foliation or contains a Reeb surface. The existence of a Reeb surface is incompatible with partially hyperbolic foliations, so we deduce from this that certain partially hyperbolic diffeomorphisms in unit tangent bundles are collapsed Anosov flows. We also conclude that every volume preserving partially hyperbolic diffeomorphism of a unit tangent bundle is ergodic.

Keywords

foliation, 3-manifold, partial hyperbolicity, Anosov flow, ergodicity

Mathematical Subject Classification

Primary: 57R30, 37C86, 37D30

Secondary: 53C12, 57K30, 37C15

Milestones

Received: 11 June 2024

Revised: 30 October 2025

Accepted: 26 February 2026

Published: 29 April 2026