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On transverse $R$-covered minimal foliations

Thierry Barbot, Sergio R. Fenley, Rafael Potrie

TL;DR

This work advances the understanding of minimal $\mathbb{R}$-covered foliations in 3-manifolds by linking transverse foliations through a developing map on the leaf-space product. It shows a dichotomy: either the foliations intersect in a way that yields a flow foliation of an Anosov flow, or the intersection is constrained by conjugate $\pi_1(M)$-actions on leaf spaces, which, under orientability and uniformity hypotheses, forces Anosov-type structure. A central novel tool is the phase-space develop map and the analysis of invariant monotone graphs, which allows reduction to 2D dynamics and yields Reeb-annulus obstructions when graphs intersect. The results extend to translation foliations, product Anosov foliations, and circle-bundle settings, with implications for partially hyperbolic diffeomorphisms and bi-contact structures in 3-manifolds.

Abstract

We study minimal transverse foliations which are $R$-covered. If in addition the dimension of the ambient manifold is $3$, and the foliations are Anosov foliations we give necessary and sufficient conditions for the intersected foliation to be the orbit foliation of an Anosov flow.

On transverse $R$-covered minimal foliations

TL;DR

This work advances the understanding of minimal -covered foliations in 3-manifolds by linking transverse foliations through a developing map on the leaf-space product. It shows a dichotomy: either the foliations intersect in a way that yields a flow foliation of an Anosov flow, or the intersection is constrained by conjugate -actions on leaf spaces, which, under orientability and uniformity hypotheses, forces Anosov-type structure. A central novel tool is the phase-space develop map and the analysis of invariant monotone graphs, which allows reduction to 2D dynamics and yields Reeb-annulus obstructions when graphs intersect. The results extend to translation foliations, product Anosov foliations, and circle-bundle settings, with implications for partially hyperbolic diffeomorphisms and bi-contact structures in 3-manifolds.

Abstract

We study minimal transverse foliations which are -covered. If in addition the dimension of the ambient manifold is , and the foliations are Anosov foliations we give necessary and sufficient conditions for the intersected foliation to be the orbit foliation of an Anosov flow.
Paper Structure (31 sections, 40 theorems, 23 equations, 4 figures)

This paper contains 31 sections, 40 theorems, 23 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Foliations and transversality
  4. $\hbox{${\mathbb R}$}$-covered foliations
  5. Topological equivalence versus homotopic equivalence
  6. Translation foliations
  7. The phase space of a pair of transverse foliations
  8. Anosov foliations
  9. The product case
  10. The skew $\hbox{${\mathbb R}$}$-covered case
  11. Uniform foliations and slitherings
  12. On the orientability assumption
  13. Intersection of minimal foliations
  14. The case of translation foliations and product $\hbox{${\mathbb R}$}$-covered Anosov foliations
  15. The case where one foliation is a translation foliation
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $\hbox{${\mathcal{F}}$}_1, \hbox{${\mathcal{F}}$}_2$ be a pair of minimal, $\hbox{${\mathbb R}$}$-covered foliations in a manifold $M$ which are transverse to each other. Suppose that at least one of them is transversely orientable. Let $\hbox{${\widetilde{\mathcal{F}}}$}_i$ be the respective li

Figures (4)

  • Figure 1: A Reeb annulus in the universal cover and its projection to an annular leaf.
  • Figure 2: Possibilities for the image of the developing map. When the developing map is not surjective, the image is bounded by graphs corresponding to conjugacies between the $\pi_1(M)$-actions on the leaf spaces (see Proposition \ref{['pro.alphafini']}). In the first case (see § \ref{['sec.intersect']}), we will find Reeb surfaces in the preimage of the invariant graph, while in the second (see § \ref{['sec.nointersect']}), assuming that there are no Reeb surfaces, we will show that the intersected foliation is homeomorphic to an Anosov foliation.
  • Figure 3: Depiction of the proof of Claim \ref{['claim-monotone']}.
  • Figure 4: The path between $c_1$ and $c_2$ consists of the curves which separate $c_1$ from $c_2$, in the picture, these are represented by the black curves intersected by the red curve (which represents $\alpha_0$).

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • ...and 90 more