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On transversely holomorphic partially hyperbolic flows

Mounib Abouanass

TL;DR

The paper advances the classification of seven-dimensional, transversely holomorphic partially hyperbolic flows by analyzing a flow-invariant, compact subcenter foliation with trivial holonomy. It proves that the flow induces a smooth fiber bundle projection to a five-dimensional base carrying a transversely holomorphic Anosov flow, with the base dynamics determined by a complex torus automorphism suspension or the geodesic flow on a hyperbolic 3-manifold (up to finite covers) in the topologically transitive case. The approach hinges on establishing holomorphicity and smoothness properties of leaves, using non-stationary linearization, global $su\Phi$-holonomy maps, and Gibbs measure techniques to obtain a holomorphic, $C^ abla$-smooth foliation structure and a robust Fubini-type disintegration. Combining these with Journé’s lemma and holomorphic foliations yields a smooth transverse holomorphic structure on center, center-stable, and center-unstable foliations, enabling the fiber-bundle description and the final classification result. The work extends discrete holomorphic-structure results to flows in dimension seven and provides a rigorous bridge between complex-geometry techniques and hyperbolic dynamics to obtain a sharp geometric/dynamical dichotomy for the seven-manifold setting.

Abstract

In this paper, we study transversely holomorphic partially hyperbolic flows, i.e. those whose holonomy pseudo-group is given by biholomorphic maps. We prove in the seven-dimensional case that under the assumption that the subcenter distribution is integrable to a flow invariant compact foliation with trivial holonomy, then the flow projects, by a smooth fiber bundle map, to a transversely holomorphic Anosov flow on a smooth five-dimensional manifold which is, in case of topological transitivity, either $C^\infty$ orbit equivalent to the suspension of a hyperbolic automorphism of a complex torus, or, up to finite covers, $C^\infty$-orbit equivalent to the geodesic flow of a compact hyperbolic manifold.

On transversely holomorphic partially hyperbolic flows

TL;DR

The paper advances the classification of seven-dimensional, transversely holomorphic partially hyperbolic flows by analyzing a flow-invariant, compact subcenter foliation with trivial holonomy. It proves that the flow induces a smooth fiber bundle projection to a five-dimensional base carrying a transversely holomorphic Anosov flow, with the base dynamics determined by a complex torus automorphism suspension or the geodesic flow on a hyperbolic 3-manifold (up to finite covers) in the topologically transitive case. The approach hinges on establishing holomorphicity and smoothness properties of leaves, using non-stationary linearization, global -holonomy maps, and Gibbs measure techniques to obtain a holomorphic, -smooth foliation structure and a robust Fubini-type disintegration. Combining these with Journé’s lemma and holomorphic foliations yields a smooth transverse holomorphic structure on center, center-stable, and center-unstable foliations, enabling the fiber-bundle description and the final classification result. The work extends discrete holomorphic-structure results to flows in dimension seven and provides a rigorous bridge between complex-geometry techniques and hyperbolic dynamics to obtain a sharp geometric/dynamical dichotomy for the seven-manifold setting.

Abstract

In this paper, we study transversely holomorphic partially hyperbolic flows, i.e. those whose holonomy pseudo-group is given by biholomorphic maps. We prove in the seven-dimensional case that under the assumption that the subcenter distribution is integrable to a flow invariant compact foliation with trivial holonomy, then the flow projects, by a smooth fiber bundle map, to a transversely holomorphic Anosov flow on a smooth five-dimensional manifold which is, in case of topological transitivity, either orbit equivalent to the suspension of a hyperbolic automorphism of a complex torus, or, up to finite covers, -orbit equivalent to the geodesic flow of a compact hyperbolic manifold.
Paper Structure (31 sections, 69 theorems, 70 equations)

This paper contains 31 sections, 69 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Reminders and definitions
  3. Foliations
  4. Holonomy
  5. Compact foliations
  6. Almost complex structure
  7. Transversely holomorphic foliations
  8. Partially hyperbolic flows
  9. Dynamical coherence
  10. Global $su\Phi$-holonomy maps
  11. Subcenter bunching
  12. Disintegration and Gibbs measures
  13. Quasiconformality
  14. Sketch of proof of the results
  15. Some fundamental results
  16. ...and 16 more sections

Key Result

Proposition 2.4

Let $\mathcal{F}$ a topological foliation with $C^1$ leaves. Then $\mathcal{F}$ has uniformly $C^1$ leaves if and only if it is integral. Any foliation with holomorphic leaves is integral and has uniformly $C^\infty$ leaves.

Theorems & Definitions (129)

  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Definition 2.7
  • Definition 2.8
  • Proposition 2.9
  • Theorem 2.10
  • Theorem 2.11: Generalized Reeb Stability
  • ...and 119 more