Anosov flows and Liouville pairs in dimension three

Massoni, Thomas

doi:10.2140/agt.2025.25.1793

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Anosov flows and Liouville pairs in dimension three

Thomas Massoni

Algebraic & Geometric Topology 25 (2025) 1793–1838

DOI: 10.2140/agt.2025.25.1793

Abstract

Building upon the work of Mitsumatsu and Hozoori, we establish a complete homotopy correspondence between three-dimensional Anosov flows and certain pairs of contact forms that we call Anosov Liouville pairs. We show a similar correspondence between projectively Anosov flows and bicontact structures, extending the work of Mitsumatsu and Eliashberg–Thurston. As a consequence, every Anosov flow on a closed oriented three-manifold $M$ gives rise to a Liouville structure on $ℝ \times M$ which is well-defined up to homotopy, and which only depends on the homotopy class of the Anosov flow. Our results also provide a new perspective on the classification problem of Anosov flows in dimension three.

Keywords

Anosov flows, projectively Anosov flows, bicontact structures, Liouville pairs

Mathematical Subject Classification

Primary: 53E50, 57K33, 57K43

Secondary: 37D20, 37D30

References

Publication

Received: 8 June 2023

Revised: 27 August 2023

Accepted: 23 September 2023

Published: 20 June 2025

Authors

Thomas Massoni
Department of Mathematics Princeton University Princeton, NJ United States	Department of Mathematics Stanford University Stanford, CA United States

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Volume 25, issue 3 (2025)