Anosov flows and Liouville pairs in dimension three

Massoni, Thomas¹

¹Department of Mathematics, Princeton University, Princeton, NJ, United States, Department of Mathematics, Stanford University, Stanford, CA, United States

Algebraic and Geometric Topology, Tome 25 (2025) no. 3, pp. 1793-1838

Résumé

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 × 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.

Détail

DOI : 10.2140/agt.2025.25.1793

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

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