arXiv subjects

Thomas Massoni

Publications and source records attributed to Thomas Massoni.

7 recordsLinked to original sources

arXiv2026

Taut foliations through a contact lens

This survey explores the rich interplay between (taut) foliations and (tight) contact structures in dimension three, highlighting recent work of the author. We outline a new method for constructing taut foliations from suitable pairs of contact structures, and discuss some applications and future research directions. In particular, we give a brief account of work in progress with Jonathan Zung on ziggurats for taut foliations transverse to pseudo-Anosov flows, and we propose a (mostly speculative) contact perspective on the $L$-space conjecture.

math.SG↗

arXiv2026

Ziggurats, taut foliations, and contact structures

We study the geography of taut foliations on compact $3$-manifolds with toroidal boundary components. The central object of our work is the set of boundary multislopes realized by foliations transverse to a fixed flow on such a manifold. We prove that these sets exhibit remarkable structural properties (rationality, rigidity, and convexity) which motivate the name ziggurats. Our main tool, of independent interest, is a two-way correspondence between foliations and contact structures on $3$-manifolds with boundary: we generalize the Eliashberg-Thurston theorem, which produces pairs of positive and negative contact structures from foliations, and a construction of the first author, which builds foliations from such contact pairs. Using both directions of this correspondence, we bring contact-geometric methods to bear on the architecture of ziggurats.

math.GT↗

arXiv2025

Topological invariance of Liouville structures for taut foliations and Anosov flows

Building on the work of Eliashberg and Thurston, we associate to a taut foliation on a closed oriented $3$-manifold $M$ a Liouville structure on the thickening $[-1,1] \times M$, under suitable hypotheses. Our main result shows that this Liouville structure is a topological invariant of the foliation: two such foliations which are topologically conjugate induce Liouville structures that are exact symplectomorphic (after completion). Specializing to the case of weak foliations of Anosov flows, we obtain that under natural orientability conditions, the Liouville structures originally introduced by Mitsumatsu are invariant under orbit equivalence. Our methods also imply that two orbit equivalent Anosov flows are deformation equivalent through projectively Anosov flows. The proofs combine two main technical ingredients: (1) a careful smoothing scheme for topological conjugacies between $C^1$-foliations, and (2) a refinement of a deep result of Vogel on the uniqueness of contact structures approximating a foliation. In an appendix, this smoothing scheme is used to construct new examples of collapsed Anosov flows, providing a key step to complete the classification of transitive partially hyperbolic diffeomorphisms in dimension three.

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arXiv2025

A symplectic viewpoint on Anosov flows

This survey explores the geometry of three-dimensional Anosov flows from the perspective of contact and symplectic geometry, following the work of Mitsumatsu, Eliashberg-Thurston, Hozoori, and the author. We also present a few original results and discuss various open questions and conjectures.

math.SG↗

arXiv2024

Taut foliations and contact pairs in dimension three

We present a new construction of codimension-one foliations from pairs of contact structures in dimension three. This constitutes a converse result to a celebrated theorem of Eliashberg and Thurston on approximations of foliations by contact structures. Under suitable hypotheses on the initial contact pairs, the foliations we construct are taut, allowing us to characterize the existence of taut foliations entirely in terms of contact geometry. This viewpoint reveals some surprising flexibility phenomena for taut foliations, and provides new insight into the $L$-space conjecture. The first part of the proof builds upon the work of Colin and Firmo on positive contact pairs. The second part involves a wide generalization of a technical result of Burago and Ivanov on the construction of branching foliations tangent to continuous plane fields, and might be of independent interest.

math.SG↗

arXiv2022

Anosov flows and Liouville pairs in dimension three

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 bi-contact 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 $\mathbb{R} \times M$ 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.

math.SG↗

arXiv2022

Floer theory of Anosov flows in dimension three

A smooth Anosov flow on a closed oriented three manifold $M$ gives rise to a Liouville structure on the four manifold $[-1,1]\times M$ which is not Weinstein, by a construction of Mitsumatsu and Hozoori. We call it the associated Anosov Liouville domain. It is well defined up to homotopy and only depends on the homotopy class of the original Anosov flow; its symplectic invariants are then invariants of the flow. We study the symplectic geometry of Anosov Liouville domains, via the wrapped Fukaya category, which we expect to be a powerful invariant of Anosov flows. The Lagrangian cylinders over the simple closed orbits span a natural $A_\infty$-subcategory, the orbit category of the flow. We show that it does not satisfy Abouzaid's generation criterion; it is moreover "very large", in the sense that is not split-generated by any strict sub-family. This is in contrast with the Weinstein case, where critical points of a Morse function play the role of the orbits. For the domain corresponding to the suspension of a linear Anosov diffeomorphism on the torus, we show that there are no closed exact Lagrangians which are either orientable, projective planes or Klein bottles. By contrast, in the case of the geodesic flow on a hyperbolic surface of genus $g \geq 2$ (corresponding to the McDuff example), we construct an exact Lagrangian torus for each embedded closed geodesic, thus obtaining at least $3g-3$ tori which are not Hamiltonian isotopic to each other. For these two prototypical cases of Anosov flows, we explicitly compute the symplectic cohomology of the associated domains, as well as the wrapped Floer cohomology of the Lagrangian cylinders, and several pair-of-pants products.

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