Articles de Patrick Massot

Geodesible contact structures

Geometry and Topology 12 (2008) 1729-1776

In this paper, we study and almost completely classify contact structures on closed 3– manifolds which are totally geodesic for some Riemannian metric. Due to previously known results, this amounts to classifying contact structures on Seifert manifolds which are transverse to the fibers. Actually, we obtain the complete classification of contact structures with negative (maximal) twisting number (which includes the transverse ones) on Seifert manifolds whose base is not a sphere, as well as partial results in the spherical case.

(version arxiv, version publiée).

Infinitely many universally tight torsion free contact structures with vanishing Ozsváth-Szabó invariants

Mathematische Annalen 353 (2012), n° 4, 1351-1376

Non-vanishing of the Ozsváth-Szabó contact invariant is a powerful way to prove tightness of contact structures but this invariant is known to vanish in the presence of Giroux torsion. In this note, we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsváth-Szabó invariant neverthe- less vanishes. Along the way, we prove a conjecture of K Honda, W Kazez and G Matić about their contact topological quantum field theory.

(version arxiv, version publiée).

Tightness in contact metric 3-manifolds

avec John Etnyre et Rafał Komendarczyk

Inventiones Mathematicae 188 (2012), n° 3, 621-657

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξ is tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S³. We also describe geometric conditions in dimension three for ξ to be universally tight in the nonpositive curvature setting.

(version arxiv, version publiée).

Weak and strong fillability of higher dimensional contact manifold

avec Klaus Niederkrüger et Chris Wendl

Inventiones Mathematicae 192 (2013) n°3, 287-373

For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five), while also being obstructed by all known manifestations of "overtwistedness". We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.

(version arxiv, version publiée).

Quantitative Darboux theorems in contact geometry

avec John Etnyre et Rafał Komendarczyk

Transactions of the AMS 368 (2016) n°11, 7845-7881

This paper begins the study of relations between Riemannian geometry and contact topology on (2n+1)-manifolds and continues this study on 3-manifolds. Specifically we provide a lower bound for the radius of a geodesic ball in a contact (2n+1)-manifold (M,ξ) that can be embedded in the standard contact structure on ℝ2n+1, that is on the size of a Darboux ball. The bound is established with respect to a Riemannian metric compatible with an associated contact form α for ξ. In dimension three, it further leads us to an estimate of the size for a standard neighborhood of a closed Reeb orbit. The main tools are classical comparison theorems in Riemannian geometry. In the same context, we also use holomorphic curves techniques to provide a lower bound for the radius of a PS-tight ball.

(version arxiv, version publiée).

Examples of non-trivial contact mapping classes in all dimensions

avec Klaus Niederkrüger

International Mathematics Research Notices (IMRN) 2016 (15), 4784-4806

We give examples of contactomorphisms in every dimension that are smoothly isotopic to the identity but that are not contact isotopic to the identity. In fact, we prove the stronger statement that they are not even symplectically pseudo-isotopic to the identity. We also give examples of pairs of contactomorphisms which are smoothly conjugate to each other but not by contactomorphisms.

(version arxiv, version publiée).

On the contact mapping class group of Legendrian circle bundles

avec Emmanuel Giroux

Compositio Mathematica 2017 153(2), 294-312

In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of nonpositive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain spaces of embeddings of surfaces into contact 3-manifolds. In the third section, this connectedness question is studied in more details with a number of (hopefully instructive) examples.

(version arxiv, version publiée).

Contactomorphism groups and Legendrian flexibility

avec Sylvain Courte

prépublication

We explain a connection between the algebraic and geometric properties of groups of contact transformations, open book decompositions, and flexible Legendrian embeddings. The main result is that, if a closed contact manifold (V,ξ) has a supporting open book whose pages are flexible Weinstein manifolds, then both the connected component of identity in its automorphism group and its universal cover are uniformly simple groups: for every non-trivial element g, every other element is a product of at most 128(dimV+1) conjugates of g±1. In particular any conjugation invariant norm on these groups is bounded.

(version arxiv).

mis à jour le 03 février 2017.