Some remarks on the size of tubular neighborhoods in contact topology and fillability

Abstract

The well-known tubular neighborhood theorem for contact submanifolds states that a small enough neighborhood of such a submanifold N is uniquely determined by the contact structure on N, and the conformal symplectic structure of the normal bundle. In particular, if the submanifold N has trivial normal bundle then its tubular neighborhood will be contactomorphic to a neighborhood of Nx{0} in the model space NxR^{2k}. In this article we make the observation that if (N,ξ_N) is a 3-dimensional overtwisted submanifold with trivial normal bundle in (M,ξ), and if its model neighborhood is sufficiently large, then (M,ξ) does not admit an exact symplectic filling.

19 pages, 2 figures; added example of manifold that is not fillable by neighborhood criterium; typos

References in corpus (4)

• Geometry of contact transformations and domains: orderability versus squeezing
• The plastikstufe - a generalization of the overtwisted disk to higher dimensions
• A class of non-fillable contact structures
• Every contact manifold can be given a non-fillable contact structure

Cited by in corpus (7)

• Weak and strong fillability of higher dimensional contact manifolds
• Loose Legendrians and the plastikstufe
• On some examples and constructions of contact manifolds
• On plastikstufe, bordered Legendrian open book and overtwisted contact structures
• The Weinstein conjecture in the presence of submanifolds having a Legendrian foliation
• Examples of non-trivial contact mapping classes for overtwisted contact manifolds in all dimensions
• An overtwisted convex hypersurface in higher dimensions