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

TL;DR

This paper examines the size of tubular neighborhoods in contact topology, showing that large neighborhoods of certain overtwisted submanifolds prevent the ambient manifold from having an exact symplectic filling.

Contribution

It highlights a new obstruction to fillability based on the size of tubular neighborhoods of overtwisted submanifolds in contact manifolds.

Findings

Large neighborhoods of overtwisted submanifolds obstruct fillability

The size of tubular neighborhoods influences symplectic fillability

Overtwisted submanifolds with trivial normal bundle impact global properties

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,\xi_N) is a 3-dimensional overtwisted submanifold with trivial normal bundle in (M,\xi), and if its model neighborhood is sufficiently large, then (M,\xi) does not admit an exact symplectic filling.