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 $N\times \left\{0\right\}$ in the model space $N\times {ℝ}^{2k}$.

In this article we make the observation that if $\left(N,{\xi }_N\right)$ is a $3$–dimensional overtwisted submanifold with trivial normal bundle in $\left(M,\xi \right)$, and if its model neighborhood is sufficiently large, then $\left(M,\xi \right)$ does not admit a symplectically aspherical filling.