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×{0} in the model space N×R2k.
In this article we make the observation that if (N,ξN) 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 a symplectically aspherical filling.
Click here to watch a talk about this article given at MSRI.