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 conformally 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 an exact symplectic filling. In symplectic geometry many invariants are known that in some way measure the “size ” of a symplectic manifold. The most obvious one is the total volume, but this is usually discarded, because by rescaling the symplectic form one can change the volume (in case it is finite) without changing any other fudamental property of the manifold. The first non-trivial example of an invariant based on size is the symplectic capacity [Gro85]. It relies on the fact that the size of a symplectic ball that can be embedded into a symplectic manifold does not only depend on its total volume but also on the volume of its intersection with the symplectic 2–planes.