Francisco Presas

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

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 Nf 0g in the model space N R 2k . In this article we make the observation that if .N; N/ is a 3‐dimensional overtwisted submanifold with trivial normal bundle in .M;/ , and if its model neighborhood is sufficiently large, then .M;/ does not admit a symplectically aspherical filling. 57R17; 53D35 In symplectic geometry, many invariants are known that measure in some way the “size” of a symplectic manifold. The most obvious one is the total volume, but this is usually discarded, because one can change the volume (in case it is finite) by rescaling the symplectic form without changing any other fundamental property of the manifold. The first non-trivial example of an invariant based on size is the symplectic capacity (see Gromov [15]). 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. Contact geometry does not give a direct generalization of these invariants. The main difficulties stem from the fact that one is only interested in the contact structure, and not in the contact form, so that the total volume is not defined, and to make matters worse the whole Euclidean space R 2nC1 with the standard structure can be compressed by a contactomorphism into an arbitrarily small open ball in R 2nC1 . A more successful approach consists in studying the size of the neighborhood of submanifolds. This can be considered to be a generalization of the initial idea since contact balls are just neighborhoods of points. In the literature this idea has been pursued by looking at the tubular neighborhoods of transverse circles. Let .N; N/ be a closed contact manifold. The product N R 2k carries a contact structure given as

On the strong orderability of overtwisted 3–folds

In this article we address the existence of positive loops of contactomorphisms in overtwisted contact 3-folds. We present a construction of such positive loops in the contact fibered connected sum of certain contact 3-folds along transverse knots. In particular, we obtain positive loops of contactomorphisms in overtwisted contact structures.

Codimension one symplectic foliations

We define the concept of symplectic foliation on a symplectic manifold and provide a method of constructing many examples, by using asymptotically holomorphic techniques.