Georgia Institute of Technology College of Sciences

We show that an embedding of a (small) ball into a contact manifold is contact if and only if it preserves the (modified) shape invariant. The latter is, in brief, the set of all cohomology classes that can be represented by the pull-back (to a closed one-form) of a contact form by a coisotropic embedding of a fixed manifold (of maximal dimension) and of a given homotopy type. The proof is based on displacement information about (non)-Lagrangian submanifolds that comes from J-holomorphic curve methods (and gives topological invariants), and the construction of a coisotropic torus whose image (under a given embedding that is not contact) admits a transverse contact vector field (i.e. a convex surface in dimension 3). The definition of shape preserving does not involve derivatives and is preserved by uniform convergence (on compact subsets). As a consequence, we prove C^0-rigidity of contact embeddings (and diffeomorphisms). The underlying ideas are adaptations of symplectic techniques to contact manifolds that, in contrast to symplectic capacities, work well in the contact setting; the heart of the proof however uses purely contact topological methods.

A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. An important question that has been circulating among contact and symplectic topologist for some time asks: whether every contractible smooth 4-manifold admits a Stein structure? In this talk we will provide examples that answer this question negatively. Moreover, along the way we will provide new evidence to a closely related conjecture of Gompf, which asserts that a nontrivial Brieskorn homology sphere, with either orientation, cannot be embedded in complex 2-space as the boundary of a Stein submanifold.

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism. (This is a joint work with Youlin Li.)