On contact type hypersurfaces in 4-space

Mark TE, Tosun B. On contact type hypersurfaces in 4-space. Inventiones mathematicae. 2021.


We consider constraints on the topology of closed 3-manifolds that can arise as hypersurfaces of contact type in standard symplectic \({\mathbb R}^4\). Using an obstruction derived from Heegaard Floer homology we prove that no Brieskorn homology sphere admits a contact type embedding in \({\mathbb R}^4\), a result that has bearing on conjectures of Gompf and Koll\'ar. This implies in particular that no rationally convex domain in \({\mathbb C}^2\) has boundary diffeomorphic to a Brieskorn sphere. We also give infinitely many examples of contact 3-manifolds that bound Stein domains but not symplectically convex ones; in particular we find Stein domains in \({\mathbb C}^2\) that cannot be made Weinstein with respect to the ambient symplectic structure while preserving the contact structure on their boundaries.

Last updated on 03/06/2022