Stein domains inside complex surfaces

Robert Gompf

Eliashberg's topological characterization of Stein surfaces leads to a practical method for locating Stein surfaces biholomorphically embedded in a preassigned complex surface X. When X is C^2 or another Stein surface, these are called "domains of holomorphy" and are a classical object of study in complex analysis. Applications include: Domains of holomorphy in C^2 realizing uncountably many exotic smoothings, compact Stein domains embedded with pseudoconvex boundary, pseudoconvex embeddings of Brieskorn spheres, pseudoconCAVE fillings with controlled topology, and pseudoconcave, compact, contractible manifolds inside any closed, simply connected complex surface. (1 hour long talk)

http://scgp.stonybrook.edu/video_portal/video.php?id=1327

Simons- Workshop on Topology and Invariants of 4-Manifolds