How to recognise a 4-ball when you see one

TL;DR

This paper uses holomorphic disc filling techniques to analyze 4-dimensional symplectic cobordisms with contact 3-sphere boundaries, establishing a dichotomy related to their topology and Reeb dynamics.

Contribution

It introduces a unified approach to study symplectic cobordisms, linking their topology to Reeb orbit properties and applications in symplectic geometry.

Findings

Either the cobordism is diffeomorphic to a ball or has a short-period Reeb orbit.

Provides a unified framework for Reeb dynamics, symplectic fillability, and non-squeezing results.

Shows non-existence of exact Lagrangian surfaces in standard symplectic 4-space.

Abstract

We apply the method of filling with holomorphic discs to a 4-dimensional symplectic cobordism with the standard contact 3-sphere as a convex boundary component. We establish the following dichotomy: either the cobordism is diffeomorphic to a ball, or there is a periodic Reeb orbit of quantifiably short period in the concave boundary of the cobordism. This allows us to give a unified treatment of various results concerning Reeb dynamics on contact 3-manifolds, symplectic fillability, the topology of symplectic cobordisms, symplectic non-squeezing, and the non-existence of exact Lagrangian surfaces in standard symplectic 4-space.