Disk-Like Surfaces of Section and Symplectic Capacities

Abstract

We prove that the cylindrical capacity of a dynamically convex domain in \({\mathbb{R}}^{4}\) agrees with the least symplectic area of a disk-like global surface of section of the Reeb flow on the boundary of the domain. Moreover, we prove the strong Viterbo conjecture for all convex domains in \({\mathbb{R}}^{4}\) which are sufficiently C³ close to the round ball. This generalizes a result of Abbondandolo-Bramham-Hryniewicz-Salomão establishing a systolic inequality for such domains.