Disk-like surfaces of section and symplectic capacities

TL;DR

This paper establishes a deep connection between the cylindrical capacity and disk-like global surfaces of section in convex domains of , and proves the strong Viterbo conjecture for domains close to the round ball.

Contribution

It proves the equality of cylindrical capacity and minimal symplectic area of disk-like surfaces in  and confirms the strong Viterbo conjecture for nearly spherical convex domains.

Findings

Cylindrical capacity equals the least symplectic area of disk-like surfaces of section.

Strong Viterbo conjecture holds for convex domains close to the sphere.

Generalizes systolic inequality for convex domains.

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 $C^3$ close to the round ball. This generalizes a result of Abbondandolo-Bramham-Hryniewicz-Salom\~{a}o establishing a systolic inequality for such domains.