Disk-Like Surfaces of Section and Symplectic Capacities

Abstract

We prove that the cylindrical capacity of a dynamically convex domain in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathbb{R}}^{4}$\\end{document} 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 \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathbb{R}}^{4}$\\end{document} 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.