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Symplectic capacities of domains close to the ball and Banach-Mazur geodesics in the space of contact forms

Symplectic capacities of domains close to the ball and Banach-Mazur geodesics in the space of contact forms
Sandwiching an invariant convex body between a ball and a cylinder of the same width. The tangent locus is a periodic orbit.

Alberto Abbondandolo, Gabriele Benedetti, and Oliver Edtmair

We prove that all normalized symplectic capacities coincide on smooth domains in Cn which are C2-close to the Euclidean ball, whereas this fails for some smooth domains which are just C1-close to the ball. We also prove that all symplectic capacities whose value on ellipsoids agrees with that of the nth Ekeland–Hofer capacity coincide in a C2-neighborhood of the Euclidean ball of Cn. These results are deduced from a general theorem about contact forms which are C2-close to Zoll ones, saying that these contact forms can be pulled back to suitable quasi-invariant contact forms. We relate all this to the question of the existence of minimizing geodesics in the space of contact forms equipped with a Banach–Mazur pseudometric. Using some new spectral invariants for contact forms, we prove the existence of minimizing geodesics from a Zoll contact form to any contact form which is C2-close to it. This paper also contains an appendix in which we review the construction of exotic ellipsoids by the Anosov–Katok conjugation method, as these are related to the above-mentioned pseudometric.

Duke Mathematical Journal, Vol. 174, No. 8, 2025.
DOI 10.1215/00127094-2024-0066

ArXiv version: 2312.07364