On the local systolic optimality of Zoll contact forms

TL;DR

This paper establishes that Zoll contact forms are local maximizers of the systolic ratio, leading to new inequalities and generalizations in symplectic geometry and contact topology.

Contribution

It introduces a normal form for near-Zoll contact forms and proves their local optimality for the systolic ratio, impacting several geometric inequalities.

Findings

Zoll contact forms are local maximizers of the systolic ratio.

Derivation of sharp local systolic inequalities for metrics near Zoll forms.

Extension of Gromov's non-squeezing theorem to intermediate dimensions.

Abstract

We prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (i) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (ii) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (iii) a generalization of Gromov's non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones.