On the local systolic optimality of Zoll contact forms

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: (1) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (2) the pert...

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Published inGeometric and functional analysis Vol. 33; no. 2; pp. 299 - 363
Main Authors Abbondandolo, Alberto, Benedetti, Gabriele
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.04.2023
Springer Nature B.V
Subjects
Analysis
Canonical forms
Mathematics
Mathematics and Statistics
Online AccessGet full text
ISSN1016-443X
1420-8970
DOI10.1007/s00039-023-00624-z

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Summary: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: (1) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (2) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (3) a generalization of Gromov’s non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones.
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ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-023-00624-z