On the local systolic optimality of Zoll contact forms

Alberto Abbondandolo*, Gabriele Benedetti

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

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: (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.

Original languageEnglish
Pages (from-to)299-363
Number of pages65
JournalGeometric and Functional Analysis
Volume33
Issue number2
Early online date3 Feb 2023
DOIs
Publication statusPublished - Apr 2023

Bibliographical note

Funding Information:
We would like to thank Viktor Ginzburg for making us aware of Bottkol’s theorem and for several discussions about it. Our gratitude goes also to Barney Bramham, Umberto Hryniewicz, Jungsoo Kang and Pedro Salomão, with whom we have discussed the topic of this paper so many times that their contribution goes surely beyond what we even realize. A. Abbondandolo is partially supported by the Deutsche Forschungsgemeinschaft under the Collaborative Research Center SFB/TRR 191 - 281071066 (Symplectic Structures in Geometry, Algebra and Dynamics). G. Benedetti is partially supported by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster), the Collaborative Research Center SFB/TRR 191 - 281071066 (Symplectic Structures in Geometry, Algebra and Dynamics), and the Research Training Group RTG 2229 - 281869850 (Asymptotic Invariants and Limits of Groups and Spaces).

Publisher Copyright:
© 2023, The Author(s).

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