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Integrable Magnetic Flows on the Two-Torus: Zoll Examples and Systolic Inequalities

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Abstract

© 2020, Mathematica Josephina, Inc.In this paper, we study some aspects of integrable magnetic systems on the two-torus. On the one hand, we construct the first non-trivial examples with the property that all magnetic geodesics with unit speed are closed. On the other hand, we show that those integrable magnetic systems admitting a global surface of section satisfy a sharp systolic inequality.
Original languageEnglish
Pages (from-to)2924-2940
JournalJournal of Geometric Analysis
Volume31
Issue number3
DOIs
Publication statusPublished - 1 Mar 2021
Externally publishedYes

Funding

We thank Alberto Abbondandolo and Serge Tabachnikov for fruitful discussions, and Mattia Carlo Sormani for the precious help with the numerical integration. We are indebted to the referees for their valuable comments on the manuscript. Luca Asselle is partially supported by the DFG-Grant AS 546/1-1 “Morse theoretical methods in Hamiltonian dynamics”. Gabriele Benedetti was partially supported by the National Science Foundation under Grant No. DMS-1440140 while in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester for the program “Hamiltonian systems, from topology to applications through analysis”. We thank Alberto Abbondandolo and Serge Tabachnikov for fruitful discussions, and Mattia Carlo Sormani for the precious help with the numerical integration. We are indebted to the referees for their valuable comments on the manuscript. Luca Asselle is partially supported by the DFG-Grant AS 546/1-1 ?Morse theoretical methods in Hamiltonian dynamics?. Gabriele Benedetti was partially supported by the National Science Foundation under Grant No.?DMS-1440140 while in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester for the program ?Hamiltonian systems, from topology to applications through analysis?.

FundersFunder number
DFG-GrantAS 546/1-1
National Science FoundationDMS-1440140, 1440140
Deutsche Forschungsgemeinschaft

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