Birkhoff Conjecture for Nearly Centrally Symmetric Domains

In this paper we prove a perturbative version of a remarkable Bialy–Mironov (Ann. Math. 196(1):389–413, 2022 ) result. They prove non perturbative Birkhoff conjecture for centrally-symmetric convex domains, namely, a centrally-symmetric convex domain with integrable billiard is ellipse. We combine t...

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Published inGeometric and functional analysis Vol. 34; no. 6; pp. 1973 - 2007
Main Authors Kaloshin, V., Koudjinan, C. E., Zhang, Ke
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.12.2024
Springer Nature B.V
Subjects
Analysis
Mathematics
Mathematics and Statistics
Birkhoff conjecture
Integrability
Birkhoff billiard
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Summary:In this paper we prove a perturbative version of a remarkable Bialy–Mironov (Ann. Math. 196(1):389–413, 2022 ) result. They prove non perturbative Birkhoff conjecture for centrally-symmetric convex domains, namely, a centrally-symmetric convex domain with integrable billiard is ellipse. We combine techniques from Bialy–Mironov (Ann. Math. 196(1):389–413, 2022 ) with a local result by Kaloshin–Sorrentino (Ann. Math. 188(1):315–380, 2018 ) and show that a domain close enough to a centrally symmetric one with integrable billiard is ellipse. To combine these results we derive a slight extension of Bialy–Mironov (Ann. Math. 196(1):389–413, 2022 ) by proving that a notion of rational integrability is equivalent to the C 0 -integrability condition used in their paper.
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ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-024-00695-6