The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in [3], where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action- angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains.

Kaloshin, V., & Sorrentino, A. (2018). On the local Birkhoff conjecture for convex billiards. ANNALS OF MATHEMATICS, 188(1), 315-380 [10.4007/annals.2018.188.1.6].

On the local Birkhoff conjecture for convex billiards

Sorrentino, A.
2018-07

Abstract

The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in [3], where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action- angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains.
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - Analisi Matematica
Settore MAT/03 - Geometria
Settore MAT/07 - Fisica Matematica
English
Con Impact Factor ISI
http://annals.math.princeton.edu/2018/188-1/p06
Kaloshin, V., & Sorrentino, A. (2018). On the local Birkhoff conjecture for convex billiards. ANNALS OF MATHEMATICS, 188(1), 315-380 [10.4007/annals.2018.188.1.6].
Kaloshin, V; Sorrentino, A
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/195394
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