On the local Birkhoff conjecture for convex billiards

By Vadim Kaloshin, A. Sorrentino. Annals of Mathematics, 2016

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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 local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends the result in [3], where only unperturbed ellipses of small eccentricities 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 prove certain spectral rigidity results for elliptic domains.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/24ab8317ad868969e61e253358d58bd9eb20ddc0