An integrable deformation of an ellipse of small eccentricity is an ellipse

TL;DR

This paper proves that small perturbations of ellipses with small eccentricity remain integrable and are still ellipses, supporting a version of the classical Birkhoff conjecture.

Contribution

It demonstrates that for small eccentricity, integrable convex domains close to ellipses are themselves ellipses, confirming a specific case of the Birkhoff conjecture.

Findings

Small perturbations of ellipses with small eccentricity are still ellipses.

Supports the Birkhoff conjecture for a class of nearly elliptical domains.

Shows stability of integrability under small perturbations.

Abstract

The classical Birkhoff conjecture says that the only integrable convex domains are circles and ellipses. In the paper we show that this a version of this conjecture is true for small perturbations of ellipses of small eccentricity.