Spectral rigidity among ellipses, Bialy's conjecture and local extrema of Mather's beta function

TL;DR

This paper proves Bialy's conjecture linking Mather beta functions and ellipse identity, demonstrating that matching beta functions at specific rotation numbers imply the ellipses are identical, with implications for extremizers of the beta function.

Contribution

It establishes a proof of Bialy's conjecture and explores conditions under which ellipses are uniquely determined by their Mather beta functions.

Findings

Proved Bialy's conjecture for two nonzero rotation numbers.

Showed that one rotation number and equal perimeter suffice for ellipse coincidence.

Discussed implications for local extremizers of Mather's beta function.

Abstract

In this paper we prove Bialy's conjecture which states that if the Mather beta functions of two ellipses coincide at two nonzero rotation numbers then the ellipses coincide. We also show that the same conclusion holds when only one rotation number is prescribed, provided the two ellipses have the same perimeter. Finally we discuss consequences for local extremizers of Mathers beta function building on a recent result of Baranzini, Bialy and Sorrentino.