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

Corentin Fierobe

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.

Spectral rigidity among ellipses, Bialy's conjecture and local extrema 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.
Paper Structure (6 sections, 11 theorems, 84 equations, 1 figure)

This paper contains 6 sections, 11 theorems, 84 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. First variations of Mather's beta function
  4. Proof of Bialy's conjecture -- Theorem \ref{['thm:main_bialy_conj']}
  5. Proof of Theorem \ref{['thm:main_cst_perimeter']}
  6. Local maximizers of $\beta$ - proof of Theorem \ref{['thm:main_local_max']}

Key Result

Theorem 1

Let $\rho_0,\rho_1\in(0,1/2]$ be distinct. Assume that $\mathscr E$ and $\mathscr E'$ are two ellipses satisfying Then $\mathscr E=\mathscr E'$ up to isometries.

Figures (1)

  • Figure 1: An oriented line $\ell$ with coordinates $(\varphi,p)$: $\ell$ is at at distance $p$ from the origin $O$ and its right normal makes an angle $\varphi$ with the $x$-axis.

Theorems & Definitions (20)

  • Conjecture 1: Bialy Bialyellipses
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3: Bialy-Baranzini-Sorrentino BBS
  • Theorem 4
  • Remark 1
  • Remark 2
  • Corollary 2
  • Definition 1
  • ...and 10 more