The Isospectral Problem for p-widths: An Application of Zoll Metrics

TL;DR

This paper investigates whether a Riemannian manifold can be uniquely identified by its sequence of p-widths, providing counterexamples on the sphere using Zoll metrics and geodesic properties.

Contribution

It introduces the isospectral problem for p-widths and constructs counterexamples on $S^2$ demonstrating non-uniqueness using Zoll metrics.

Findings

Counterexamples on $S^2$ show non-uniqueness of manifolds from p-widths

p-widths are unions of immersed geodesics in Zoll metrics

The isospectral problem for p-widths has negative solutions

Abstract

We pose the isospectral problem for the $p$-widths: Is a riemannian manifold $(M^n, g)$ uniquely determined by its $p$-widths, $\{\omega_p(M,g)\}_{p=1}^{\infty}$? We construct many counterexamples on $S^2$ using Zoll metrics and the fact that geodesic $p$-widths are given by unions of immersed geodesics.