Isoparametric foliations and the Pompeiu problem

TL;DR

This paper investigates the Pompeiu property in Riemannian manifolds with isoparametric foliations, identifying spectral conditions under which certain domains fail the property, with detailed analysis on spheres and generalizations of Ungar's Freak theorem.

Contribution

It provides new spectral criteria for the failure of the Pompeiu property in manifolds with isoparametric foliations, extending known results on spheres and related geometries.

Findings

Level domains of isoparametric functions can fail the Pompeiu property under specific spectral conditions.

Explicit calculations are provided for the case of the round sphere.

The paper discusses generalizations of Ungar's Freak theorem in this context.

Abstract

A bounded domain $\Omega$ in a Riemannian manifold $M$ is said to have the Pompeiu property if the only continuous function which integrates to zero on $\Omega$ and on all its congruent images is the zero function. In some respects, the Pompeiu property can be viewed as an overdetermined problem, given its relation with the Schiffer problem. It is well-known that every Euclidean ball fails the Pompeiu property while spherical balls have the property for almost all radii (Ungar's Freak theorem). In the present paper we discuss the Pompeiu property when $M$ is compact and admits an isoparametric foliation. In particular, we identify precise conditions on the spectrum of the Laplacian on $M$ under which the level domains of an isoparametric function fail the Pompeiu property. Specific calculations are carried out when the ambient manifold is the round sphere, and some consequences are derived. Moreover, a detailed discussion of Ungar's Freak theorem and its generalizations is also carried out.