## “Isoparametricity and the Pompeiu property”

Venerdì 16 Settembre 2022, ore 11:30 - Aula 2AB40 e Zoom - Luigi Provenzano (Sapienza Università di Roma)

Abstract

A bounded domain $\Omega$ in $\mathbb{R}^n$ has the Pompeiu property if the only continuous function which has zero integral over all congruent copies of $\Omega$ is the zero function. It is known that balls fail the Pompeiu property, and the famous Pompeiu conjecture states that the only domains homeomorphic to a ball doing so are balls.

We investigate the analogous problem in the case of domains in compact Riemannian manifolds $M$ admitting an isoparametric foliation (a term conied by Tullio Levi-Civita), with particular attention to spherical domains. In particular, we relate the Pompeiu property with the spectrum of the Laplacian on $M$ , and find precise spectral conditions under which the level domains of an isoparametric function fail the Pompeiu property. In this context, we will also discuss Ungar’s “freak theorem about functions on the sphere”, for which we provide a simple and intuitive explanation.

Joint work with Alessandro Savo (Sapienza Universit`a di Roma)