Isoparametric foliations and the Pompeiu property

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 to have 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 to have 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.