Isoparametric foliations and the Pompeiu problem

A bounded domain Ω in a Riemannian manifold M is said to have the Pompeiu property if the only continuous function which integrates to zero on Ω 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.

[1]  L. Zalcman.  Analyticity and the Pompeiu problem , 1972 .

[2]  C. Berenstein,et al.  Pompeiu's problem on symmetric spaces , 1980 .

[3]  P. Ungar.  Freak Theorem About Functions on a Sphere , 1954 .

[4]  A. Savo.  Geometric rigidity of constant heat flow , 2017, Calculus of Variations and Partial Differential Equations.

[5]  Alessandro Savo.  On the heat content functional and its critical domains , 2021, Calculus of Variations and Partial Differential Equations.

[6]  Zizhou Tang,et al.  Isoparametric foliation and Yau conjecture on the first eigenvalue , 2012, 1211.2533.

[7]  A. Savo.  Heat flow, heat content and the isoparametric property , 2014, 1406.2835.

[8]  E. Cartan.  Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques , 1939 .

[9]  Tsunero Takahashi,et al.  Minimal immersions of Riemannian manifolds , 1966 .

[10]  A. Beck.  Heat Flow , 2015 .

[11]  Jianquan Ge,et al.  Isoparametric functions and exotic spheres , 2010, 1003.0355.

[12]  George B. Arfken,et al.  Sturm-Liouville Theory , 2012 .

[13]  Victor Alexandrov,et al.  Problem section , 2007 .

[14]  Qi-Ming Wang,et al.  Isoparametric functions on Riemannian manifolds. I , 1987 .

[15]  Isoparametric Hypersurfaces with Four Principal Curvatures , 2004, math/0402272.

[16]  Stephen A. Williams,et al.  A partial solution of the Pompeiu problem , 1976 .

[17]  Some results in E. Cartan’s theory of isoparametric families of hypersurfaces , 1973 .

[18]  GinéM Evarist,et al.  The addition formula for the eigenfunctions of the Laplacian , 1975 .

[19]  L. Zalcman.  A Bibliographic Survey of the Pompeiu Problem , 1992 .

[20]  H. Münzner.  Isoparametrische Hyperflächen in Sphären , 1980 .

[21]  E. Cartan.  Familles de surfaces isoparamétriques dans les espaces à courbure constante , 1938 .

[22]  S. Helgason.  Differential Geometry and Symmetric Spaces , 1964 .

[23]  Hsien-Chtjng Wang,et al.  TWO-POINT HOMOGENEOUS SPACES , 1952 .

[24]  Carlos A. Berenstein,et al.  An inverse spectral theorem and its relation to the Pompeiu problem , 1980 .

[25]  V. Shklover.  Schiffer problem and isoparametric hypersurfaces , 2000 .

[26]  C. Berenstein,et al.  Pompeiu’s problem on spaces of constant curvature , 1976 .