Planar Normal Sections on Isoparametric Hypersurfaces and the infinity laplacian

Abstract

The present article is devoted to present a new characterization of the Cartan isoparametric hypersurfaces in terms of properties of the polynomial, that determines the algebraic set of planar normal sections on the homogeneous isoparametric hypersurfaces in spheres. We show that Cartan isoparametric hypersurfaces are the only homo- geneous isoparametric hypersurfaces in spheres for which the Innity Laplacian of the polynomial that denes the algebraic set of planar normal sections is the polynomial multiplied by the squared norm of the tangent vector. Since it is required for our work, we also give these polynomials for all homogeneous isoparametric hypersurfaces in spheres.