Quartic isoparametric hypersurfaces and quadratic forms
Every known quartic isoparametric family can be defined by a sum of squares of harmonic quadratic forms, whose common zero set is one focal variety of the family. We show that in fact, any harmonic quadratic form which vanishes on a focal variety produces an eigenfunction of the laplacian on each leaf of the isoparametric family. On the unique minimal hypersurface in each family, the corresponding eigenvalue is always exactly \(2n\) --- twice the hypersurface dimension. This number fills an obvious gap in the sequence of previously known eigenvalues on a minimal quartic isoparametric hypersurface: \(0,n,3n\) and \(4n\). \textit{D. Ferus}, \textit{H. Karcher}, and \textit{H.-F. Münzner} constructed most of the known quartic isoparametric families in Math. Z. 177, 479-502 (1981; Zbl 0443.53037). By studying the quadratic forms associated with these examples in [loc. cit.], we discover infinite ``rational'' sequences of eigenvalues on the leaves of these examples. We also discover, for each family in [loc. cit.] \((1,k)\) and \((2,2k+1)\) series, an entirely new space of quadratic forms that can be used to construct that family, and which therefore produces more eigenfunctions on the family's leaves, as indicated above. In contrast to the 2- or 3- dimensional spaces of forms ascribed to these series in [loc. cit.], the spaces we exhibit are large, their dimensions proportional to \(k^ 2\).
scientific article; zbMATH DE number 4216068