On a classification of minimal cubic cones in R^n

We establish a classification of cubic minimal cones in case of the so-called radial eigencubics. Our principal result states that any radial eigencubic is either a member of the infinite family of eigencubics of Clifford type, or belongs to one of 18 exceptional families. We prove that at least 12 of the 18 families are non-empty and study their algebraic structure. We also establish that any radial eigencubic satisfies the trace identity $\det \mathrm{Hess}^3 (f)=\alpha f$ for the Hessian matrix of $f$, where $\alpha\in \R{}$. Another result of the paper is a correspondence between radial eigencubics and isoparametric hypersurfaces with four principal curvatures.