On a classification of minimal cubic cones in R^n

TL;DR

This paper classifies cubic minimal cones called radial eigencubics, revealing they are either of Clifford type or belong to 18 exceptional families, and explores their algebraic structure and geometric connections.

Contribution

It provides a comprehensive classification of radial eigencubics, identifying their algebraic properties and establishing links to isoparametric hypersurfaces with four principal curvatures.

Findings

Radial eigencubics are either Clifford type or in 18 exceptional families.

At least 12 of the 18 families are non-empty.

Radial eigencubics satisfy a specific trace identity involving the Hessian.

Abstract

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.