Minimal cubic cones via Clifford algebras

TL;DR

This paper constructs two infinite families of algebraic minimal cones in Euclidean spaces using Clifford algebras, providing new explicit examples and answering longstanding questions about algebraic minimal cones.

Contribution

It introduces explicit constructions of minimal cones via Clifford systems and classifies their congruence classes, expanding the known examples of algebraic minimal cones.

Findings

Existence of minimal cones in R^n for all n ≥ 4, n ≠ 16k+1.

Construction of minimal cones in R^{m^2} using irreducible homogeneous polynomials.

Provides answers to questions on algebraic minimal cones posed by Wu-Yi Hsiang.

Abstract

We construct two infinite families of algebraic minimal cones in $R^{n}$. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byproduct, we prove that for any $n\ge4$, $n\ne 16k+1$, there is at least one minimal cone in $R^{n}$ given by an irreducible homogeneous cubic polynomial. The second family consists of minimal cones in $R^{m^2}$, $m\ge2$, defined by an irreducible homogeneous polynomial of degree $m$. These examples provide particular answers to the questions on algebraic minimal cones posed by Wu-Yi Hsiang in the 1960's.