Minimal Piecewise Linear Cones in $\mathbb{R}^{4}$

TL;DR

This paper classifies all three-dimensional piecewise linear cones in four-dimensional space that are mass minimizing, revealing exactly five such cones and extending understanding of their geometric properties.

Contribution

The authors provide a complete classification of minimal piecewise linear cones in \,R^4, identifying all possible configurations and proving the non-existence of others.

Findings

Identified five minimal piecewise linear cones in R^4.

Proved no additional minimal cones exist beyond these five.

Extended the classification of mass minimizing cones in higher dimensions.

Abstract

We consider three dimensional piecewise linear cones in $\mathbb{R}^4$ that are mass minimizing w.r.t. Lipschitz maps in the sense of \cite{almgren1976existence} as in \cite{Taylor76}. There are three that arise naturally by taking products of $\mathbb{R}$ with lower dimensional cases and earlier literature has demonstrated the existence of two with 0-dimensional singularities. We classify all possible candidates and demonstrate that there are no p.l. minimizers outside these five.