Almgren-minimality of unions of two almost orthogonal planes in $\mathbb R^4$

TL;DR

This paper proves that the union of two nearly orthogonal planes in four-dimensional space is Almgren-minimal, introducing a new example of minimal cones that expands understanding of singularities in higher dimensions.

Contribution

It establishes the Almgren-minimality of unions of two almost orthogonal planes in R4, providing a novel example of minimal cones in four dimensions.

Findings

Union of two almost orthogonal planes in R4 is Almgren-minimal.

Introduces a new one-parameter family of minimal cones in R4.

Uses advanced techniques like stopping time arguments and harmonic extensions.

Abstract

In this article we prove that the union of two almost orthogonal planes in R4 is Almgren-minimal. This gives an example of a one parameter family of minimal cones, which is a phenomenon that does not exist in R3. This work is motivated by an attempt to classify the singularities of 2-dimensional Almgren-minimal sets in R4. Note that the traditional methods for proving minimality (calibrations and slicing arguments) do not apply here, we are obliged to use some more complicated arguments such as a stopping time argument, harmonic extensions, Federer-Fleming projections, etc. that are rarely used to prove minimality (they are often used to prove regularity). The regularity results for 2-dimensional Almgren minimal sets ([5],[6]) are also needed here.