Uniqueness of 2-dimensional minimal cones in $\mathbb{R}^3$

TL;DR

This paper investigates the uniqueness of 2-dimensional minimal cones in three-dimensional space, establishing properties that enable the construction of new minimal cones through unions and analyzing the stability of these structures.

Contribution

It proves the upper semi continuity property for Almgren minimal sets and confirms the uniqueness of all 2D minimal cones in , facilitating the generation of new minimal cones via unions.

Findings

Upper semi continuity for Almgren minimal sets established

All 2D minimal cones in  are unique

Almost orthogonal unions of stable minimal cones remain minimal

Abstract

In this article we treat two closely related problems: 1) the upper semi continuity property for Almgren minimal sets in regions with regular boundary, which guanrantees that the uniqueness property is well defined; and 2) the Almgren (resp. topological) uniqueness property for all the 2-dimensional Almgren (resp. topological) minimal cones in $\mathbb{R}^3$. As proved in \cite{2T}, when several 2-dimensional Almgren (resp. topological) minimal cones are measure and Almgren (resp. topological) sliding stable, and Almgren (resp. topological) unique, the almost orthogonal union of them stays minimal. As consequence, the results of this article, together with the measure and sliding stability properties proved in \tb{\cite{stablePYT} and \cite{stableYXY}}, permit us to use all known 2-dimensional minimal cones in $\mathbb{R}^n$ to generate new families of minimal cones by taking their almost orthogonal unions. The upper semi continuity property is also helpful in various circumstances: when we have to carry on arguments using Hausdorff limits and some properties do not pass to the limit, the upper semi continuity can serve as a link. As an example, it plays a very important role throughout \cite{2T}.