Measure and sliding stability for 2-dimensional minimal cones in Euclidean spaces

Abstract

In this article we prove the measure stability for all 2-dimensional Almgren minimal cones in $\mathbb{R}^n$, and the Almgren (resp. topological) sliding stability for 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 uniqueness properties proved in \cite{uniquePYT}, permit us to use all 2-dimensional minimal cones in $\mathbb{R}^3$ to generate new families of minimal cones by taking their almost orthogonal unions.