Translational and Sliding Stability for Two-Dimensional Minimal Cones in Euclidean Spaces

In this article, we prove the translational stability for all two-dimensional Almgren minimal cones in ${mathbb{R}}^n$ and the Almgren (resp. topological) sliding stability for the two-dimensional Almgren (resp. topological) minimal cones in ${mathbb{R}}^3$. As proved in [19], when several two-dimensional Almgren (resp. topological) minimal cones are translational, Almgren (resp. topological) sliding stable, and Almgren (resp. topological) unique, their almost orthogonal union stays minimal. As a consequence, the results of this article, together with the uniqueness properties proved in [14], permit us to use all two-dimensional minimal cones in ${mathbb{R}}^3$ to generate new families of minimal cones by taking their almost orthogonal unions.