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

Abstract

In this article we treat two closely related problems: 1) the upper semicontinuity property for Almgren minimal sets in regions with regular boundary; and 2) the uniqueness property for all the 2-dimensional minimal cones in R3 . Given an open set Ω ⊂ Rn, a closed set E ⊂ Ω is said to be Almgren minimal of dimension d in Ω if it minimizes the d-Hausdorff measure among all its Lipschitz deformations in Ω. We say that a d-dimensional minimal set E in an open set Ω admits upper semi-continuity if, whenever {fn(E)}n is a sequence of deformations of E in Ω that converges to a set F, then we have Hd(F) ≥ lim supn Hd(fn(E)). This guarantees in particular that E minimizes the d-Hausdorff measure, not only among all its deformations, but also among limits of its deformations. As proved in [19], when several 2-dimensional minimal cones are all translational and sliding stable, and admit the uniqueness property, then their almost orthogonal union stays minimal. As a consequence, the uniqueness property obtained in the present paper, together with the translational and sliding stability properties proved in [18] and [20] permit us to use all known 2-dimensional minimal cones in Rn 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 [19].