Global regularity for minimal sets near a $\mathbb{T}$-set and counterexamples

Abstract

We discuss the global regularity of 2-dimensional minimal sets that are near a $\mathbb{T}$-set (i.e., the cone over the 1-skeleton of a regular tetrahedron centered at the origin), that is, whether every global minimal set in ${\mathbb{R}}^n$ that looks like a $\mathbb{T}$-set at infinity is a $\mathbb{T}$-set or not. The main point is to use the topological properties of a minimal set at a large scale to control its topology at smaller scales. This is how one proves that all 1-dimensional Almgren-minimal sets in ${\mathbb{R}}^n$ and all 2-dimensional Mumford–Shah-minimal sets in ${\mathbb{R}}^3$ are cones. In this article we discuss two types of 2-dimensional minimal sets: Almgren-minimal sets in ${\mathbb{R}}^3$ whose blow-in limits are $\mathbb{T}$-sets, and topological minimal sets in ${\mathbb{R}}^4$ whose blow-in limits are $\mathbb{T}$-sets. For the former we eliminate a potential counterexample that was proposed by several people, and show that a genuine counterexample should have a more complicated topological structure; for the latter we construct a potential example using a Klein bottle.