Global regularity for minimal sets near a T-set and counterexamples

We discuss the global regularity of 2-dimensional minimal sets that are near a 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 Rn that looks like a T-set at infinity is a 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 Rn and all 2-dimensional Mumford-Shah-minimal sets in ℝ³ are cones. In this article we discuss two types of 2-dimensional minimal sets: Almgren-minimal sets in ℝ³ whose blow-in limits are T-sets, and topological minimal sets in ℝ⁴ whose blow-in limits are 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.