Global regularity for minimal sets near a $\mathbb{T}$-set and counterexamples
Xiangyu Liang · Revista Matemática Iberoamericana · 2014
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.