We discuss the global regularity of 2-dimensional minimal sets that are near a -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 that looks like a -set at infinity is a -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 and all 2-dimensional Mumford–Shah-minimal sets in are cones. In this article we discuss two types of 2-dimensional minimal sets: Almgren-minimal sets in whose blow-in limits are -sets, and topological minimal sets in whose blow-in limits are -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.
Cite this article
Xiangyu Liang, Global regularity for minimal sets near a -set and counterexamples. Rev. Mat. Iberoam. 30 (2014), no. 1, pp. 203–236DOI 10.4171/RMI/775