JournalsrmiVol. 30, No. 1pp. 203–236

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

  • Xiangyu Liang

    University of Warwick, Coventry, UK
Global regularity for minimal sets near a $\mathbb{T}$-set and counterexamples cover
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We discuss the global regularity of 2-dimensional minimal sets that are near a T\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 Rn\mathbb{R}^n that looks like a T\mathbb{T}-set at infinity is a T\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 Rn\mathbb{R}^n and all 2-dimensional Mumford–Shah-minimal sets in R3\mathbb{R}^3 are cones. In this article we discuss two types of 2-dimensional minimal sets: Almgren-minimal sets in R3\mathbb{R}^3 whose blow-in limits are T\mathbb{T}-sets, and topological minimal sets in R4\mathbb{R}^4 whose blow-in limits are T\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.

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Xiangyu Liang, Global regularity for minimal sets near a T\mathbb{T}-set and counterexamples. Rev. Mat. Iberoam. 30 (2014), no. 1, pp. 203–236

DOI 10.4171/RMI/775