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Global regularity for minimal sets near a $\mathbb{T}$-set and counterexamples
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Liang, Xiangyang (2014) Global regularity for minimal sets near a $\mathbb{T}$-set and counterexamples. Revista Matematica Iberoamericana, Volume 30 (Number 1). pp. 203-236. doi:10.4171/rmi/775 ISSN 0213-2230.
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Official URL: http://dx.doi.org/10.4171/RMI/775
Abstract
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.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Revista Matematica Iberoamericana | ||||
Publisher: | European Mathematical Society Publishing House | ||||
ISSN: | 0213-2230 | ||||
Official Date: | 2014 | ||||
Dates: |
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Volume: | Volume 30 | ||||
Number: | Number 1 | ||||
Page Range: | pp. 203-236 | ||||
DOI: | 10.4171/rmi/775 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published |
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