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2021 Uniqueness property for $2$-dimensional minimal cones in $\mathbb R^3$
Xiangyu Liang
Publ. Mat. 65(1): 3-59 (2021). DOI: 10.5565/PUBLMAT6512101

Abstract

In this article we treat two closely related problems: 1) the upper semi-continuity property for Almgren minimal sets in regions with regular boundary; and 2) the uniqueness property for all the $2$-dimensional minimal cones in $\mathbb R^3$.

Given an open set $\Omega\subset\mathbb R^n$, a closed set $E\subset \Omega$ is said to be Almgren minimal of dimension $d$ in $\Omega$ if it minimizes the $d$-Hausdorff measure among all its Lipschitz deformations in $\Omega$. We say that a $d$-dimensional minimal set $E$ in an open set $\Omega$ admits upper semi-continuity if, whenever $\{f_n(E)\}_n$ is a sequence of deformations of $E$ in $\Omega$ that converges to a set $F$, then we have ${\mathcal H}^d(F)\ge \limsup_n {\mathcal H}^d(f_n(E))$. This guarantees in particular that $E$ minimizes the $d$-Hausdorff measure, not only among all its deformations, but also among limits of its deformations.

As proved in [19], when several $2$-dimensional minimal cones are all translational and sliding stable, and admit the uniqueness property, then their almost orthogonal union stays minimal. As a consequence, the uniqueness property obtained in the present paper, together with the translational and sliding stability properties proved in [18] and [20] permit us to use all known $2$-dimensional minimal cones in $\mathbb R^n$ to generate new families of minimal cones by taking their almost orthogonal unions.

The upper semi-continuity property is also helpful in various circumstances: when we have to carry on arguments using Hausdorff limits and some properties do not pass to the limit, the upper semi-continuity can serve as a link. As an example, it plays a very important role throughout [19].

Citation

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Xiangyu Liang. "Uniqueness property for $2$-dimensional minimal cones in $\mathbb R^3$." Publ. Mat. 65 (1) 3 - 59, 2021. https://doi.org/10.5565/PUBLMAT6512101

Information

Received: 5 April 2019; Revised: 30 March 2020; Published: 2021
First available in Project Euclid: 11 December 2020

MathSciNet: MR4185826
Digital Object Identifier: 10.5565/PUBLMAT6512101

Subjects:
Primary: 28A75
Secondary: 49K21 , 49Q20

Keywords: Hausdorff measure , minimal cones , Plateau's problem , uniqueness

Rights: Copyright © 2021 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.65 • No. 1 • 2021
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