An upper Minkowski dimension estimate for the interior singular set of area minimizing currents

IF 3.1 1区数学 Q1 MATHEMATICS Communications on Pure and Applied Mathematics Pub Date : 2023-09-18 DOI:10.1002/cpa.22165

Anna Skorobogatova

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Abstract

We show that for an area minimizing m-dimensional integral current T of codimension at least two inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most $m - 2$ . This provides a strengthening of the existing $(m - 2)$ -dimensional Hausdorff dimension bound due to Almgren and De Lellis & Spadaro. As a by-product of the proof, we establish an improvement on the persistence of singularities along the sequence of center manifolds taken to approximate T along blow-up scales.

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面积最小电流内部奇异集的上闵可夫斯基维估计

我们证明了在一个充分正则黎曼流形中，对于余维至少为2的m维积分电流T的面积最小化，内部奇异集的上闵可夫斯基维不超过m-2$ m-2$。这提供了现有的(m−2)$ (m-2)$维Hausdorff维界由于Almgren和De Lellis &;斯巴达罗。作为证明的一个副产品，我们建立了一个关于沿膨胀尺度近似T的中心流形序列上奇点持久性的改进。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Communications on Pure and Applied Mathematics 数学-数学

CiteScore

6.70

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.