Minkowski content estimates for generic area minimizing hypersurfaces

IF 2.1 2区数学 Q1 MATHEMATICS

Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-13 DOI:10.1007/s00526-024-02791-9

Xuanyu Li

引用次数: 0

Abstract

Let \(\Gamma \) be a smooth, closed, oriented, \((n-1)\)-dimensional submanifold of \(\mathbb {R}^{n+1}\). It was shown by Chodosh–Mantoulidis–Schulze [6] that one can perturb \(\Gamma \) to a nearby \(\Gamma '\) such that all minimizing currents with boundary \(\Gamma '\) are smooth away from a set with Hausdorff dimension less than \(n-9\). We prove that the perturbation can be made such that the singular set of the minimizing current with boundary \(\Gamma '\) has Minkowski dimension less than \(n-9\).

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一般面积最小超曲面的闵科夫斯基内容估计值

让 \(\Gamma \)是 \(\mathbb {R}^{n+1}\) 的一个光滑、封闭、定向、（(n-1)\）维的子满面。Chodosh-Mantoulidis-Schulze [6]证明，我们可以扰动\(\Gamma \)到附近的\(\Gamma '\)，使得所有边界为\(\Gamma '\)的最小化流都平滑地远离一个豪斯多夫维度小于\(n-9\)的集合。我们证明，扰动可以使得边界为（\Gamma '\）的最小化电流的奇异集合的闵科夫斯基维度小于（n-9）。

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

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.