论欧拉-迪尔克斯-惠斯肯变分问题

IF 1.3 2区 数学 Q1 MATHEMATICS
Hongbin Cui, Xiaowei Xu
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引用次数: 0

摘要

本文研究了欧几里得空间中与 f 加权面积函数 $$\begin{aligned} 相关的高次元曲面的稳定性和最小化特性。\d \mathcal {H}_k \end{aligned}$$带有密度函数 \(f(x)=g(|x|)\) 且 g(t) 为非负,这发展了 U. Dierkes 和 G. Huisken(《数学年刊》,2023 年 10 月 20 日)最近关于带有密度函数 \(|x|^\alpha \) 的超曲面的研究。在关于 g(t) 的适当假设下,我们证明了具有全局平坦法向束的极小圆锥是 f 稳定的,我们还证明了满足劳勒曲率准则的极小圆锥、行列式变种和普法方变种在没有某些特殊情况下是 f 最小化的。作为应用,我们证明了球积上的k维圆锥对于\(\alpha \ge -k+2\sqrt{2(k-1)}\)是\(|x|^\alpha \)稳定的,定向稳定的最小超圆锥对于\(\alpha \ge 0\) 是\(|x|^\alpha \)稳定的、而且我们还证明了对于(dim \mathcal {C} \ge 7\), (k_i>;1) and\(\alpha \ge 0\), the Simons cones \(C(S^{p} \times S^{p})\) are \(|x|^\alpha \)-minimizing for \(\alpha \ge 1\), which relaxes the assumption \(1\le \alpha \le 2p\) in Dierkes and Huisken ( Math Ann, https://doi.org/10.1007/s00208-023-02726-3, 2023).最近,迪尔克斯(Rend Sem Mat Univ Padova, 2024)证明了\(C(S^{p} \times S^{p})\)对于\(\alpha \ge 3-p\)是\(|x|^\alpha \)最小化的,这改进了我们对于\(p\ge 3\)的假设\(\alpha \ge 1\).
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On Euler–Dierkes–Huisken variational problem

In this paper, we study the stability and minimizing properties of higher codimensional surfaces in Euclidean space associated with the f-weighted area-functional

$$\begin{aligned} \mathcal {E}_f(M)=\int _M f(x)\; d \mathcal {H}_k \end{aligned}$$

with the density function \(f(x)=g(|x|)\) and g(t) is non-negative, which develop the recent works by U. Dierkes and G. Huisken (Math Ann, 20 October 2023) on hypersurfaces with the density function \(|x|^\alpha \). Under suitable assumptions on g(t), we prove that minimal cones with globally flat normal bundles are f-stable, and we also prove that the minimal cones satisfy the Lawlor curvature criterion, the determinantal varieties and the Pfaffian varieties without some exceptional cases are f-minimizing. As an application, we show that k-dimensional cones over product of spheres are \(|x|^\alpha \)-stable for \(\alpha \ge -k+2\sqrt{2(k-1)}\), the oriented stable minimal hypercones are \(|x|^\alpha \)-stable for \(\alpha \ge 0\), and we also show that the cones over product of spheres \(\mathcal {C}=C \left( S^{k_1} \times \cdots \times S^{k_{m}}\right) \) are \(|x|^\alpha \)-minimizing for \(\dim \mathcal {C} \ge 7\), \(k_i>1\) and \(\alpha \ge 0\), the Simons cones \(C(S^{p} \times S^{p})\) are \(|x|^\alpha \)-minimizing for \(\alpha \ge 1\), which relaxes the assumption \(1\le \alpha \le 2p\) in Dierkes and Huisken ( Math Ann, https://doi.org/10.1007/s00208-023-02726-3, 2023). Recently, Dierkes (Rend Sem Mat Univ Padova, 2024) prove that \(C(S^{p} \times S^{p})\) are \(|x|^\alpha \)-minimizing for \(\alpha \ge 3-p\), which has improved our assumption \(\alpha \ge 1\) for \(p\ge 3\).

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来源期刊
Mathematische Annalen
Mathematische Annalen 数学-数学
CiteScore
2.90
自引率
7.10%
发文量
181
审稿时长
4-8 weeks
期刊介绍: Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin. The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.
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