The rigidity of minimal Legendrian submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices

IF 2.1 2区数学 Q1 MATHEMATICS Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-19 DOI:10.1007/s00526-024-02822-5

Pei-Yi Wu, Ling Yang

引用次数: 0

Abstract

In this paper, we study the rigidity problem for compact minimal Legendrian submanifolds in the unit Euclidean spheres via eigenvalues of fundamental matrices, which measure the squared norms of the second fundamental form on all normal directions. By using Lu’s inequality (Lu in J Funct Anal 261:1284–1308, 2011) on the upper bound of the squared norm of Lie brackets of symmetric matrices, we establish an optimal pinching theorem for such submanifolds of all dimensions, giving a new characterization for the Calabi tori. This pinching condition can also be described by the eigenvalues of the Ricci curvature tensor. Moreover, when the third large eigenvalue of the fundamental matrix vanishes everywhere, we get an optimal rigidity theorem under a weaker pinching condition.

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通过基本矩阵的特征值求欧几里得球中最小 Legendrian 子满足的刚性

本文通过基本矩阵的特征值研究单位欧几里得球内紧凑极小 Legendrian 子满面的刚性问题，基本矩阵的特征值度量所有法向上第二基本形式的平方法。利用卢氏不等式（Lu in J Funct Anal 261:1284-1308, 2011）关于对称矩阵的列括号平方法的上界，我们为这种所有维度的子平面建立了最优捏合定理，给出了卡拉比环形的新特征。这种捏合条件也可以用里奇曲率张量的特征值来描述。此外，当基本矩阵的第三个大特征值在任何地方都消失时，我们会在一个较弱的捏合条件下得到一个最优刚性定理。

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.