Willmore超曲面与极值超曲面的谱特征

数学物理学报 ›› 2023, Vol. 43 ›› Issue (1): 35-42.

Willmore超曲面与极值超曲面的谱特征

杨登允^1,^*,张金国¹(),陶永芊²()

¹江西师范大学数学与统计学院南昌 330022
²南昌大学数学系南昌 330031

收稿日期:2021-11-24 修回日期:2022-10-17 出版日期:2023-02-26 发布日期:2023-03-07
通讯作者: ^*杨登允, E-mail: yangdengyun@139.com
作者简介:张金国, E-mail: jgzhang@jxnu.edu.cn|陶永芊, E-mail: taoyongqian@ncu.edu.cn
基金资助:
国家自然科学基金(12061036);国家自然科学基金(11761049);江西省自然科学基金重点项目(20202ACB201001)

Spectral Geometry of Willmore and Extremal Hypersurfaces

Yang Dengyun^1,^*,Zhang Jinguo¹(),Tao Yongqian²()

¹School of Mathematics and Statistic, Jiangxi Normal University, Nanchang 330022
²Department of Mathematics, Nanchang University, Nanchang 330031

Received:2021-11-24 Revised:2022-10-17 Online:2023-02-26 Published:2023-03-07
Supported by:
National Natural Science Foundation of China(12061036);National Natural Science Foundation of China(11761049);Jiangxi Provincial Natural Science Foundation(20202ACB201001)

摘要/Abstract

摘要：

设 $M$ 为单位球面 $S^{n+1}$ 中的Willmore超曲面(或极值超曲面). 该文证明了, 若 $M$ 与Willmore环面 $W_{m,n-m}$ (或Clifford环面$C_{m,n-m}$)具有相同的第二基本形式模长, 并且 $Spec^p(M)=Spec^p(W_{m,n-m})$ (或$Spec^p(M)=Spec^p(C_{m,n-m})$),其中 $p=0,1,2$, 则有 $M=W_{m,n-m}$ (或$M=C_{m,m}$).

关键词: 拉普拉斯算子, 谱, Willmore超曲面, 极值超曲面, 第二基本形式

Abstract:

Let $M$ be a Willmore (or extremal) hypersurface in $S^{n+1}$ with the same squared length of the second fundamental form of Willmore torus $W_{m,n-m}$ (or Clifford torus $C_{m,n-m}$). In this article the authors proved that if $Spec^p(M)=Spec^p(W_{m,n-m})$ (or $Spec^p(M)=Spec^p(C_{m,n-m})$) for $p=0,1,2$, then $M$ is $W_{m,n-m}$ (or $C_{m,m}$).

Key words: Spectrum, Laplace operator, Extremal hypersurface, Willmore hypersurface, The second fundamental form

中图分类号:

O186.12

杨登允, 张金国, 陶永芊. Willmore超曲面与极值超曲面的谱特征[J]. 数学物理学报, 2023, 43(1): 35-42.

Yang Dengyun, Zhang Jinguo, Tao Yongqian. Spectral Geometry of Willmore and Extremal Hypersurfaces[J]. Acta mathematica scientia,Series A, 2023, 43(1): 35-42.

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