Research on geometryof eigenvalues of differential operators and submanifolds

微分算子和子流形特征值的几何研究

• 批准号: 18540091
• 负责人: CHENG Qing ming
• 金额: $ 2.55万
• 依托单位: Saga University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540091/
• 关键词: Bucking problem; eigenvalue; universal inequalit; Laplacian; Biharmonic operator; submanifold; sacalar curvature; Jacobi operator; Dirichlet eigenvalue problem; hypersurface; constant scalar curvature; Riemannian manifold; Gauss-Kronecker curv

In this project, we mainly investigated eigenvalues of the eigenvalue problem of differential operators and the differential geometry of submanifolds. It is our purpose to research estimates for eigenvalues of the eigenvalue problems of differential operators and the differential geometry of submanifolds by means of many different methods. (1) 50 years ago, Payne, Polya and Weinberger proposed to derive a universal inequality for eigenvalues of the buckling problem, which is a very hard problem. By initiating a new method for constructing appropriated trial functions, we solve the hard problem of Payne, Polya and Weinberger. Our results become one of the most main contributions in research for eigenvalues of the buckling problem. (2) For the research on universal inequalities for eigenvalues of a Dirichlet eigenvalue problem of the biharmonic operator, we have solved a problem proposed by Ashbaugh in 1999. (3) The optimal estimates for eigenvalues of the Laplacian on a domain in comple … More x projective spaces and in complex submanifolds of complex projective spaces are obtained. (4) Since it is very difficult to derive an upper bound for the kth eigenvalue of the Laplacian on a domain in Euclidean space of dimension n, there are no any known results about it almost. We prove an algebraic recursion formula, firstly, and then we derive an upper bound for the kth eigenvalue of the Laplacian, which is best possible in the meaning of the order of k. (5) By making use of a theorem of Nash, we construct trial functions, successfully, for the eigenvalue problem of the Laplacian on a domain in a complete Riemannian manifold. By using our trial functions, we obtain universal bounds for eigenvalues of this eigenvalue problem. Our universal bounds are best possible. (6) We study the pinching problem for compact submanifolds with constant Mobius scalar curvature in a unit sphere. Furthermore, we give a classification of this kind of submanifolds. (7) We give an optimal estimate for the first eigenvalue of Jacobi operator of compact hypersurfaces with constant scalar curvature in a unit sphere. Less

在这个项目中，我们主要研究了差异操作员的特征值问题和亚曼群的差异几何形状。我们的目的是研究差异操作员特征值问题的特征值以及通过许多不同方法的副算法的特征值问题和子序列的差异几何形状。 （1）50年前，Payne，Polya和Weinberger提议为屈曲问题的特征值带来普遍的不平等，这是一个非常困难的问题。通过启动一种构建适当试验功能的新方法，我们解决了Payne，Polya和Weinberger的严重问题。我们的结果成为屈曲问题特征值研究中最主要的贡献之一。 (2) For the research on universal inequalities for eigenvalues ​​of a Dirichlet eigenvalue problem of the biharmonic operator, we have solved a problem proposed by Ashbaugh in 1999. (3) The optimal estimates for eigenvalues ​​of the Laplacian on a domain in complete … More x projective spaces and in complex submanifolds of complex projective spaces are obtained. （4）由于在欧几里得空间中的域中，在laplacian的kth特征值中得出上限非常困难，因此几乎没有任何已知结果。我们证明了一个代数递归公式，首先，然后为laplacian的kth特征值提供了上限，这在k的顺序含义上是最好的。 （5）通过利用纳什定理，我们成功地构建了试验功能，以解决完整的riemannian歧管中的域上拉普拉斯的特征值问题。通过使用我们的试验功能，我们获得了该特征值问题的特征值的通用界限。我们的普遍界限是最好的。 （6）我们研究了单位球体中具有恒定Mobius标量曲率的紧凑型亚曲线的捏合问题。此外，我们对这种子手法进行了分类。 （7）我们对单位球体中恒定标态曲率的Jacobi操作员的第一个特征值给出了最佳估计。较少的

项目成果

Estimates for eigenvalues of Laplacian on Riemannian manifolds

黎曼流形上拉普拉斯算子特征值的估计

• 发表时间: 2007
• 作者: Qing-Ming;Cheng
• 通讯作者: Cheng

Inequalites for eigenvalues of Laplation with any order

任意阶 Laplation 特征值的不等式

• 发表时间: 2008
• 期刊: Commun.Contemporary Math. (in press)
• 作者: Susumu;Hirose;Akira;Yasuhara;Qing-Ming Cheng
• 通讯作者: Qing-Ming Cheng

非コンパクト多様体のデイラック作用素の真性スペクトル

非紧流形狄拉克算子的本征谱

• 发表时间: 2007
• 作者: Qing-Ming;Cheng;Yasuhiko Kamiyama;Qing-Ming Cheng;Yasuhiko Kamiyama;成 慶明;Yasuhiko Kamiyama;河合 茂生;Yasuhiko Kamiyama;河合 茂生
• 通讯作者: 河合 茂生

Universal inequalities for eigenvaluesq

特征值的普遍不等式q

• 发表时间: 2006
• 作者: Qing-Ming;Cheng
• 通讯作者: Cheng

INEQUALITIES FOR EIGENVALUES OF LAPLACIAN WITH ANY ORDER

• DOI: 10.1142/s0219199709003533
• 发表时间: 2009-08
• 期刊: Communications in Contemporary Mathematics
• 影响因子: 1.6
• 作者: Q. Cheng;T. Ichikawa;S. Mametsuka