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

基本信息

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

项目摘要

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，我们通过许多不同的方法来调查差异操作员的特征值。。获得复杂的投影空间。 K.使用我们的试验功能，在一个完整的riemnian歧管中，laplacian的特征值问题。单位球体的分类

项目成果

期刊论文数量（0）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

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

DOI：
发表时间：
2007
期刊：
影响因子：
0
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

DOI：
发表时间：
2008
期刊：
Commun.Contemporary Math. (in press)
影响因子：
0
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

DOI：
发表时间：
2007
期刊：
影响因子：
0
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