2007 Fiscal Year Final Research Report Summary
Research on geometryof eigenvalues of differential operators and submanifolds
| Project/Area Number |
18540091
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Saga University |
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Principal Investigator |
CHENG Qing ming Saga University, Fac. of Sci. and Eng., Prof. (50274577)
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| Co-Investigator(Kenkyū-buntansha) |
KAWAI Shigeo Saga Univ., Fac. of Cul and Edu., Prof. (30186043)
HIROSE Susumu Saga Univ., Fac. of Sci. and Eng., Ass. Prof. (10264144)
MASHIKO Yukihiro Saga Univ., Fac. of Sci. and Eng., Lecturer (00315178)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Bucking problem / eigenvalue / universal inequalit / Laplacian / Biharmonic operator / submanifold / sacalar curvature / Jacobi operator |
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| Research Abstract |
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
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Research Products
(86 results)
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[Presentation] 固有値の上限について2007
Author(s)
成 慶明
Organizer
リーマン幾何と幾何解析研究集会
Place of Presentation
筑波大学
Year and Date
20070305-07
Description
「研究成果報告書概要(和文)」より
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[Book] 線形代数学入門2006
Author(s)
石川 晋
Total Pages
230
Publisher
丸善出版
Description
「研究成果報告書概要(和文)」より
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