2010 Fiscal Year Final Research Report
The development and fusion on the geometry of submanifolds and theory for eigenvalues of the differential operators in Riemannian manifolds
| Project/Area Number |
20540082
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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, 大学院・工学系研究科, 教授 (50274577)
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| Co-Investigator(Kenkyū-buntansha) |
KAWAI Shigeo 佐賀大学, 文化教育学部, 教授 (30186043)
MAEDA Sadahiro 佐賀大学, 大学院・工学系研究科, 教授 (40181581)
SHODA Toshihiro 佐賀大学, 文化教育学部, 准教授 (10432957)
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| Project Period (FY) |
2008 – 2010
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| Keywords | Laplace作用素の固有値問題 / the clamped plate problem / 固有値に関する普遍不等式 / 固有値の下限 / 固有値の上限 / スカラー曲率 / 超曲面 / 埋め込み超曲面 / 平均曲率 / ヤコビ作用素の固有値 / 安定性指数 |
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| Research Abstract |
In this project, by making use of fusion on the research methods in the geometry of submanifolds and the research methods for eigenvalues of the differential operators on bounded domains in Riemannian manifolds, we can construct trial functions with good properties such that we obtain a sharp universal inequalityfor eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on bounded domainsin Riemannian manifolds and a universal inequality for eigenvalues of the clamped plate problem. Furthermore, by combining the universal inequality for eigenvalues with therecursion formula of Cheng and Yang, we give the upper bounds for the k-th eigenvalueof the Dirichlet eigenvalue problem of the Laplacian on bounded domains in Riemannian manifolds. It is optimal in the sense of the order of k. By using a new and original method replacing the method of the Fourier transform, we obtain a Li-Yau type lower bound for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on bounded domains in Riemannian manifolds. About the study on lower bounds for eigenvalues of the clamped plate problem on bounded domains in a Euclidean space, we improve the inequality of Levine and Protter by using the Fourier transform and the symmetry rearrangement of a domain. On the other hand, we study structures of curvatures and topological structures of submanifolds according to several different view points. An optimal upper bound for the first eigenvalue of Jacobi operator of compact hypersurfaces with constant scalar curvature in the unit sphere is given. Many embedded compact hypersurfaces with constantk-th mean curvature in the unit sphere are constructed.
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