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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

Research Project

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Project/Area Number 20540082
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionSaga University

Principal Investigator

CHENG Qing-Ming  Saga University, 大学院・工学系研究科, 教授 (50274577)

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)
Project Period (FY) 2008 – 2010
KeywordsLaplace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
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.

  • Research Products

    (49 results)

All 2011 2010 2009 2008 Other

All Journal Article (20 results) (of which Peer Reviewed: 20 results) Presentation (28 results) Remarks (1 results)

  • [Journal Article] A lower bound for eigenvalues of a clamped plate problem2011

    • Author(s)
      Qing-Ming Cheng Li, Guoxin Wei
    • Journal Title

      Calculus of Variations and PDEs

      Pages: DOI:10.1007/s00526-011-0399-6

    • Peer Reviewed
  • [Journal Article] Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space2011

    • Author(s)
      Qing-Ming Cheng, Hongcang Yang
    • Journal Title

      Proc.Amer.Math.Soc. 139

      Pages: 461-471

    • Peer Reviewed
  • [Journal Article] Embedded hypersurfaces with constant mth mean curvature in a unit sphere2010

    • Author(s)
      Qing-Ming Cheng, Haizhong Li, Guoxin Wei
    • Journal Title

      Comm.Contemporary Math. 12

      Pages: 997-1013

    • Peer Reviewed
  • [Journal Article] On some rigidity results of hypersurfaces in a sphere2010

    • Author(s)
      Qing-Ming Cheng, Haizhong Li, Guoxin Wei
    • Journal Title

      Proc.Roy.Soc.Edinburgh 140A

      Pages: 477-493

    • Peer Reviewed
  • [Journal Article] A classification of hypersurfaces with parallel para-Blaschke tensor in S^{m+1}2010

    • Author(s)
      Qing-Ming Cheng, Xingxiao Li, Xuerong Qi
    • Journal Title

      International J.Math. 21

      Pages: 297-316

    • Peer Reviewed
  • [Journal Article] Estimates for lower order eigenvalues of a clamped plate problem2010

    • Author(s)
      Qing-Ming Cheng, Guangyue Huang, Guoxin Wei
    • Journal Title

      Calculus of Variations and PDEs 38

      Pages: 409-416

    • Peer Reviewed
  • [Journal Article] Estimates for eigenvalues of a clamped plate problem on Riemannian manifolds2010

    • Author(s)
      Qing-Ming Cheng, Takamichi Ichikawa, Shinji Mametsuka
    • Journal Title

      J.Math.Soc.Japan 62

      Pages: 673-686

    • Peer Reviewed
  • [Journal Article] Estimates for eigenvalues on Riemannian manifolds2009

    • Author(s)
      Qing-Ming Cheng, Hongcang Yang
    • Journal Title

      J.Diff.Eqns. 47

      Pages: 2270-2281

    • Peer Reviewed
  • [Journal Article] Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere2009

    • Author(s)
      Qing-Ming Cheng, Takamichi Ichikawa, Shinji Mametsuka
    • Journal Title

      Calculus of Variations and PDEs 36

      Pages: 507-523

    • Peer Reviewed
  • [Journal Article] Inequalities for eigenvalues of Laplacian with any order2009

    • Author(s)
      Qing-Ming Cheng, Takamichi Ichikawa, Shinji Mametsuka
    • Journal Title

      Comm.Contemp.Math. 11

      Pages: 639-655

    • Peer Reviewed
  • [Journal Article] Scalar curvature of hypersurfaces with constant mean curvature in a sphere2009

    • Author(s)
      Qing-Ming Cheng, Yujun He, Haizhong Li
    • Journal Title

      Glasg.Math.J. 51

      Pages: 413-423

    • Peer Reviewed
  • [Journal Article] Universal inequalities for eigenvalues of a system of elliptic equations2009

    • Author(s)
      Qing-Ming Cheng, Hongcang Yang
    • Journal Title

      Proc.Royal Soc.Edinburgh 139A

      Pages: 273-285

    • Peer Reviewed
  • [Journal Article] Characterization of parallel isometric immersions of space froms into space forms in the calss of isotropic immersions2009

    • Author(s)
      Sadahiro Maeda, Seiichi Udagawa
    • Journal Title

      Canad.J.Math. 61

      Pages: 641-655

    • Peer Reviewed
  • [Journal Article] A characterization of the homogeneous minimal ruled real hypersurface in a complex hyperbolic space2009

    • Author(s)
      Sadahiro Maeda, Toshiaki Adachi, Young Ho Kim
    • Journal Title

      J.Math.Soc.Japan 61

      Pages: 315-325

    • Peer Reviewed
  • [Journal Article] An example of Jacobian variety and its applications to minimal surfaces2009

    • Author(s)
      Toshihiro Shoda
    • Journal Title

      Results Math. 56

      Pages: 23-39

    • Peer Reviewed
  • [Journal Article] Complete harmonic stable minimal hypersurfaces in a Riemannian manifold2008

    • Author(s)
      Qing-Ming Cheng, Young Jin Suh
    • Journal Title

      Mh.Math. 154

      Pages: 121-134

    • Peer Reviewed
  • [Journal Article] First eigenvalue of Jacobi operator of hypersurfaces with constant Scalar curvature2008

    • Author(s)
      Qing-Ming Cheng
    • Journal Title

      Proc.Amer.Math.Soc. 136

      Pages: 3309-3318

    • Peer Reviewed
  • [Journal Article] Extrinsic estimates for eigenvalues of the Laplace operator2008

    • Author(s)
      Daguang Chen, Qing-Ming Cheng
    • Journal Title

      J.Math.Soc.Japan 60

      Pages: 325-339

    • Peer Reviewed
  • [Journal Article] Lower order eigenvalues of Dirichlet Laplacian2008

    • Author(s)
      Hejun Sun, Qing-Ming Cheng, Hongcang Yang
    • Journal Title

      Manuscripta Math. 152

      Pages: 139-156

    • Peer Reviewed
  • [Journal Article] Etrinsic geodesics and hypersurfaces of type (A) in a complex projective space2008

    • Author(s)
      Sadahiro Maeda, Toshiaki Adachi
    • Journal Title

      Tohoku Math.J. 60

      Pages: 597-605

    • Peer Reviewed
  • [Presentation] Estimates for eigenvalues on complete Riemannian manifolds2011

    • Author(s)
      成慶明
    • Organizer
      International Workshop : Differential Geometry and Geometric Analysis
    • Place of Presentation
      Osaka City University.
    • Year and Date
      20110313-20110315
  • [Presentation] バックリング問題の固有値に関する普遍不等式について2010

    • Author(s)
      成慶明
    • Organizer
      金沢大学幾何学セミナー
    • Place of Presentation
      金沢大学
    • Year and Date
      2010-11-24
  • [Presentation] A new process on universal estimates for eigenvalues of the buckling problem2010

    • Author(s)
      成慶明
    • Organizer
      Fudan University
    • Place of Presentation
      Shanghai, China.
    • Year and Date
      2010-09-10
  • [Presentation] Can one hear the shape of a drum?2010

    • Author(s)
      成慶明
    • Organizer
      Xianyang Normal University
    • Place of Presentation
      Xianyang, China.
    • Year and Date
      2010-09-01
  • [Presentation] バックリング問題の固有値に関する予想及びそれの新展望について2010

    • Author(s)
      成慶明
    • Organizer
      九州大学幾何学セミナー
    • Place of Presentation
      九州大学
    • Year and Date
      2010-07-23
  • [Presentation] バックリング問題の固有値に関する普遍不等式の研究についての新展望2010

    • Author(s)
      成慶明
    • Organizer
      東北大学幾何学セミナー
    • Place of Presentation
      東北大学
    • Year and Date
      2010-07-13
  • [Presentation] A conjecture on eigenvalues of the buckling problem2010

    • Author(s)
      成慶明
    • Organizer
      Wuhan University
    • Place of Presentation
      Wuhan, China.
    • Year and Date
      2010-04-30
  • [Presentation] On eigenvalues of Laplacian2010

    • Author(s)
      成慶明
    • Organizer
      Hubei University
    • Place of Presentation
      Wuhan, China.
    • Year and Date
      2010-04-29
  • [Presentation] Laplace作用素の固有値下限及び上限に関する研究2010

    • Author(s)
      成慶明
    • Organizer
      金沢大学幾何学セミナー
    • Place of Presentation
      金沢大学
    • Year and Date
      2010-02-17
  • [Presentation] Indices of hypersurfaces in a sphere2009

    • Author(s)
      成慶明
    • Organizer
      湯沢研究集会「部分多様体論」
    • Place of Presentation
      湯沢グランドホテル
    • Year and Date
      20091126-20091128
  • [Presentation] A Jacobi operator of hypersurfaces with constant scalar curvature2009

    • Author(s)
      成慶明
    • Organizer
      幾何学阿蘇研究集会
    • Place of Presentation
      阿蘇休暇村
    • Year and Date
      20090913-20090916
  • [Presentation] Recent developments in universal estimates for eigenvalues of a biharmonic operator2009

    • Author(s)
      成慶明
    • Organizer
      Perspectives and open problems in geometric analysis I
    • Place of Presentation
      Zhejiang University, Hangzhou, China.
    • Year and Date
      20090706-20090711
  • [Presentation] A new process on estimates for eigenvalues of the buckling problem2009

    • Author(s)
      成慶明
    • Organizer
      Tsinghua University
    • Place of Presentation
      Beijing, China.
    • Year and Date
      20090500
  • [Presentation] A solution of the PPW problem, The Symposium on Geometry and Analysis on Manifolds2009

    • Author(s)
      成慶明
    • Organizer
      Chongqing University of Science and Technology
    • Place of Presentation
      Chongqing, China.
    • Year and Date
      20090425-20090429
  • [Presentation] Indices of compact hypersurfaces2009

    • Author(s)
      成慶明
    • Organizer
      Beijing Normal Univ.
    • Place of Presentation
      Beijing, China.
    • Year and Date
      2009-11-04
  • [Presentation] Biharmonic作用素の固有値評価の研究について2009

    • Author(s)
      成慶明
    • Organizer
      東京理科大学幾何学セミナー
    • Place of Presentation
      東京理科大学
    • Year and Date
      2009-06-11
  • [Presentation] Hypersurfaces in spheres2009

    • Author(s)
      成慶明
    • Organizer
      Southwest University
    • Place of Presentation
      Chongqing, China.
    • Year and Date
      2009-04-25
  • [Presentation] Universal inequalities for eigenvalues of a biharmonic operator2008

    • Author(s)
      成慶明
    • Organizer
      The 4th Geometry Conference for the Friendship of China and Japan
    • Place of Presentation
      Nankai University, Tenjin, China.
    • Year and Date
      20081222-20081227
  • [Presentation] Upper and lower bounds for eigenvalues of Laplacian2008

    • Author(s)
      成慶明
    • Organizer
      Integrable System, Geometry and Visualization 2008
    • Place of Presentation
      Kyushu University.
    • Year and Date
      20081211-20081213
  • [Presentation] 双調和作用素の固有値に関する普遍不等式について2008

    • Author(s)
      成慶明
    • Organizer
      多様体上の微分方程式
    • Place of Presentation
      金沢市いしかわシティカレッジ
    • Year and Date
      20081201-20081203
  • [Presentation] Eigenvalues of Laplacian on Riemannian manifolds, Perspectives in Geometric Analysis2008

    • Author(s)
      成慶明
    • Organizer
      BICMR, Peking University
    • Place of Presentation
      Beijing, China.
    • Year and Date
      20081027-20081031
  • [Presentation] Estimates for eigenvalues of Laplacian2008

    • Author(s)
      成慶明
    • Organizer
      Capital Normal Univ.
    • Place of Presentation
      Beijing, China.
    • Year and Date
      20081000
  • [Presentation] Universal estimates for eigenvalues of Laplacian2008

    • Author(s)
      成慶明
    • Organizer
      Wuhan University
    • Place of Presentation
      Wuhan, China.
    • Year and Date
      20081000
  • [Presentation] リーマン多様体における固有値の普遍評価について2008

    • Author(s)
      成慶明
    • Organizer
      第1回佐賀大学数学研究交流会
    • Place of Presentation
      佐賀大学
    • Year and Date
      20080900
  • [Presentation] Estimates for eigenvalues of the buckling problem2008

    • Author(s)
      成慶明
    • Organizer
      東北大学 幾何学セミナー
    • Place of Presentation
      東北大学
    • Year and Date
      20080700
  • [Presentation] A generalized conjecture of Polya2008

    • Author(s)
      成慶明
    • Organizer
      International Conference on Differential Geometry
    • Place of Presentation
      Hefei, China
    • Year and Date
      20080523-20080527
  • [Presentation] Universal estimates for eigenvalues of a biharmonic operator2008

    • Author(s)
      成慶明
    • Organizer
      One Day Workshop on Differential Geometry
    • Place of Presentation
      Saga University.
    • Year and Date
      2008-11-20
  • [Presentation] Estimates for eigenvalues of a biharmonic operator2008

    • Author(s)
      成慶明
    • Organizer
      Zhejiang University
    • Place of Presentation
      Hongzhou, China
    • Year and Date
      2008-05-30
  • [Remarks] ホームページ等

    • URL

      http://www.ms.saga-u.ac.jp/~cheng/index_j.html

URL: 

Published: 2012-01-26   Modified: 2016-04-21  

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