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GEOMETRY OF LAPLACE OPERATOR OR ITS VARIATION TYPE OPERATOR ON RIEMANNIAN MANIFOLD (2003)

Research Project

Project/Area Number 12640078
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionSAGA UNIVERSITY

Principal Investigator

ISHIKAWA Susumu  SAGA UNIV., MATHEMATICS, PROFESSOR, 理工学部, 教授 (10039258)

Co-Investigator(Kenkyū-buntansha) SHIOHAMA Katsuhiro  SAGA UNIV., MATHEMATICS, PROFESSOR, 理工学部, 教授 (20016059)
KUROGI Tetsunori  FUKUI UNIV., MATHEMATICS, PROFESSOR, 教育地域, 教授 (90022681)
KAWAI Shigeo  SAGA UNIV., MATHEMATICS, PROFESSOR, 文化教育学部, 教授 (30186043)
CHENG Qing-ming  SAGA UNIV., MATHEMATICS, PROFESSOR, 理工学部, 教授 (50274577)
YAMADA Kotaro  KYUSHU UNIV., MATHEMATICS, PROFESSOR, 数理学, 教授 (10221657)
猿子 幸弘  佐賀大学, 理工学部, 講師 (00315178)
Project Period (FY) 2000 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 2003: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2001: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2000: ¥1,900,000 (Direct Cost: ¥1,900,000)
KeywordsFinite type submanifold / Biharmonic submanifold / Minimal submanifold / 部分多様体論 / 極小曲面論 / 共形幾何学 / 有限型多様体 / 二重調和多様体
Research Abstract

We tried to investigate the following themes or questions with concernig of the finite type submanifolds or biharmonic submanifolds :
(i)Are there the finite type surfaces with given mean curvature H in the family of surfaces of revolution x.(u, v)=(u cos v, u sin v, f(u)) generated by the periodic function z=f(u) ?
(ii)Are there the finite type surfaces with given Gauss curvature K in the family of surfaces of revolution x(u, v)=(u cos v, u sin v, f(u)) generated by the function z=f(u) ?
(iii)Are there the finite type surfaces with constant mean curvature in the family of surfaces ?
(iv)Are there the Willmore surfaces of finite type ?
(v-1)The finite type submanifolds in the space with D` Atri metric.
(v-2)The biharmonic submanifolds in the space with D` Atri metric
We will continue to study in future the following still open conjectures of Prof. Bang-yen Chen
1.To determine all of the finite type surfaces in Eucldean space of dimension 3
(Chen conjectur 1 : The only finite type surfaces in Eucldean space of dimension 3 are the minimal suefaces, spheres and right cylinders.)
2.To determine all of the biharmonic submanifolds in Eucldean space of dimension n
(Chen conjectur 2 : The only in Eucldean space of dimension n (n>3) are the harmonc ones.)
3.To determine all of the biharmonic submanifolds in Minkowsky space of dimension 4.

Report

(5 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (35 results)

All 2005 2003 2002 2001 2000 Other

All Journal Article (32 results) Book (1 results) Publications (2 results)

  • [Journal Article] Complete space-like hypersurfaces…2003

    • Author(s)
      Q.M.Cheng, Y.J.suh
    • Journal Title

      J.Geom.and Phy.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Compact hypersurfaces in a unit sphere with infinite fundamental group2003

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

      Pacific J.Math. vol.212

      Pages: 49-56

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Complete space-like hypersurfaces in locally symmetric Lorentz spaces2003

    • Author(s)
      J.O.Baek, Q.-M.Cheng, Y.J.Suh
    • Journal Title

      J.Geom.and Phy, (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Flat fronts in hyperbolic 3-space2003

    • Author(s)
      Masatoshi Kokubu, Masaaki Umehara, Kotaro Yamada
    • Journal Title

      Pacific Journal of Mathematics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Symmetric CMC-1 surfaces with irregular ends2003

    • Author(s)
      Wayne Rossmman, Masaaki Umehara, Kotaro Yamada
    • Journal Title

      Clay Mathematics Proceedings vol.3(to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] An elementary proof of Small's formula for null curves in PSL(2,C) and an analogy( for Legendrian. curves in PSL(2,C)2003

    • Author(s)
      Masatoshi Kokubu, Masaaki Umehara, Kotaro Yamada
    • Journal Title

      Osaka Journal of Matheamtics vol 40

      Pages: 687-715

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature II2003

    • Author(s)
      Wayne Rossman, Masaaki Umehara, Kotaro Yamada
    • Journal Title

      Tohoku Mathematical Journal vol.55

      Pages: 375-395

    • NAID

      110000027016

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Some homological invariants of a mapping class group of a 3-dimensional handlebody2003

    • Author(s)
      Susumu Hirose
    • Journal Title

      Tohoku Mathematical Journal 55 Issue 4

      Pages: 543-549

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] A four dimensional analogy of torus links2003

    • Author(s)
      Susumu Hirose
    • Journal Title

      Topology and its applications 133 Issue 3

      Pages: 199-207

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Erratum to conformally flat…2002

    • Author(s)
      Q.M.Cheng, S.Ishikawa
    • Journal Title

      Journ.of Math.Soc.Japan 54

      Pages: 997-1000

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Submanifolds with constant scalar…2002

    • Author(s)
      Q.M.Cheng
    • Journal Title

      Proc.Royal Soc.Edinbergh 132A

      Pages: 1163-1168

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Erratum to Conformally flat 3-manifolds with constant scalar curvature2002

    • Author(s)
      Qing-Ming Cheng, Susumu Ishikawa, Katsuhiro Shiohama
    • Journal Title

      J.Math.Soc.Japan 54

      Pages: 997-100

    • NAID

      10010355856

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Complete hypersurfaces in a Euclidean space R^<n+1> with constant scalar curvature2002

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

      Indiana Univ.Math.J. 51

      Pages: 53-68

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Submanifolds with constant scalar curvature2002

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

      Proc.Royal Soc.Edinbergh 132A

      Pages: 1163-1183

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Minimal surfaces that attain equality in the Chern-Osserman inequality in "Differential Geometry and Integrable Systems"2002

    • Author(s)
      Masatoshi Kokubu, Masaaki Umehara, Kotaro Yamada
    • Journal Title

      Contemporary Mathematics, (Ed.M.Guest, R.Miyaoka., Y.Ohnita)(American Mathematical Society) 308

      Pages: 223-228

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] A complex of curves and a presentation for the mapping class group of a surface2002

    • Author(s)
      Susumu Hirose
    • Journal Title

      Osaka Journal of Mathematics 39

      Pages: 795-820

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Diffeomorphisms over surfaces trivially embedded in the 4-sphere2002

    • Author(s)
      Susumu Hirose
    • Journal Title

      Algebraic and Geometric Topology 2

      Pages: 791-824

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Con formally flat 3-manifolds…2001

    • Author(s)
      Q.M.Cheng, S.Ishikawa
    • Journal Title

      Journ.Japanese Math. 27

      Pages: 387-404

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Conformally flat 3-manifolds with constant scalar curvature II2001

    • Author(s)
      Qing-Ming Cheng, Susumu Ishikawa, Katsuhiro Shiohama
    • Journal Title

      J.Japanese Math. 27

      Pages: 387-404

    • NAID

      10007537382

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Hypersurfaces in Euclidean spaces2001

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

      Hypersurfaces in Euclidean spaces Proc.Fifth Pacific Rim Geom.Conference, Tohoku University, Sensai, Japan.(Tohoku Math.Publ.) 20

      Pages: 33-42

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Compact locally conformally flat Riemannian manifolds2001

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

      Bulletin of the London Mathemathical Society 33

      Pages: 459-465

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Complete submanifolds in Euclidean spaces with parallel mean curvature vector2001

    • Author(s)
      Qing-Ming Cheng, Kazuhiro Nonaka
    • Journal Title

      Manuscripta Math. 105

      Pages: 353-366

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Hypersurfaces in a unit sphere S^<n+1> (1) with constant scalar curvature2001

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

      Journal of the London Mathematical Society 64

      Pages: 755-768

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Complete maximal spacelike surfaces…2000

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

      Glasgow Math. Jour. 42

      Pages: 139-156

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Metrics of constant curvature 1…2000

    • Author(s)
      M.Umehara, K.Yamada
    • Journal Title

      Illinois Journ.of Math. 44

      Pages: 72-94

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Sphere Theorems2000

    • Author(s)
      Katsuhiro Shiohama
    • Journal Title

      Handbook of Differential Geometry, Chapter 8, (Edited by G.Dillen, P.Verstraelen) Vol.1

      Pages: 867-903

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Rigidity and sphere theorems for submanifolds II2000

    • Author(s)
      Katsuhiro Shiohama, Hongwei Xu
    • Journal Title

      Kyushu Journal of Mathematics Vol.54

      Pages: 103-109

    • NAID

      130000062908

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Complete maximal spacelike surfaces in anti-de Sitter space H^4_2(c)2000

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

      Glasgow Math.J. 42

      Pages: 139-156

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Mean curvature 1 surfaces with low total curvature in hyperbolic 3-space2000

    • Author(s)
      Wayne Rossman, Masaaki Umehara, Kotaro Yamada
    • Journal Title

      Advanced Studies in Pure Mathematics, Minimal surfaces, Geometric Analysis and Symplictic Geometry 29

      Pages: 279-297

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] Metrics of constant curvature 1 with three conical singularities on 2-sphere2000

    • Author(s)
      Masaaki Umehara, Kotaro Yamada
    • Journal Title

      Illinois Journal of Mathematics vol.44

      Pages: 72-94

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux2000

    • Author(s)
      Shin Kato, Masaaki Umehara, Kotaro Yamada
    • Journal Title

      Communications in Analysis and Geometry vol.8

      Pages: 83-114

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Journal Article] An analogue of minimal surface theory in SL(n, C)/SU(n)

    • Author(s)
      Masatoshi Kokubu, Masaro Takahashi, Masaaki Umehara, Kotaro Yamada
    • Journal Title

      Transactions of the American Mathematical Society

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Book] 線形代数学入門(行列と行列式)2005

    • Author(s)
      石川晋, 成慶明
    • Total Pages
      150
    • Publisher
      丸善出版
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Q.M.Cheng, S.Ishikawa, K.Shiohama: "Conformably flat 3-manifolds with constant scales cnvvatitc II"Japanese. Tournal of Marhenatics. 27(2). 387-404 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Q.M.Cheng,K.Shiohana: "Conformally flat 3-manifolds with constant curvatin II"Japanese Journal of Math. Vol26. (2001)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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