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Research on constant mean curvature surfaces

Research Project

Project/Area Number 12440012
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

KENMOTSU Katsuei  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60004404)

Co-Investigator(Kenkyū-buntansha) MASHIMO Katsuya  Tokyo University of Agriculture and Technology, Professor, 工学部, 教授 (50157187)
NAGASAWA Takeyuki  Saitama University, Faculty of Science, Professor, 理学部, 教授 (70202223)
NISHIKAWA Seiki  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60004488)
TASAKI Hiroyuki  Tsukuba University, Dep. of Math., Associate Professor, 数学系, 助教授 (30179684)
OHNITA Yoshihiro  Tokyo Metropolitan University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90183764)
小谷 元子  東北大学, 大学院・理学研究科, 教授 (50230024)
清水 悟  東北大学, 大学院・理学研究科, 助教授 (90178971)
砂田 利一  東北大学, 大学院・理学研究科, 教授 (20022741)
Project Period (FY) 2000 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥15,300,000 (Direct Cost: ¥15,300,000)
Fiscal Year 2003: ¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2002: ¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 2001: ¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 2000: ¥5,000,000 (Direct Cost: ¥5,000,000)
Keywordsminimal surface / constant mean curvature surface / mean curvature vector / complex space form / periodic surface of revolution / Delaunay surface / Bezier curve / harmonic map / 平均曲率一定 / C-多様体 / エネルギー最小写像 / 周期的平均曲率 / 周期的回転超曲面 / ウィルモア汎関数 / 多重Kaehler角度 / Poincareの公式 / 極小ラグランジアン部分多様体 / 積分幾何学 / グラスマン幾何学 / 全測地的部分多様体 / 対称空間 / 格子モデル / グラフ
Research Abstract

1.We classified these isometric minimal immersions of real two dimensional Riemannian manifold with constant Gaussian curvature into a complex two dimensional complex space form. In this research, it was important to prove the constancy of the Kaehler angles of these immersions.
2.We studied periodic surfaces of revolution in the three dimensional Euclidean space in order to understand the importance of constant mean curvature surfaces and got the criterion for a periodic function to be the mean curvature of a periodic surface of revolution. Moreover, we found an interesting relation between periodic mean curvature function and Bezier curves.
3.Since the theory of non zero constant mean curvature surfaces is well developed on these twenty years, I have written a textbook about this subject in order to make a unified explanation of the theory. Recently, this book was translated into English by American Mathematical Society.

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

    (36 results)

All Other

All Publications (36 results)

  • [Publications] Katsuei Kenmotsu: "The classification of the surfaces with parallel mean curvature-vector in two-dimensional complex space forms"Amer.Jour.Math.. 122・2. 295-317 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Katsuei Kenmotsu: "On minimal surfaces of constant curvature in two-dimensional complex space form"J.reine angew.Math.. 523・1. 69-101 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Katsuei Kenmotsu: "Periodic mean curvature and Bezier curve"Differential Geometry and Related Topics(論文集). 135-146 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Katsuei Kenmotsu: "Surfaces of revolution with periodic mean curvature"Osaka Jour.Math.. 40・3. 687-696 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Katsuei Kenmotsu: "Energy minimizer maps on C-manifolds"Diff.Geom.Appl.. (印刷中). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Seiki Nishikawa: "Harmonic maps in complex Finsler geometry"Geometric Variational Problems, Scans and Geometric Flows, Birkhauser(論文集). (印刷中). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Katsuei Kenmotsu: "Surfaces with constant mean curvature, Translations of Math.Monographs."Amer.Math.Soc.. 142 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kenmotsu: "The classification of the surfaces with parallel mean curvature vector in two-dimensional complex apace forms"Amer.Jour.Math.. 122. 295-317 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kenmotsu: "On minimal surfaces of constant curvature in two-dimensional complex space form"J. reine angew.Math.. 523. 69-101 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kenmotsu: "Periodic mean curvature and Bezier curve"Differential Geometry and Related Topics(C.Gu, H.Hesheng and T.Li(eds)))(World Scientific). 135-146 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kenmotsu: "Surfaces of revolution with periodic mean curvature"Osaka Jour.Math.. 40. 687-696 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kenmotsu: "Energy minimizer maps on C-manifolds"Diff.Geom.Appl.. (in press). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Nishikawa: "Harmonic maps in complex Finsler geometry, Geometric Variational Problems"Scans and Geometric Flows, Birkhauser. (in press). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kenmotsu: "Surfaces with constant mean curvature(Translations of Math. Monographs)(Vol.221)"American Mathematical society. 142 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Kenmotsu: "Surfaces of revolution with periodic mean curvature"Osaka Jour.Math. 40. 687-696 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Kenmotsu: "Energy minimizer maps on C-manifolds"Diff.Geom.and Appl.. (掲載予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Kotani: "Lipschitz continuity of the spectra of the magnetic transition"J.Geom.Phys. 47. 323-342 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Ohnita: "Hamiltonian stability of certain minimal Lagrangian submanifolds"Tohoku Math.J.. 55. 583-610 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 間下 克哉: "Chenの方程式をみたす6次元球面内の3次元CR部分多様体"数理解析研究所講究録. 1346. 91-98 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Tasaki: "Generalization of Kaehler angles and integral geometry II"Math.Nachr.. 252. 106-112 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Kenmotu: "Surfaces with constant mean curvature"American Math. Society. 142 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Kenmotsu: "Surfaces of revolution with periodic mean curvature"Osaka Journ. Math.. (未定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 剱持 勝衛: "周期的回転面の構成について"部分多様体論湯沢2002記録集. (未定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Nishikawa: "Harmonic maps between Carnot spaces"Proceeding of the International Conference on Modern Mathematics and the International Symposium on Differential Geometry. (未定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Nagawawa: "Bifurcating critical points of bending energy with constraints related to the shane of red blood cell"Cal. Var. Partial Differential Equations. 16(1). 63-111 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Tasaki: "Generalization of Kaehler angle and integral geometry in complex proiective spaces II"Math. Nachr. (未定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 剱持 勝衛: "E. Cartan in the Bonnet Problem"数理解析研究所講究録. 1206. 45-54 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Kenmotsu: "Periodic mean curvature and Bezier curve"Proceedings of the Conference and the Symposium on Differentaial Geometry in honor of Prof. Su on the occasion of his 100^<th> birthday. (未定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Katsuei Kenmotsu: "The classification of the surfaces with parallel mean curvature vector in two-dimensional complex space forms"American Journal of Mathematics. 122・2. 295-317 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Katsuei Kenmotsu: "On minimal surfaces of constant curvature in two-dimensional complex space form"J.reine angew,Math.. 523・1. 69-101 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Motoko Kotani: "Jacobian tori associated with a finite graph and its abelian covering graphs"Advances in Appl.Math.. 24・1. 89-110 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Takeyuki Nagasawa: "Closed surfaces minimizing the bending energy"International Conference on Diff.Eq.Berlin 1999. 1. 561-563 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Satoui Shimizu: "A classification of two-dimensional tube domains"American Journal of Mathematics. 122・6. 1289-1308 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] O.Ikawa: "Totally geodesic submanifolds of maximal rank in sym.Spaces"Japanese J.Math.. 26・1. 1-19 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 剱持勝衛: "曲面論講義"培風館. 150 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 砂田利一: "分割の幾何学"日本評論社. 140 (2000)

    • Related Report
      2000 Annual Research Report

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

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