• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Construction of submanifold with constant mean curvature, and its applications

Research Project

Project/Area Number 10440024
Research Category

Grant-in-Aid for Scientific Research (B).

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKyushu University (2000)
Kumamoto University (1998-1999)

Principal Investigator

YAMADA Kotaro  Kyushu University, Faculty of Math., Prof., 大学院・数理学研究院, 教授 (10221657)

Co-Investigator(Kenkyū-buntansha) ROSSMAN Wayne  Kobe Univ., Fac.of Sci., Assoc.Prof., 理学部, 助教授 (50284485)
CHO Koji  Kyushu University, Faculty of Math., Assoc.Prof., 大学院・数理学研究院, 助教授 (10197634)
YAMAGUCHI Takao  Kyushu University, Faculty of Math., Prof., 大学院・数理学研究院, 教授 (00182444)
INOUE Hisao  Kumamoto Univ., Fac.of Sci., Lect., 理学部, 講師 (40145272)
KUROSE Takashi  Fukuoka Univ., Fac.of Sci., Assoc.Prof., 理学部, 助教授 (30215107)
大脇 信一  熊本大学, 理学部, 教授 (50040506)
伊藤 仁一  熊本大学, 教育学部, 助教授 (20193493)
前橋 敏之  熊本大学, 理学部, 教授 (90032804)
原岡 喜重  熊本大学, 理学部, 助教授 (30208665)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥5,900,000 (Direct Cost: ¥5,900,000)
Fiscal Year 2000: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1999: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 1998: ¥1,700,000 (Direct Cost: ¥1,700,000)
Keywordsminimal surfaces / Weierstrass representation / CMC-1 surface / Gauss map / Osserman inequality / Total curvature / 非コンパクト型対称空間 / キーワード7 / キーワード8 / 低平均曲率曲面 / フラックス / モノドロミー問題 / 定平均曲率曲面 / 常微分方程式 / 可積分系
Research Abstract

We investigated properties of minimal surfaces in the three dimensional euclidean space using the Weierstrass representation formula, and generalizations of them. First, we gave an affirmative result for an inverse problem of flux for minimal surfaces in the three dimensional euclidean space. Moreover, as a generalization of (a complex analytic) flux, we defined a new homology invariant, which is also called as "flux", for surfaces of constant mean curvature one in the hyperbolic three space. Using the balancing formula of the flux, we proved some non-existence results for constant mean curvature one surface in hyperbolic space.
As a continuation of this non-existence results, we tried to classify the complete constant mean curvature one surface in hyperbolic space with low total absolute curvature, and we obtained the complete classification for surfaces with total absolute curvature less than or equal to 4π.
On the other hand, as a generalization of the Weierstrass-type representation formula for minimal surface with higher dimensional euclidean space, we defined a notion of surfaces with holomorphic right gauss map in some non-compact type symmetric space, and obtained the Weierstrass-Bryant type representation formula. As an application of this formula, we obtained an Osserman-type inequality for total absolute curvature.

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] W.Rossman,K.Yamada et.al: "Mean curvature 1 surfaces with low total curvature in hyperbolic 3-space"Advanced Studies in Pure Mathematics. 29. 279-297 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Umehara and K.Yamada: "Metrics of constant curvature 1 with three conical singularities on 2-sphere"Illinois Journal of Mathematics. 44. 72-94 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] W.Rossman,K.Yamada, et.al: "A new flux for mean curvature 1 surfaces in hyperbolic 3-space, and applications"Proceedings of the American Mathematical Society. 12. 2147-2154 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Kato,K.Yamada, et.al: "General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux"Communications in Analysis and Geometry. 8. 83-114 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kurose: "Two characterizations of the standard affine connection on a sphere"Fukuoka University Science Reports. 30. 179-180 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kurose: "Conformal-projective geometry of statistical manifolds"the Interdisciplinary Information Sciences. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] W.Rossman, M.Umehara and K.Yamada: "Mean curvature 1 surfaces with low total curvature in hyperbolic 3-space"Advanced Studies in Pure Mathematics. vol.29. 279-297 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Umehara and K.Yamada: "Metrics of constant curvature 1 with three conical singulaerities on 2-sphere"Illinois Journal of Mathematics. vol.44. 72-94 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] W.Rossman, M.Umehara and K.Yamada: "A new flux for mean curvature 1 surfaces in hyperbolic 3-space, and applications"Proceedings of the American Mathematical Society. vol.12. 21247-2154 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Kato, M.Umehara and K.Yamada: "General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux"Communications in Analysis and Geometry. vol.8. 83-114 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kurose: "Two charactrerizations of the standard affine connection on a sphere"Fukuoka University Science Reprorts. vol.30. 179-180 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kurose: "Conformal-projective geometry of statistical manifold"The Interdisciplinary Information Sciences. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] W.Rossman,K.Yamada et.al: "Mean curvature 1 surfaces with low total curvature in hyperbolic 3-space"Advanced Studies in Pure Mathematics. 29. 279-297 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Umehara and K.Yamada: "Metrics of constant curvature 1 with three conical singularities on 2-sphere"Illinois Journal of Mathematics. 44. 72-94 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Kato,K.Yamada, et.al: "General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux"Communications in Analysis and Geometry. 8. 83-114 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Kurose: "Two Characterizations of the standard affine connection on a sphere"Fukuoka University Science Reports. 30. 179-180 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Kurose: "Conformal-projective geometry of statistical manifolds"the Interdisciplinary Information Science. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Masaaki Umehara and Kotaro Yamada: "Metrics of constant curvature 1 with three conical singularities on 2-sphere"Illinuis Journal of Mathematics. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] Shin Kato,Masaaki Umehara and Kotaro Yamada: "General existence of minimal surfaces of genus zero of catenoidal ends and prescribed flux"Communications in Analysis and Geometry. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] Wayne Rossman,Masaaki Umehara and Kotaro Yamada: "A new flux for mean curvature 1 surfaces in hyperbolic 3-space, and applications"Proceedings of the American Mathematical Soc.. 127. 2147-2154 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Umehara and K.Yamada: "Metrics of constant curvature 1 with three conical singularities on 2-sphere" Inidiana Journal of Mathematics. (To appear). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] W.Rossman et.al.: "A new flux formean curvature 1 surfaces in hyperbolic 3-space, and applications" Proceedings of American Math. Soc.(To appear). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] J.Itoh and M.Tanaka: "The dimension of a cut locus on a smooth riemannian manifold" Tohoku Mathematical Journal. 50. 571-575 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Haraoka: "Quadratic relations for confluent hypergeometric functions on Z_<2,n+1>" Funkcialaj Ekvacioj. (To appear). (1999)

    • Related Report
      1998 Annual Research Report

URL: 

Published: 1998-04-01   Modified: 2021-04-07  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi