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Research on surfaces of constant mean curvature one in hyperbolic space and its application

Research Project

Project/Area Number 13640075
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

UMEHARA Masaaki  Hiroshima Univ. Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90193945)

Co-Investigator(Kenkyū-buntansha) WAYNE Rossman  Kobe Univ. Faculty of Science, Associate Professor, 理学部, 助教授 (50284485)
YAMADA Kotaro  Hiroshima Univ. Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (10221657)
MATSUMOTO Takao  Hiroshima Univ. Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50025467)
INOGUCHI Junichi  Utsunomiya Univ., Faculty of Education, Associate Professor, 教育学部, 助教授 (40309886)
KOKUBU Masatoshi  Tokyo Denki Univ., School of Engineering, Associate Professor, 工学部, 助教授 (50287439)
土井 英雄  広島大学, 大学院・理学研究科, 講師 (50197993)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥4,000,000 (Direct Cost: ¥4,000,000)
Fiscal Year 2002: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2001: ¥2,100,000 (Direct Cost: ¥2,100,000)
KeywordsHyperbolic space / Mean curvature / Gauss map / Gaussian curvature
Research Abstract

We get the following results:
1. The head investigator Masaaki Umehara proved that the equality of the Osserman inequality for minimal surfaces in Euclidean n-space holds if and only if each end has no self-intersection and asymptotic to a catenoid or a plane, which is a joint work with M. Kokubu and K. Yamada. Moreover, we construct a new example which attains the equality.
2. The head investigator Masaaki Umehara gave countable many new irreducible constant mean curvature one (i.e. CMC-1) surfaces in hyperbolic 3-space whose ends all irregular, which is a joint work with W. Rossman and K. Yamada.
3. The head investigator Masaaki Umehara gave an elementary proof of the Small's representation formula for CMC-1 surfaces in hyperbolic 3-space and also got a similar representation formula for flat surfaces in hyperbolic 3-space, which is a joint work M. Kokubu and K. Yamada. A flat surface in hyperbolic 3-space called a flat front if it admits singularity but can be lifted to a Legendrian immersion into the unit cotangent bundle of hyperbolic 3-space. We showed flat fronts with complete ends satisfy an Osserman-type inequality with respect to the degree of the hyperbolic Gauss maps. The equality holds if and only if all ends have no self intersection. Furthermore, we classify flat front of genus zero with embedded regular 3-ends and also construct an example of genus one with five regular embedded ends.

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] A.I.Bobenko M.Umehara: "Monodromy of isometric deformation of CMC-surfaces"Hiroshima Math. J.. 31. 291-297 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kokubu, M.Takahashi, M.Umehara K.Yamada: "An analogue of minimal surface theory in SL(n, C)/Su(n)"Trans. Amer. Math. Soc.. 354. 1299-1325 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kokubu, M.Umehara K.Yamada: "Minimal surfaces that attain equality in the Chern-Osserman inequality"Contemporary Mathematics. 308. 223-228 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] W.Rossman, M.Umehara K.Yamada: "Constant mean curvature 1 surfaces with low total curvature in hyperbolic 3-space"Advanced studies in Pure Math., Minimal Surfaces, Geometric Analysis and Symplectic Geometry. 34. 245-253 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A. I. Bobenko and M. Umehara: "Monodromy of isometric deformation of CMC-surfaces"Hiroshima Math. J.. 31. 291-297 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Kokubu, M. Takahashi, M. Umehara, and K. Yamada,: "An analogue of minimal surface theory in SL(n, C) / SU(n)"Trans. Amer. Math. Soc.,. 354. 1299-1325 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Kokubu, M. Umehara, and K. Yamada: "Minimal surfaces that attain equality in the Chern-Osserman inequality"Contemporary Mathematics. 303. 223-228 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] W. Rossman, M. Umehara, and K. Yamada: "Constant mean curvature I surfaces with low total curvature in hyperbolic 3-space"Advanced Studies in Pure Math., Minimal Surfaces, Geometric Analysis and Simplistic Geometry. 34. 245-253 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Kokubu, M.Takahashi, M.Umehara, K.Yamada: "An analogue of minimal surface theory in SL(n, C)/SU(n)"Trans. Amer. Math. Soc.. 354. 1299-1325 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] G.Thorbergsson, M.Umehara: "Sectactic points on a simple closed curve"Nagoya Math. J.. 167. 55-94 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Kokubu, M.Umehara, K.Yamada: "Minimal surfaces that attain equality in the Chern-Osserman inequality"Contemporary Mathematics. 308. 223-228 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.I.Bobenko: "Monodromy of isometric deformation of CMC surfaces"Hiroshima Mathematical Journal. 31・2. 291-297 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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