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Generalizations of Weierstrass-type representation formulae and applications

Research Project

Project/Area Number 14340024
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

YAMADA Kotaro  Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (10221657)

Co-Investigator(Kenkyū-buntansha) MIYAOKA Reiko  Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (70108182)
SAEKI Osamu  Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (30201510)
UMEHARA Masaaki  Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90193945)
KUROSE Takashi  Fukuoka University, Faculty of Science, Associate Professor, 理学部, 助教授 (30215107)
TAKAHASHI Masaro  Kurume National College of Technology, Associate Professor, 助教授 (70311107)
井ノ口 順一  宇都宮大学, 教育学部, 助教授 (40309886)
長友 康行  九州大学, 大学院・数理学研究院, 助教授 (10266075)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥9,200,000 (Direct Cost: ¥9,200,000)
Fiscal Year 2005: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2004: ¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2003: ¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2002: ¥2,200,000 (Direct Cost: ¥2,200,000)
KeywordsWeierstrass representation / singularities / front / maximal surface / flat surface / cuspidal edge / cuspidal cross cap / swallowtail / ワイエルストラス型表現 / CMC-1曲面 / 全曲率 / 平坦フロント / 極小曲面 / 定平均曲率曲面 / ワイエルストラス表現公式 / オサーマン型不等式
Research Abstract

1.W rewrote the Weierstrass-type representation formula for flat surfaces in hyperbolic 3-space in the form without integration (Darboux-type formula), and classified complete flat surfaces with small numbers of ends. 2.We pointed out the class of ambient spaces for which an analogue of Weierstrass-type (Bryant) representation formula for mean curvature one surfaces in hyperbolic 3-space holds. 3.We found criteria for singularities (cuspidal edges, swallowtails, cuspidal cross caps) which are generic singularities of fronts or frontals. 4.We established fundamental notions of flat fronts in hyperbolic 3-space, and investiagted properties of singularities of such surfaces. 5.We defined a certain class of maximal surfaces with singularities in Minkowski 3-space (called maxface), and investigated their singularities.

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (29 results)

All 2006 2005 2004 2003 2002 Other

All Journal Article (18 results) Publications (11 results)

  • [Journal Article] Maximal surfaces with singularities in Minkowski space2006

    • Author(s)
      Masaaki UMEHARA, Kotaro YAMADA et al.
    • Journal Title

      Hokkaido Mathematical Journal (in press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Singularities'' of flat fronts on hyperbolic 3-space2005

    • Author(s)
      Masatoshi KOKUBU, Kotaro YAMADA et al.
    • Journal Title

      Pacific Journal of Mathematics 211・2

      Pages: 303-351

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Symmetric CMC-1 surfaces with invequla ends2005

    • Author(s)
      Wayne ROSSMAN, Kotaro YAMADA et al.
    • Journal Title

      Clay Mathematical Proceedirys 2

      Pages: 561-584

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Singularities of flat fronts in hyperbolic 3-space2005

    • Author(s)
      Masatoshi KOKUBU, Wayne ROSSMAN, Kentaro SAJI, Masaaki UMEHARA, Kotaro YAMADA
    • Journal Title

      Pacific Journal of Mathematics volume 221 no.2

      Pages: 303-351

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Symmetric CMC-1 surfaces with irregular ends2005

    • Author(s)
      Wayne ROSSMAN, Masaaki UMEHARA, Kotaro YAMADA
    • Journal Title

      Clay Mathematical Proceedings volume 2

      Pages: 561-584

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Singularities of flat fronts in hyperbolic 3-space2005

    • Author(s)
      Masaaki UMEHARA, Kotaro YAMADA et al.
    • Journal Title

      Pacific Journal of Mathematics 221・2

      Pages: 303-351

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Constructing mean curvature 1 surfaces in H^3 with irregular ends2005

    • Author(s)
      Wayne ROSSMAN, Kotaro YAMADA et al.
    • Journal Title

      Clay Mathematical Proceedings 2

      Pages: 687-715

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Singularities of flat fronts in hyperbolic 3-space2005

    • Author(s)
      Masaaki UMEHARA, Kotaro YAMADA et al.
    • Journal Title

      Pacific Journal of Mathematics (in press)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Maximal surfaus in Minkowski space2005

    • Author(s)
      Masaaki UMEHARA, Kotaro YAMADA
    • Journal Title

      Hokkaido Jounal of Mathematics (in press)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Flat fronts in hyperbolic 3-space2004

    • Author(s)
      Masatoshi KOKUBU, Kotaro YAMADA et al.
    • Journal Title

      Pacific Journal of Mathematics 216・1

      Pages: 149-175

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Flat fronts in hyperbolic 3-space2004

    • Author(s)
      Masatoshi KOKUBU, Masaaki UMEHARA, Kotaro YAMADA
    • Journal Title

      Pacific Journal of Mathematics volume 216, no.1

      Pages: 149-175

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Flat fronts in hyperbolic 3-space2004

    • Author(s)
      Masaaki UMEHARA, Kotaro YAMADA et al.
    • Journal Title

      Pacific Journal of Mathematics 206-1

      Pages: 149-175

    • Related Report
      2004 Annual Research Report
  • [Journal Article] An elementary proof of small's formula for null canals in PSL(2.β)and an analogue for leqendrian curves in PSL(2.α)2003

    • Author(s)
      Masaaki UMEHARA, Kotaro YAMADA et al.
    • Journal Title

      Osaka Journal of Mathematical 40

      Pages: 681-715

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Mean curoatuve 1 surfaces in hyperbolic 3-space with low total curuatnne2003

    • Author(s)
      Masaaki UMEHARA, Kotaro YAMADA et al.
    • Journal Title

      Tohoku Mathematical Journal 55

      Pages: 375-395

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C)2003

    • Author(s)
      Masatoshi KOKUBU, Masaaki UMEHARA, Kotaro YAMADA
    • Journal Title

      Osaka Journal of Mathematics volume 55

      Pages: 375-395

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature2003

    • Author(s)
      Wayne ROSSMAN, Masaaki UMEHARA, Kotaro YAMADA
    • Journal Title

      Tohoku Mathematical Journal volume 55

      Pages: 375-395

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An analogue of minimal sin face theory in SL(n.C)/SU(h)2002

    • Author(s)
      Nasaro TAKAMASHI, Kotaro YAMADA et al.
    • Journal Title

      Transactions of the American Mathematical Society 354

      Pages: 1299-1325

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An analogue of minimal surface theory in SL(2,C)/ SU(n)2002

    • Author(s)
      Masatoshi KOKUBU, Masaro TAKAHASHI, Masaaki UMEHARA, Kotaro YAMADA
    • Journal Title

      Transactions of the American Mathematical Society volume 354

      Pages: 1299-1325

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Publications] W.Rossman, M.Umehara, K.Yamada: "Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature II"Tohoku Mathematical Journal. 55. 375-395 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Kokubu, M.Umehara, K.Yamada: "An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C)"Osaka Journal of Mathematics. 40. 687-715 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] W.Rossman, M.Umehara, K.Yamada: "Constructing mean curvature 1 surfaces in H^3 with irregular ends"Cray Mathematics Proceedings. (to appear). (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Kokubu, M.Umehara, K.Yamada: "Flat fronts in hyperbolic 3-space"Pacific Journal of Mathematics. (to appear). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] W.Rossman, M.Umehara, K.Yamada: "Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature II"Hiroshima Journal of Mathematics. (to appear). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Kokubu, M.Takahashi, M.Umehara, K.Yamada: "An analogue of minimal surface theory in SL(e^n, C)/Su(n)"Transactions of the American Mathematical Society. 354. 1299-1325 (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(Amer. Math. Soc). 308. 223-228 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] W.Rossman, M.Umehara, K.Yamada: "Meen curvature 1 surbaces in Aypelalic 3-space with low total curvature"Tohoku Mathematical Journal. (To appear) (In press).

    • Related Report
      2002 Annual Research Report
  • [Publications] W.Rossman, M.Umehara, K.Yamada: "Period Probleme for mean curvature one surfaces in H^3"the Proceedings of MSJ-ZRI 2000. (To appear) (In press).

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Kokubu, M.Umehara, K.Yamada: "An elementary proof of Small's formula for null ousves in PSL(2, C) and an analogue for Legendsian anues in PSL(2, C)"Osaka Journal of Mathematics. (To appear) (In press).

    • Related Report
      2002 Annual Research Report
  • [Publications] W.Rossman, M.Umehara, K.Yamada: "Constructing Mean Curuature 1 Sunbaces in H^3 with irrelulor ends"the Proceedinegs of workshop of MSRI, 2001. (To appear) (In press).

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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