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Generalizations of Weierstrass-type representation formula and their applications to theory of surface with singularities

Research Project

Project/Area Number 18340019
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 (KOTARO Yamada)  Kyushu University, 大学院・数理学研究院, 教授 (10221657)

Co-Investigator(Kenkyū-buntansha) MIYAOKA Reiko  東北大学, 大学院・理学研究科, 教授 (70108182)
SAEKI Osamu  九州大学, 大学院・数理学研究院, 教授 (30201510)
OTSU Yukio  九州大学, 大学院・数理学研究院, 准教授 (80233170)
NAGATOMO Yasuyuki  九州大学, 大学院・数理学研究院, 准教授 (10266075)
TAKAYAMA Haruko  九州大学, 大学院・数理学研究院, 助教 (90274430)
UMEHARA Masaaki  大阪大学, 大学院・理学研究科, 教授 (90193945)
KUROSE Takashi  福岡大学, 理学部, 教授 (30215107)
KOKUBU Masatoshi  東京電機大学, 工学部, 准教授 (50287439)
Co-Investigator(Renkei-kenkyūsha) FUJIMORI Shoichi  福岡教育大学, 教育学部, 准教授 (00452706)
SHODA Toshihiro  佐賀大学, 文化教育学部, 准教授 (10432957)
TAKAHASHI Masaro  久留米工業高等専門学校, 准教授 (70311107)
Project Period (FY) 2006 – 2008
Project Status Completed (Fiscal Year 2008)
Budget Amount *help
¥8,740,000 (Direct Cost: ¥7,300,000、Indirect Cost: ¥1,440,000)
Fiscal Year 2008: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2007: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2006: ¥2,500,000 (Direct Cost: ¥2,500,000)
Keywords微分幾何 / ワイエルストラス表現 / フロント / 特異点 / ワイエルストラス型表現公式 / 特異点の微分幾何 / 双曲空間の平坦曲面 / ド・ジッター空間のCMC1曲面 / 特異点に微分幾何
Research Abstract

Properties of certain classes of surfaces with singularities are investigated with Weirstrass-type representation formula. For example, global behavior of flat fronts, and behavior of singularities of maximal surfaces in Lorentz-Minkowski 3-space and mean curvature one surfaces in de Sitter 3-space are investigated.
On the other hand, as a general theory of differential geometry of singularities, a notion of singular curvature of the singular points of wave fronts is defined, and Gauss-Bonnet type formulas are obtained.

Report

(5 results)
  • 2008 Annual Research Report   Self-evaluation Report ( PDF )   Final Research Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • Research Products

    (41 results)

All 2009 2008 2007 2006 Other

All Journal Article (20 results) (of which Peer Reviewed: 12 results) Presentation (18 results) Remarks (3 results)

  • [Journal Article] The geometry of fronts2009

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

      Annals of Mathematics vol.169

      Pages: 491-529

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Hyperbolic Schwarz map for the hypergeometric differential equation2008

    • Author(s)
      T. Sasaki, K. Yamada, M. Yoshida
    • Journal Title

      Experimental Mathematics vol.17

      Pages: 269-282

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Derived Schwarz map of the hypergeometric differential equation and a parallel family of flat fronts2008

    • Author(s)
      T. Sasaki, K. Yamada, M. Yoshida
    • Journal Title

      International Journal of Mathematics vol.19

      Pages: 847-863

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Confluence of swallowtail singularities of the hyperboic Schwarz map defined by the hypergeometric differential equation2008

    • Author(s)
      M. Noro, T. Sasaki, K. Yamada, M. Yoshida
    • Journal Title

      Experimental Mathematics vol.17

      Pages: 191-204

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Singularities of maximal surfaces2008

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

      Mathematische Zeitschrift vol.259

      Pages: 1827-848

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Behavior of corank one singular points on weve fronts2008

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

      Kyushu J. Math. 62

      Pages: 259-280

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Confluence of swallowtail singularities of the hyperbolic Schwarz maps defined by the hypergeometric differential equation2008

    • Author(s)
      M.Noro, T. Sasaki, K. Yamada and M. Yoshida
    • Journal Title

      Experimentarl Math. 17

      Pages: 191-204

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Hyperbolic Schwarz maps for the hypergeoometric differential equation2008

    • Author(s)
      T.Sasaki, M. Yoshida and K. Yamada
    • Journal Title

      Exp. Math. 17

      Pages: 269-282

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Singularities of Maxial surfaces2008

    • Author(s)
      S. Fujimori, K. Saji, M. Umehara and K. Yamada
    • Journal Title

      Math. Z. 259

      Pages: 827-848

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Singularities of maximal surfaces2008

    • Author(s)
      Shoichi Fujimoori, et. al.
    • Journal Title

      Mathematicse Zeitschrift 259

      Pages: 827-848

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Behavior of corank one singularities of wave fronts2008

    • Author(s)
      Kentaro Saji, et. al.
    • Journal Title

      Kyushu Journal of Mathematics 62

      Pages: 259-280

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Hyperbolic Schwarz map for the hypergeometric differential equation2008

    • Author(s)
      Takeshi Sasaki, et. al.
    • Journal Title

      Experimental Mathematics 17

      Pages: 269-282

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Flat fronts in hyperbolic 3-space and their caustics2007

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

      Journal of Mathematical Society of Japan vol.59(1)

      Pages: 265-299

    • NAID

      10019540696

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Flat fronts in hyperbolic 3-space and their caustics2007

    • Author(s)
      M.Kokubu , W. Rossman, M. Umehara and K. Yamada
    • Journal Title

      J. of Math. Soc. Japan 50

      Pages: 265-299

    • NAID

      10019540696

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Flat fronts in hyperbolic 3-space and their caustics2007

    • Author(s)
      Masatoshi Kokubu, et. al.
    • Journal Title

      Journal of the Mathematical Society of Japan 59

      Pages: 265-299

    • NAID

      10019540696

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Flat fronts in hyperbdic 3-space and their caustics2007

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

      Hournal of the Mathematical Society of Japan 59.1

      Pages: 265-299

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The qaometry of fronts2007

    • Author(s)
      Kentaro SAJI, Kotaro TAMADA et al.
    • Journal Title

      Amals of Mathematics 未定(掲載決定)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Maximal surfaces with singularities in Minkowski space2006

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

      Hokkaido Mathematical Journal vol.35(1)

      Pages: 13-40

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Maximal surfaces with singularities in Minkowski space2006

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

      Hokkaido Mathematical Journal 35.1

      Pages: 13-40

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Behavior of corank one singular points on wave front

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

      Kyushu Journal of Mathematics vol.62(1)

      Pages: 259-280

    • NAID

      110006648027

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Presentation] Asymptotic behavior of flat surfaces in hyperbolic 3-space2008

    • Author(s)
      K. Yamada
    • Place of Presentation
      Unversidad de Granada, Spain.
    • Year and Date
      2008-06-06
    • Related Report
      2008 Final Research Report
  • [Presentation] Asymptotic behavior of flat surfaces in hyperbolic 3-space2008

    • Place of Presentation
      Universidad de Granada
    • Year and Date
      2008-06-06
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Asymtotic behavior of flat surfaces in hyperbolic 3-space2008

    • Author(s)
      Kotaro Yamada
    • Organizer
      Geometry Seminar
    • Place of Presentation
      Universidad de Granada
    • Year and Date
      2008-06-06
    • Related Report
      2008 Annual Research Report
  • [Presentation] Spacelike mean curvature one surface in de Sitter 3-space2007

    • Author(s)
      K. Yamada
    • Place of Presentation
      Geometry Seminar, Northeastern University, Shenyang, China.
    • Year and Date
      2007-10-11
    • Related Report
      2008 Final Research Report
  • [Presentation] Spacelike mean curvature one surfaces in de Sitter 3-space2007

    • Place of Presentation
      Northeastern University, Shenyang
    • Year and Date
      2007-10-11
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Flat fronts in hyperbolic 3-space2007

    • Organizer
      S. S. Chern Institute of Math.
    • Place of Presentation
      Nankai University,Tianjin
    • Year and Date
      2007-08-15
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Spacelike mean curvature one surfaces in de Sitter 3-space2007

    • Author(s)
      山田光太郎
    • Organizer
      部分多様体論と可積分系および幾何解析とのつながり
    • Place of Presentation
      京都大学数理解析研究所.
    • Year and Date
      2007-07-11
    • Related Report
      2008 Final Research Report
  • [Presentation] Spacelike mean curvature one surfaces in de Sitter 3-space2007

    • Place of Presentation
      京都大学数理解析研究所( 研究集会)
    • Year and Date
      2007-07-11
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Spacelike mean curvature one surfaces in de Sitter 3-space2007

    • Author(s)
      K. Yamada
    • Place of Presentation
      Universidad de Granada
    • Year and Date
      2007-05-10
    • Related Report
      2008 Final Research Report
  • [Presentation] Spacelike mean curvautre one surfaces in de Sitter 3-space2007

    • Place of Presentation
      Universidad de Granada
    • Year and Date
      2007-05-10
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Spacelike mean curvature 1 surfaces in de Sitter 3-space2007

    • Author(s)
      K. Yamada
    • Place of Presentation
      Mathematische Forschungsinstitut Oberwolfach, Germany
    • Year and Date
      2007-05-02
    • Related Report
      2008 Final Research Report
  • [Presentation] Spacelike mean curvature one surfaces in de Sitter 3-space2007

    • Place of Presentation
      Mathematische Forschungsinstitute Overwolfach
    • Year and Date
      2007-05-02
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Spacelike mean curvature 1 surfaces in de Sitter 3-space2007

    • Author(s)
      Kotaro Yamada
    • Organizer
      Progress in Surface Theory
    • Place of Presentation
      Math. Forschungsinst. Oberwolfach
    • Year and Date
      2007-05-02
    • Related Report
      2007 Annual Research Report
  • [Presentation] Global properties of flat fronts in hyperbolic 3-space2006

    • Author(s)
      山田光太郎
    • Organizer
      特異点論セミナー
    • Place of Presentation
      北海道大学
    • Year and Date
      2006-11-17
    • Related Report
      2008 Final Research Report
  • [Presentation] Global properties of flat fronts in hyperbolic 3-space2006

    • Place of Presentation
      北海道大学
    • Year and Date
      2006-11-17
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Flat fronts in hyperbolic 3-space2006

    • Author(s)
      山田光太郎
    • Organizer
      北海道大学集中講義オーバーヴュー
    • Place of Presentation
      北海道大学
    • Year and Date
      2006-11-13
    • Related Report
      2008 Final Research Report
  • [Presentation] Flat fronts in hyperbolic 3-space2006

    • Place of Presentation
      北海道大学
    • Year and Date
      2006-11-13
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Flat fronts in hyperbolic 3-space-behavior of ends2006

    • Author(s)
      山田光太郎
    • Organizer
      幾何学阿蘇研究集会
    • Place of Presentation
      休暇村南阿蘇
    • Year and Date
      2006-09-12
    • Related Report
      2008 Final Research Report
  • [Remarks] ホームページ等

    • URL

      http://www.math.titech.ac.jp/~kotaro/

    • Related Report
      2008 Final Research Report
  • [Remarks] 研究成果に関するwebページ:

    • URL

      http://kotaro.math.kyushu-u.ac.jp/research/

    • Related Report
      2008 Annual Research Report
  • [Remarks] 研究成果に関するwebページ

    • URL

      http://kotaro.math.kyushu-u.ac.jp/research/

    • Related Report
      2007 Annual Research Report

URL: 

Published: 2006-04-01   Modified: 2016-04-21  

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