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Development of value distribution theory of Gauss maps of immersed surfaces in space forms and their applications to global property of surfaces

Research Project

Project/Area Number 15K04840
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionKanazawa University

Principal Investigator

Kawakami Yu  金沢大学, 数物科学系, 准教授 (60532356)

Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords幾何学 / 複素解析学 / 曲面論 / 値分布論 / 極小曲面 / ガウス写像 / 等角計量 / 解析的延長 / 表現公式 / カテノイド / 除外値 / 一意性定理 / 極大曲面 / ラグランジアン曲面 / 全曲率 / 解析的拡張
Outline of Final Research Achievements

We perform a systematic study of the images of the Gauss maps of complete minimal surfaces in Euclidean 4-space. In particular, we give a geometric interpretation of value-distribution theoretic properties for the Gauss maps of complete minimal surfaces in Euclidean 4-space, for example, the maximal number of exceptional values and unicity theorem. We also provide optimal results of the size of the image under the Gauss map of a complete minimal Lagrangian surface in the complex 2-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean 4-space. We also study the analytic extensions and related global properties of maximal surfaces in the Lorentz-Minkowski 3-space and CMC-1 surfaces in the de-Sitter 3-space.

Academic Significance and Societal Importance of the Research Achievements

曲率条件をもつ空間内の曲面は,現実社会の物理的現象としてあらわれるものの数学的モデルになっていることが多い.このような事情から古くから研究され,その成果は物理学や工学など様々な分野の研究に応用されている.本課題の手法は,その実現性をガウス写像という視点で調べるものである.本研究成果は,曲率条件をもつ空間内の曲面の実現性の問題の本質を深めるものであり,幾何学及び複素解析学の研究の発展に意義をもつと考えられる.

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (26 results)

All 2019 2018 2017 2016 2015 Other

All Int'l Joint Research (4 results) Journal Article (7 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 7 results,  Acknowledgement Compliant: 3 results) Presentation (14 results) (of which Int'l Joint Research: 4 results,  Invited: 14 results) Book (1 results)

  • [Int'l Joint Research] Hanoi National University of Education(ベトナム)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] 高麗大学(韓国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Hanoi National University of Education(ベトナム)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Hanoi National University of Education(ベトナム)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Analytic extension of exceptional constant mean curvature one catenoids in de Sitter 3-space2019

    • Author(s)
      S. Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara, K. Yamada
    • Journal Title

      Mathematical Journal of Okayama University

      Volume: 印刷中

    • NAID

      120006778830

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Analytic extension of Jorge-Meeks type maximal surfaces in Lorentz-Minkowski 3-space2017

    • Author(s)
      S .Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara, K. Yamada
    • Journal Title

      Osaka Journal of Mathematics

      Volume: 54 Pages: 249-272

    • NAID

      120006312701

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space2017

    • Author(s)
      R. Aiyama, K. Akutagawa, S. Imagawa, Y. Kawakami
    • Journal Title

      Annali di Matematica Pura ed Applicata

      Volume: 196 Issue: 5 Pages: 1863-1875

    • DOI

      10.1007/s10231-017-0643-6

    • NAID

      120006226804

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] A note on a unicity theorem for the Gauss maps2017

    • Author(s)
      Pham Hoang Ha, Yu Kawakami
    • Journal Title

      Canadian Mathematical Bulletin

      Volume: 印刷中 Issue: 2 Pages: 292-300

    • DOI

      10.4153/cmb-2017-015-0

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] 曲面のGauss写像の値分布2017

    • Author(s)
      川上 裕
    • Journal Title

      数学

      Volume: 69 Pages: 56-69

    • NAID

      130007557779

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Analytic extension of Jorge-Meeks type maximal surfaces in Lorentz-Minkowski 3-space2017

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

      Osaka Journal of Mathematics

      Volume: 印刷中

    • NAID

      120006312701

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Entite zero-mean curvature graphs of mixed type in Lorents-Minkowski 3-space2016

    • Author(s)
      S. Fujimori, Y. Kawakami, M. Kokubu, W. Rossman, M. Umehara, K. Yamada
    • Journal Title

      Quart. J. Math.

      Volume: 67 Pages: 1-37

    • DOI

      10.1093/qmath/haw038

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Value distribution theory for the Gauss maps of several classes of surfaces2019

    • Author(s)
      Yu Kawakami
    • Organizer
      The 2nd International Conference "Geometry of Submanifolds and Integrable Systems"
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Recent topics on the study of the Gauss images of minimal surfaces2018

    • Author(s)
      川上 裕
    • Organizer
      東大複素幾何セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Global properties for the Gauss images of various classes of surfaces2017

    • Author(s)
      Yu Kawakami
    • Organizer
      The 3rd Japan-China Geometry Conference
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Global properties for the Gauss images of various classes of surfaces2017

    • Author(s)
      Yu Kawakami
    • Organizer
      第23回複素幾何シンポジウム
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 曲面のガウス写像の値分布論入門2017

    • Author(s)
      川上 裕
    • Organizer
      琵琶湖特異点論ワークショップ
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 様々な曲面のガウス写像の値分布論的性質について2017

    • Author(s)
      川上 裕
    • Organizer
      研究集会「複素幾何と幾何解析」
    • Place of Presentation
      明治大学
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] On the Gauss image of complete minimal surfaces in Euclidean 4-space2016

    • Author(s)
      川上 裕
    • Organizer
      内藤博夫退職記念研究集会
    • Place of Presentation
      山口大学
    • Year and Date
      2016-03-07
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] On the Gauss images for various classes of surfaces2016

    • Author(s)
      Yu Kawakami
    • Organizer
      Japan-Austria Joint Workshop 'Transformations and Singularities'
    • Place of Presentation
      東京工業大学
    • Year and Date
      2016-02-20
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 完備極小曲面のガウス写像の値分布論的性質の幾何学的解釈について2016

    • Author(s)
      川上 裕
    • Organizer
      2016年度日本数学会秋季総合分科会函数論分科会特別講演
    • Place of Presentation
      関西大学
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 完備極小曲面のガウス写像の値分布論について2016

    • Author(s)
      川上 裕
    • Organizer
      東工大幾何学セミナー
    • Place of Presentation
      東京工業大学
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 完備極小曲面のガウス写像の値分布論的性質2016

    • Author(s)
      川上 裕
    • Organizer
      名大幾何学セミナー
    • Place of Presentation
      名古屋大学
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 完備極小曲面の研究の最近の進展について2016

    • Author(s)
      川上 裕
    • Organizer
      東大複素解析幾何セミナー
    • Place of Presentation
      東京大学
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 曲面のガウス写像の値分布論的性質について2015

    • Author(s)
      川上 裕
    • Organizer
      第18回東北複素解析セミナー
    • Place of Presentation
      東北大学
    • Year and Date
      2015-06-24
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] 曲面のガウス写像の値分布論的性質について2015

    • Author(s)
      川上 裕
    • Organizer
      研究集会「部分多様体と様々な幾何構造」
    • Place of Presentation
      京都大学
    • Year and Date
      2015-06-22
    • Related Report
      2015 Research-status Report
    • Invited
  • [Book] 極小曲面論入門 その幾何学的性質を探る2019

    • Author(s)
      川上裕,藤森祥一
    • Total Pages
      106
    • Publisher
      サイエンス社
    • Related Report
      2018 Annual Research Report

URL: 

Published: 2015-04-16   Modified: 2020-03-30  

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