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Surfaces with singularities in space-times and Weierstrass-type representation formulas

Research Project

Project/Area Number 26400066
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Geometry
Research InstitutionTokyo Institute of Technology

Principal Investigator

Yamada Kotaro  東京工業大学, 理学院, 教授 (10221657)

Co-Investigator(Renkei-kenkyūsha) Masaaki Umehara  東京工業大学, 情報理工学院, 教授 (90193945)
Project Period (FY) 2014-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2014: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywordsローレンツ多様体 / ワイエルストラス表現公式 / 特異点 / 極大曲面 / 型変化 / 零平均曲率曲面 / ミンコフスキー空間 / ド・ジッター空間 / CMC-1 面 / 因果特性の変化 / ド・ジッター空間の平均曲率1曲面 / 波面 / ミンコフスキー時空 / 平均曲率 / 折り目特異点
Outline of Final Research Achievements

A class of maximal surfaces in Lorentz-Minkowski 3-space, named "Kobayashi surfaces" is introduced. A surface in this class can be extended to a zero-mean curvature surfaces which changes causal types from space-like to time-like. Existence of infinitely many surfaces among this class which are graphs of functions over the space-like plane. On the other hand, the first example of a zero-mean curvature which contains a light-like line and changes causal types across the line is obtained. Such a property, called the light-like line theorem, are generalized for wider class of surfaces.
For a surface in Lorentzian 3-manifold which changes its causal type from space-like to time-like at a light-like point, it is shown that the mean curvature function converges to zero at the light-like point. In particuler, it is shown that a non-zero constant mean curvature surface cannot change its causal type.

Report

(5 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (15 results)

All 2017 2016 2015 2014 Other

All Int'l Joint Research (1 results) Journal Article (7 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 7 results,  Open Access: 3 results,  Acknowledgement Compliant: 6 results) Presentation (5 results) (of which Int'l Joint Research: 3 results,  Invited: 5 results) Funded Workshop (2 results)

  • [Int'l Joint Research] TU-Wien(オーストリア)

    • Related Report
      2015 Research-status Report
  • [Journal Article] Surfaces With Light-Like Points In Lorentz-Minkowski 3-Space With Applications2017

    • Author(s)
      Masaaki Umehara and Kotaro Yamada
    • Journal Title

      Springer Proceedings in Mathematics & Statistics

      Volume: 211 Pages: 253-273

    • DOI

      10.1007/978-3-319-66290-9_14

    • ISBN
      9783319662893, 9783319662909
    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Mixed type surfaces with bounded mean curvature in 3-dimensional space-times2017

    • Author(s)
      A. Honda, M. Koiso, M. Kokubu, M. Umehara and Kotaro Yamada
    • Journal Title

      Diff. Geom. and its Appl.

      Volume: 52 Pages: 64-77

    • DOI

      10.1016/j.difgeo.2017.03.009

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] ISOMETRIC DEFORMATIONS OF CUSPIDAL EDGES2016

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

      Tohoku Mathematical Journal, Second Series

      Volume: 68 Issue: 1 Pages: 73-90

    • DOI

      10.2748/tmj/1458248863

    • NAID

      130008106069

    • ISSN
      0040-8735, 2186-585X
    • Year and Date
      2016-03-30
    • 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
  • [Journal Article] Behavior of Gaussian Curvature and Mean Curvature Near Non-degenerate Singular Points on Wave Fronts2016

    • Author(s)
      Luciana F, Martins, Kentaro Saji, Masaaki Umehara and Kotaro Yamada
    • Journal Title

      Springer Proceedings in Mathematics & Statistics

      Volume: 154 Pages: 1-39

    • DOI

      10.1007/978-4-431-56021-0_14

    • ISBN
      9784431560197, 9784431560210
    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Intrinsic properties of surfaces with singularities2015

    • Author(s)
      M. Hasegawa, A. Honda, K. Naokawa, K. Saji, M. Umehara, K. Yamada
    • Journal Title

      International Journal of Mathatematics

      Volume: 26 Issue: 04 Pages: 1-34

    • DOI

      10.1142/s0129167x1540008x

    • Related Report
      2015 Research-status Report 2014 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Intrinsic Deformation of Cuspidal Edges2015

    • Author(s)
      Kosuke Naokawa, Masaaki Umehara and Kotaro Yamada
    • Journal Title

      Tohoku Mathematical Jourrnal

      Volume: -

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Type changes of spacelike maximal surfaces in Minkowski 3-space to timelike surfaces2016

    • Author(s)
      Kotaro Yamada
    • Organizer
      福岡大学幾何学研究集会
    • Place of Presentation
      福岡大学セミナーハウス(福岡市中央区)
    • Year and Date
      2016-11-05
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Type changes of spacelike maximal surfaces in Minkowski 3-space to timelike surfacesc2016

    • Author(s)
      Kotaro Yamada
    • Organizer
      8th International Meeting on Lorentzian Geometry, Malaga
    • Place of Presentation
      Universidad de Malaga (Malaga, Spain)
    • Year and Date
      2016-09-23
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Analytic extensions of spacelike maximal surfaces in Minkowski 3-space to timelike surfaces2015

    • Author(s)
      Kotaro Yamada
    • Organizer
      Geometric Analysis in Geometry and Topology, 2015
    • Place of Presentation
      東京理科大学
    • Year and Date
      2015-11-10
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Analytic extensions of spacelike maximal surfaces in Minkowski 3-space to timelike surfaces2015

    • Author(s)
      Kotaro Yamada
    • Organizer
      Singularities in Generic Geometry and applications Kobe - Kyoto 2015 (Valencia IV)
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2015-06-09
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Analytic extensions of spacelike maximal surfaces in Minkowski 3-space to timelike surfaces2014

    • Author(s)
      Kotaro Yamada
    • Organizer
      The 10th Geometry Conference of the Frendship between China and Japan
    • Place of Presentation
      Fudan University
    • Year and Date
      2014-09-10
    • Related Report
      2014 Research-status Report
    • Invited
  • [Funded Workshop] The third Spanish-Japanese Workshop on Differential Geometry2017

    • Related Report
      2017 Annual Research Report
  • [Funded Workshop] Transformations and Singularities2016

    • Place of Presentation
      東京工業大学
    • Year and Date
      2016-02-19
    • Related Report
      2015 Research-status Report

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Published: 2014-04-04   Modified: 2019-03-29  

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