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Geometry of curves, surfaces and hypersurfaces with singularities

Research Project

Project/Area Number 22244006
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTokyo Institute of Technology (2011-2013)
Osaka University (2010)

Principal Investigator

UMEHARA Masaaki  東京工業大学, 情報理工学(系)研究科, 教授 (90193945)

Co-Investigator(Kenkyū-buntansha) YAMADA Kotaro  東京工業大学, 大学院理工学研究科, 教授 (10221657)
OHNITA Yoshihiro  大阪市立大学, 大学院理学研究科, 教授 (90183764)
MASHIMO Katsuya  法政大学, 理工学部, 教授 (50157187)
HASHIMOTO Hideya  名城大学, 理工学部, 教授 (60218419)
ROSSMAN Wayne  神戸大学, 大学院理学研究科, 教授 (50284485)
満渕 俊樹  大阪大学, 理学研究科, 教授 (80116102)
Co-Investigator(Renkei-kenkyūsha) KOISO Norihito  大阪大学, 大学院理学研究科, 教授 (70116028)
GOTO Ryushi  大阪大学, 大学院理学研究科, 准教授 (30252571)
KOKUBU Masatoshi  東京電機大学, 工学部, 教授 (50287439)
FUJIMORI Syoichi  岡山大学, 理学部, 准教授 (00452706)
SAJI Kentaro  神戸大学, 大学院理学研究科, 准教授 (70451432)
MIYAOKA Reiko  東北大学, 大学院理学研究科, 教授 (70108182)
IZUMIYA Shyuichi  北海道大学, 大学院理学研究科, 教授 (80127422)
ISHIKAWA Goo  北海道大学, 大学院理学研究科, 教授 (50176161)
KAWAKAMI Yu  金沢大学, 理工研究域数物科学系, 准教授 (60532356)
AGAOKA Yoshio  広島大学, 大学院理学研究科, 教授 (50192894)
KITAGAWA Yoshihisa  宇都宮大学, 教育学部, 教授 (20144917)
MABUCHI Toshiki  大阪大学, 大学院理学研究科, 教授 (80116102)
Project Period (FY) 2010-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥30,940,000 (Direct Cost: ¥23,800,000、Indirect Cost: ¥7,140,000)
Fiscal Year 2013: ¥7,280,000 (Direct Cost: ¥5,600,000、Indirect Cost: ¥1,680,000)
Fiscal Year 2012: ¥7,280,000 (Direct Cost: ¥5,600,000、Indirect Cost: ¥1,680,000)
Fiscal Year 2011: ¥7,280,000 (Direct Cost: ¥5,600,000、Indirect Cost: ¥1,680,000)
Fiscal Year 2010: ¥9,100,000 (Direct Cost: ¥7,000,000、Indirect Cost: ¥2,100,000)
Keywords特異点 / ガウス曲率 / 微分幾何学 / 半正定値計量 / 曲面 / 曲線 / 超曲面 / 波面 / 4頂点定理 / 極大曲面 / ガウス写像 / ガウス・ボンネの定理 / 変曲点 / 4頂点定理 / 双対性
Outline of Final Research Achievements

Using the concept of coherent tangent bundles, we (the head investigator and the research group) found four new Gauss-Bonnet type formulas for closed surfaces with singularities in Euclidean 3-space. Using the fact that a spacelike maximal surface with fold singularities has an analytic extension across those singularities, we showed that the analytic extensions of the triply-periodic Schwarz D type maximal surfaces are all embedded. In a joint work with Yoshihisa Kitagawa, we proved that the Clifford torus is rigid in the class of immersed flat tori whose mean curvature functions do not change sign. Moreover, we obtained some interesting results for plane curves. For example, a simplification of the proof of Bol's conjecture on sextactic points was discovered.

Report

(5 results)
  • 2014 Final Research Report ( PDF )
  • 2013 Annual Research Report
  • 2012 Annual Research Report
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • Research Products

    (28 results)

All 2014 2013 2012 2011 2010 Other

All Journal Article (18 results) (of which Peer Reviewed: 17 results,  Open Access: 2 results) Presentation (10 results) (of which Invited: 8 results)

  • [Journal Article] Flat surfaces in hyperbolic 3-space whose hyperbolic Gauss maps are bounded2014

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

      Revista Mathemática Iberoamericana

      Volume: 30 Issue: 1 Pages: 309-316

    • DOI

      10.4171/rmi/779

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Embedded triply periodic zero mean curvature surfaces of mixed type in Lorentz-Minkowski 3-space2014

    • Author(s)
      S. Fujimori, W. Rossman, M. Umehara, K. Yamada and S.-D. Yang
    • Journal Title

      Michigan Math. J.

      Volume: 63 Issue: 1 Pages: 189-207

    • DOI

      10.1307/mmj/1395234364

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A construction of a complete bounded null curve in <b>C</b><sup>3</sup>2014

    • Author(s)
      L. Ferrer, F. Martin, M. Umehara and K. Yamada
    • Journal Title

      Kodai Mathematical Journal

      Volume: 37 Issue: 1 Pages: 59-96

    • DOI

      10.2996/kmj/1396008249

    • NAID

      130004496212

    • ISSN
      0386-5991, 1881-5472
    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Intrinsic invariants of cross caps Selecta Mathematica2013

    • Author(s)
      Masaru Hasegawa, Atsufumi Honda, Kosuke Naokawa, Masaaki Umehara and Kotaro Yamada
    • Journal Title

      Online First

      Volume: 20130730 Issue: 3 Pages: 769-785

    • DOI

      10.1007/s00029-013-0134-6

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Closed planar curves without inflections2013

    • Author(s)
      S. Ohno, T. Ozawa and M. Umehara
    • Journal Title

      Proc. Amer. Math. Soc.

      Volume: 141 Issue: 1 Pages: 281-287

    • DOI

      10.1090/s0002-9939-1991-1043406-7

    • Related Report
      2013 Annual Research Report
  • [Journal Article] Zero mean curvature surfaces in L^3 containing a light-like line2012

    • Author(s)
      Shoichi Fujimori
    • Journal Title

      Comptes Reudus de l'Academie Sciences. Serie I. Mathematique,

      Volume: 350 Issue: 21-22 Pages: 975-978

    • DOI

      10.1016/j.crma.2012.10.024

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A refinement of Foreman's four vertex theorem and its dual version2012

    • Author(s)
      Throbergsson and M. Umehara
    • Journal Title

      Kyoto J. Math.

      Volume: 52 Issue: 4

    • DOI

      10.1215/21562261-1728848

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The behavior of curvature functions at cusps and inflection points2012

    • Author(s)
      S. Shiba and M. Umehara
    • Journal Title

      Differential Geometry and its Applications

      Volume: 30 Issue: 3 Pages: 285-299

    • DOI

      10.1016/j.difgeo.2012.04.001

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts2012

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

      Journal of Geometric Analysis

      Volume: vol.22 Issue: 2 Pages: 383-409

    • DOI

      10.1007/s12220-010-9193-5

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] CMC-1 trinoids in H^3 and metrics of constant curvature one…2011

    • Author(s)
      S.Fujimori, K.Yamada, et.al.
    • Journal Title

      Proceedings of Japan Academy, Ser.A.

      Volume: 87 Issue: 8 Pages: 144-149

    • DOI

      10.3792/pjaa.87.144

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] <i>A</i><sub>2</sub>-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space2011

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

      Kodai Mathematical Journal

      Volume: 34 Issue: 3 Pages: 390-409

    • DOI

      10.2996/kmj/1320935549

    • NAID

      130004687956

    • ISSN
      0386-5991, 1881-5472
    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Extrinsic diameter of immersed flat tori in S^32011

    • Author(s)
      Y.Kitagawa, M.Umehara
    • Journal Title

      Geom.Dedicata

      Volume: 155 Issue: 1 Pages: 105-140

    • DOI

      10.1007/s10711-011-9580-5

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Orientability of linear Weingarten surfaces, spacelike CMC-1 surfaces and maximal surfaces2011

    • Author(s)
      M.Kokubu and M.Umehara
    • Journal Title

      Math. Nachr.

      Volume: 284 Issue: 14-15 Pages: 1903-1918

    • DOI

      10.1002/mana.200910176

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Applications of a completeness lemma in minimal surface theory to various classes of surfaces2011

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

      Bulletin of the London Mathematical Society

      Volume: 43 Issue: 3 Pages: 191-199

    • DOI

      10.1112/blms/bds017

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The duality of conformally flat manifolds2011

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

      Bulletin of the Brasilian Mathematical Society

      Volume: 42 Pages: 131-152

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A simplification of the proof of Bol's conjecture on sextactic points2011

    • Author(s)
      M.Umehara
    • Journal Title

      Proc.Japan Acad.Ser.A

      Volume: 87 Issue: 1 Pages: 10-12

    • DOI

      10.3792/pjaa.87.10

    • NAID

      40017436335

    • Related Report
      2011 Annual Research Report 2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The duality between singular points and inflection points on wave fronts2010

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

      Osaka J.Math.

      Volume: 47 Pages: 591-607

    • NAID

      120004844268

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Singularities of Blaschke normal maps of convex surfaces2010

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

      C.R.Acad.Sci.Paris.Ser.I

      Volume: 348 Pages: 665-668

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Presentation] 特異点をもつ曲面の微分幾何学2012

    • Author(s)
      梅原雅顕
    • Organizer
      日本数学会2012年度年会
    • Place of Presentation
      トッパン・ホール(招待講演)
    • Year and Date
      2012-03-27
    • Related Report
      2011 Annual Research Report
  • [Presentation] 変曲点を持たない平面閉曲線について2011

    • Author(s)
      梅原雅顕
    • Organizer
      RIMS研究集会(部分多様体の微分幾何学的研究)
    • Place of Presentation
      京都大学数理解析研究所(招待講演)
    • Year and Date
      2011-06-27
    • Related Report
      2011 Annual Research Report
  • [Presentation] Zero Mean curvature surfaces in Lorentz-Minkowski 3-space which changes type across a light-like line

    • Author(s)
      梅原雅顕
    • Organizer
      Internatinal workshop on special Geometry and Minimal Subminifolds
    • Place of Presentation
      東北大学・理学研究科
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] 卵形線の縮閉線の形状と特異点

    • Author(s)
      梅原雅顕
    • Organizer
      福岡大学微分幾何研究会(Geometry or Something)
    • Place of Presentation
      福岡大学セミナーハウス
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] 特異点の内的不変量と等長変形

    • Author(s)
      梅原雅顕
    • Organizer
      山口幾何学研究集会2013 (進展する曲面論 - Advances in Surface Theory )
    • Place of Presentation
      山口大学理学部
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] 交叉帽子の微分幾何学

    • Author(s)
      梅原雅顕
    • Organizer
      部分多様体と四元数構造 (RIMS研究集会)
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Differential Geometry of surfaces with singularities

    • Author(s)
      梅原雅顕
    • Organizer
      12th International Workshop on Real and Complex Singularities
    • Place of Presentation
      サンパウロ大学サンカルロス校,ブラジル
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Differential Geometry of surfaces with singularities

    • Author(s)
      梅原雅顕
    • Organizer
      The 8th Geometry Conference for the Friendship of China and Japan
    • Place of Presentation
      四川大学,中国
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] 双曲計量と3次元de Sitter 空間の平均曲率1の曲面

    • Author(s)
      梅原雅顕
    • Organizer
      福岡大微分幾何研究会(Geometry and something)
    • Place of Presentation
      福岡大学セミナーハウス
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] 3次元時空の空間的極大曲面と、その時間的極小曲面への型変化について

    • Author(s)
      梅原雅顕
    • Organizer
      部分多様体論・湯沢2012
    • Place of Presentation
      湯沢町・湯沢グランドホテル
    • Related Report
      2012 Annual Research Report
    • Invited

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Published: 2010-08-23   Modified: 2021-04-07  

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