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Value distribution theory of bounded domains

Research Project

Project/Area Number 23654021
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Geometry
Research InstitutionTohoku University

Principal Investigator

MIYAOKA Reiko  東北大学, 理学(系)研究科(研究院), 教授 (70108182)

Research Collaborator KOBAYASHI Ryoichi  名古屋大学, 大学院多元数理科学研究科, 教授 (20162034)
Project Period (FY) 2011-04-28 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords代数的極小曲面 / 全曲率有限 / ガウス写像 / 除外値 / 全分岐値数 / ネバンリンナ理論 / コーンフォッセンの不等式 / 対数微分の補題 / 値分布 / 除外値数 / Nevanlinna理論 / 極小曲面 / 除外値問題 / 双曲型曲面 / ガロア群 / 被覆変換
Outline of Final Research Achievements

Through a challenge to estimate the number of exceptional values as well as the total ramified value numbers of the Gauss map of algebraic minimal surfaces, we obtain the following:
1.We found an invariant R given by the ratio of the degree of the Gauss map and a topological quantity, and taking the period condition into account, we have an estimate of the total ramified value number, although the maximal number of the exceptional values is not yet obtained. 2. Lifting all the data to the universal covering (disk) of the surface, we found a candidate of the key number κ, which is somehow corresponds to R above. 3. Using the special properties of minimal surfaces, we consider 1-jet space of the Gauss map to which we apply a new Nevanlinna theory. On the jet space, we consider the proximity function, the counting function and try to use the lemma on logarithmic differential to obtain the final defect relation.

Report

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

    (18 results)

All 2014 2013 Other

All Journal Article (4 results) (of which Peer Reviewed: 4 results) Presentation (13 results) (of which Invited: 13 results) Remarks (1 results)

  • [Journal Article] Isoparametric hypersurfaces with (g,m)=(6,2)2013

    • Author(s)
      R.Miyaoka
    • Journal Title

      Annals of Math.

      Volume: 176 Issue: 1 Pages: 53-110

    • DOI

      10.4007/annals.2013.177.1.2

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Moment maps of the spin action and the Cartan-Munzner polynomials of degree four2013

    • Author(s)
      R.Miyaoka
    • Journal Title

      Math. Ann.

      Volume: 355 Issue: 3 Pages: 1067-1084

    • DOI

      10.1007/s00208-012-0819-8

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Transnormal functions and isoparametric hypersurfaces2013

    • Author(s)
      R.Miyaoka
    • Journal Title

      Diff. Geom. Appl.

      Volume: 31 Issue: 1 Pages: 130-139

    • DOI

      10.1016/j.difgeo.2012.10.005

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Moment map of the spin action and Cartan Munzner polynomials2013

    • Author(s)
      R. Miyaoka
    • Journal Title

      Proceedings of ``Differential Geometry and Tanaka Theory"

      Volume: -

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] 代数的極小曲面のガウス写像の除外値数について2014

    • Author(s)
      宮岡礼子
    • Organizer
      複素解析セミナー
    • Place of Presentation
      東北大学情報科学研究科
    • Year and Date
      2014-11-19
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 極小曲面のガウス写像の除外値に対するOsserman理論の計量化の試み2014

    • Author(s)
      小林亮一
    • Organizer
      大岡山談話会
    • Place of Presentation
      東京工業大学
    • Year and Date
      2014-07-16
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Localization via parabolic translation and collective Cohn-Vossen inequality2014

    • Author(s)
      小林亮一
    • Organizer
      5th International Workshop on Differential Geometry and Analysis
    • Place of Presentation
      虹ノ松原ホテル,唐津
    • Year and Date
      2014-06-04
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Hypersurface geometry, classical and modern2013

    • Author(s)
      R.Miyaoka
    • Organizer
      7th KIAS Winter School on Differential Geometry
    • Place of Presentation
      KIAS,韓国
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Hypersurface geometry and moment map

    • Author(s)
      R.Miyaoka
    • Organizer
      KWMS 2012 Conference
    • Place of Presentation
      Ehwa Univ. Korea
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Hypersurface geometry and moment map

    • Author(s)
      R.Miyaoka
    • Organizer
      UK-Japan Mathematical Forum
    • Place of Presentation
      慶応大学
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 等径超曲面とモーメント写像

    • Author(s)
      宮岡礼子
    • Organizer
      第58回幾何学シンポジウム
    • Place of Presentation
      九州大学
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] スピン作用をもつ超曲面とモーメント写像

    • Author(s)
      宮岡礼子
    • Organizer
      Submanifold Geometry and Lie Group Action
    • Place of Presentation
      東京理科大
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Expression of Cartan-Munzner Polynomials via moment map of the spin action

    • Author(s)
      R.Miyaoka
    • Organizer
      8-th Geometry Conference for the Friendship of China and Japan
    • Place of Presentation
      四川大学,中国
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Theory of isoparametric hypersurfaces and related topics

    • Author(s)
      R.Miyaoka
    • Organizer
      International Conference on Differential Geometry for Prof. Su Buqing
    • Place of Presentation
      復旦大学,中国
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Isoparametric hypersurfaces and moment map

    • Author(s)
      R.Miyaoka
    • Organizer
      The 16th International Workshop on Differential Geometry
    • Place of Presentation
      慶北大学,韓国
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Singular set of the mean curvature flow

    • Author(s)
      R.Miyaoka
    • Organizer
      Informal geometry seminar
    • Place of Presentation
      King's College London,イギリス
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Singular set of the mean curvature flow

    • Author(s)
      R.Miyaoka
    • Organizer
      GCOE小研究集会 -Minimal submanifolds and mean curvature flow-
    • Place of Presentation
      KKR 蔵王白銀荘
    • Related Report
      2012 Research-status Report
    • Invited
  • [Remarks] 宮岡礼子

    • URL

      http://www.math.tohoku.ac.jp/people/miyaoka/indexv3.html

    • Related Report
      2012 Research-status Report

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Published: 2011-08-05   Modified: 2019-07-29  

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