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Loewner theory on universal covering maps

Research Project

Project/Area Number 16K05206
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionYamaguchi University

Principal Investigator

Yanagihara Hiroshi  山口大学, 大学院創成科学研究科, 教授 (30200538)

Co-Investigator(Kenkyū-buntansha) 堀田 一敬  山口大学, 大学院創成科学研究科, 講師 (10725237)
Project Period (FY) 2016-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords普遍被覆 / 普遍被覆写像 / 正則函数 / Loewner理論 / subordination / Loewner chain / 複素解析 / レブナー理論 / 等角写像 / 連結度 / 核収束 / 時間発展 / レブナーの微分方程式 / 解析学
Outline of Final Research Achievements

In the research we showed the Loewner theory, a powerful method in the theory of univalent functions, is applicable to study analytic universal covering maps.
In 1923 Loewner found that for any bounded slit mapping of the unit disc has a parametric representations satisfying a differential equation known as the Loewner differential equation.The parametric representation method was extensively developed and generalized by Pommerenke.
We tried to generalize Loewner-Pommerenke theory on univalent functions to universal covering maps on the unit disc and succeded to show that the theory has a geometrically natural extension. We also studied a behavior on associated Fuchsian gorup and found that there exists a family of time-depend homomorphisms between them.

Academic Significance and Societal Importance of the Research Achievements

レブナー方程式は2次元的な領域を時間とともに拡大する際に現れる基本的な方程式である.レブナー理論はこの方程式の解析や応用を行うものであり, 「拡がる」という極めて自然な現象を数学的に捉えるものであるから,その応用は物理学,工学から社会学まで幅広い.本研究により従来の理論の限界である穴の空いていない領域のみ扱うことが出来るという制約を除くことに成功した. これからのさらなる研究の基盤として期待が出来る.また研究が進むほどに新たな疑問が生じてきた. 普遍被覆写像を制御する被覆変換群または像領域の基本群が時間とともにどのような挙動を示すかなどである. これらの疑問についてはこれからの課題となる.

Report

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

    (20 results)

All 2019 2018 2017 2016 Other

All Int'l Joint Research (2 results) Journal Article (8 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 7 results,  Open Access: 1 results,  Acknowledgement Compliant: 2 results) Presentation (9 results) (of which Int'l Joint Research: 5 results,  Invited: 9 results) Remarks (1 results)

  • [Int'l Joint Research] Dept. of Math., Wuerzburg Univ.(ドイツ)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] Dept. of Mat. Sci., Univ. of Cicinnati(米国)

    • Related Report
      2018 Annual Research Report
  • [Journal Article] Loewner chains with quasiconformal extensions: an approximation approach2019

    • Author(s)
      Ikkei Hotta
    • Journal Title

      J. Anal. Math.

      Volume: -

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Loewner chains for quasiconformal extensions: old and new2019

    • Author(s)
      Ikkei Hotta
    • Journal Title

      Interdiscip. Inform. Sci.

      Volume: -

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Two points distortion estimates for convex univalent functions2018

    • Author(s)
      M. Okada and H. Yanagihara
    • Journal Title

      Bull. Korean Math. Soc.

      Volume: 55 Pages: 957-965

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Two points distortion estimates for convex univalent functions2018

    • Author(s)
      Mari Okada and Hiroshi Yanagihara
    • Journal Title

      Bulletin of the Korean Mathematical Society

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Tightness results for infinite-slit limits of the chordal Loewner equation2018

    • Author(s)
      Andrea Del Monaco, Ikkei Hotta, Sebastian Schleissinger
    • Journal Title

      Comput. Methods Funct. Theory

      Volume: 18 Issue: 1 Pages: 9-33

    • DOI

      10.1007/s40315-017-0205-3

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Some remarks on Schur's theorem for bounded analytic functions2017

    • Author(s)
      Ikkei Hotta and Andrzej Michalski
    • Journal Title

      Bull. Soc. Sci. Lett. Lodz Ser. Rech. Deform.

      Volume: 67 Pages: 93-99

    • Related Report
      2017 Research-status Report
  • [Journal Article] Quasiconformal extendibility of integral transforms of Noshiro-Warschawski functions2017

    • Author(s)
      Ikkei Hotta and Li-Mei Wang
    • Journal Title

      Rocky Mountain J. Math.

      Volume: 47 Issue: 1 Pages: 185-204

    • DOI

      10.1216/rmj-2017-47-1-185

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Chordal Loewner chains with quasiconformal extensions2017

    • Author(s)
      Pavel Gumenyuk and Ikkei Hotta
    • Journal Title

      Math. Z.

      Volume: 285 Issue: 3-4 Pages: 1063-1089

    • DOI

      10.1007/s00209-016-1738-2

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] Transfinite Diameter2019

    • Author(s)
      Hiroshi Yanagihara
    • Organizer
      Colloquium Talk, School of Basic Science, IIT Bhubaneshwar, India
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] レブナー方程式による擬等角拡張の構成について2019

    • Author(s)
      堀田 一敬
    • Organizer
      日本数学会2019年度年会 特別講演
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Loewner Theory on Analytic Universal Covering Maps2018

    • Author(s)
      Hiroshi Yanagihara
    • Organizer
      International Workshop on Conformal Dynamics and Loewner Theory
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Loewner Theory on Analytic Universal Covering Maps2018

    • Author(s)
      Hiroshi Yanagihara
    • Organizer
      Colloquium Talk, Department of Mathematics, Wuerzburg University, Germany
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Hydrodynamic limit of multiple SLE2018

    • Author(s)
      Ikkei Hotta
    • Organizer
      International Workshop on Conformal Dynamics and Loewner Theory 2018
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Yamada's method on quasiconformal variations2017

    • Author(s)
      柳原宏
    • Organizer
      平成28年度等角写像論・値分布論研究集会
    • Place of Presentation
      千葉大学理学部(千葉県千葉市)
    • Year and Date
      2017-01-21
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Loewner Theory on Analytic Universal Covering Maps2017

    • Author(s)
      柳原 宏
    • Organizer
      等角写像論・値分布論合同研究集会
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Chordal Loewner chainsとその境界挙動2017

    • Author(s)
      堀田一敬
    • Organizer
      第60回函数論シンポジウム
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Linearly accessibility by flows associated with Loewner's equation2016

    • Author(s)
      Ikkei Hotta
    • Organizer
      XVIIIth Conference on Analytic Functions and Related Topics
    • Place of Presentation
      The State School of Higher Education in Chelm, Poland
    • Year and Date
      2016-06-26
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks] Hiroshi Yanagihara (柳原 宏)ホームページ

    • URL

      http://web.cc.yamaguchi-u.ac.jp/~hiroshi/

    • Related Report
      2018 Annual Research Report

URL: 

Published: 2016-04-21   Modified: 2020-03-30  

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