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A new perspective of complex manifolds from the view point of generalizations of holomorphic motions

Research Project

Project/Area Number 18K18717
Research Category

Grant-in-Aid for Challenging Research (Exploratory)

Allocation TypeMulti-year Fund
Review Section Medium-sized Section 12:Analysis, applied mathematics, and related fields
Research InstitutionKyoto Sangyo University (2019-2022)
Tokyo Institute of Technology (2018)

Principal Investigator

Shiga Hiroshige  京都産業大学, 理学部, 教授 (10154189)

Project Period (FY) 2018-06-29 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
KeywordsHolomorphic motion / Quasiconformal mapping / Riemann surface / 擬等角写像 / 正則運動 / 複素多様体 / タイヒミュラー空間
Outline of Final Research Achievements

Holomorphic motions are defined as holomorphic families of injections on subset of the Riemann sphere parametrized by complex manifolds. On this topic, we give a counter example to Chirka's statement, and also an example of a holomorphic motion which satisfies a kind of a topological trivial condition but can not be extended to a holomorphic motion over the Riemann sphere.
We find some conditions for generalized Cantor sets to be quasiconformally equivalent to each other. We also give estimate maximal dilatations. A necessary and sufficient condition for a generalized Cantor set to be quasiconformally equivalent to the standard Cantor set is obtained. Those results imply that different Cantor sets are connected via holomorphic motions.

Academic Significance and Societal Importance of the Research Achievements

The holomorphic motion is a quite simple object in mathematics, that is, it is a holomorphic family of injections on a set in the complex plane. We have found various aspects on holomorphic motions and quasiconformal mappings.

Report

(6 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (14 results)

All 2023 2022 2021 2020 2019 2018

All Journal Article (4 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 4 results,  Open Access: 3 results) Presentation (9 results) (of which Int'l Joint Research: 4 results,  Invited: 8 results) Book (1 results)

  • [Journal Article] On the quasiconformal equivalence of dynamical Cantor sets2022

    • Author(s)
      Shiga Hiroshige
    • Journal Title

      Journal d'Analyse Mathematique

      Volume: 147 Issue: 1 Pages: 1-28

    • DOI

      10.1007/s11854-022-0214-7

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A note on the extendability of holomorphic motions2020

    • Author(s)
      Shiga Hiroshige
    • Journal Title

      Kodai Mathematical Journal

      Volume: 43 Issue: 1 Pages: 162-169

    • DOI

      10.2996/kmj/1584345692

    • NAID

      130007812091

    • ISSN
      0386-5991, 1881-5472
    • Year and Date
      2020-03-15
    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] The quasiconformal equivalence of Riemann surfaces and the universal Schottky space2019

    • Author(s)
      Shiga Hiroshige
    • Journal Title

      Conformal Geometry and Dynamics of the American Mathematical Society

      Volume: 23 Issue: 11 Pages: 189-204

    • DOI

      10.1090/ecgd/343

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] TEICHMÜLLER SPACES AND TAME QUASICONFORMAL MOTIONS2018

    • Author(s)
      Jiang Yunping、Mitra Sudeb、Shiga Hiroshige、Wang Zhe
    • Journal Title

      Tohoku Mathematical Journal, Second Series

      Volume: 70 Issue: 4 Pages: 607-631

    • DOI

      10.2748/tmj/1546570827

    • ISSN
      0040-8735, 2186-585X
    • Year and Date
      2018-12-30
    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] Quasiconformal mappings and hyperbolic geometry2023

    • Author(s)
      志賀 啓成
    • Organizer
      Workshop: Quasiconformal mappings, hyperbolic geometry and Riemann surfaces
    • Related Report
      2022 Annual Research Report
  • [Presentation] Open Riemann surfacesのトリセツ2022

    • Author(s)
      志賀 啓成
    • Organizer
      第56回函数論サマーセミナー
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Quasicircles and Dirichlet finite harmonic functions on open Riemann surfaces2022

    • Author(s)
      志賀 啓成
    • Organizer
      The POSTECH Conference 2022 on Complex Geometry
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Quasiconformal mappings and quasicircles on open Riemann surfaces2022

    • Author(s)
      志賀 啓成
    • Organizer
      Conference in honor of the 65th birthday of Athanase Papadopoulos
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Quasi-circles and Dirichlet finite harmonic functions on Riemann Surfaces2021

    • Author(s)
      志賀啓成
    • Organizer
      静岡複素解析幾何セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Julia 集合の函数論2020

    • Author(s)
      志賀 啓成
    • Organizer
      複素力学系理論の総合的研究
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Dynamical Cantor sets and quasiconformal mappings2020

    • Author(s)
      志賀 啓成
    • Organizer
      Teichmuller Theory: Classical, Higher, Super and Quantum
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On dynamical Cantor sets and quasiconformal mappings2019

    • Author(s)
      志賀 啓成
    • Organizer
      First Analysis Mathematica International Conference
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On dynamical Cantor sets and quasiconformal mappings2019

    • Author(s)
      志賀 啓成
    • Organizer
      函数論シンポジウム
    • Related Report
      2019 Research-status Report
    • Invited
  • [Book] リーマンと解析学2020

    • Author(s)
      黒川 信重、志賀 啓成
    • Total Pages
      112
    • Publisher
      共立出版
    • ISBN
      9784320112353
    • Related Report
      2019 Research-status Report

URL: 

Published: 2018-07-25   Modified: 2024-01-30  

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