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New perspective on Riemann surfaces of infinite genus and their deformation spaces

Research Project

Project/Area Number 26610014
Research Category

Grant-in-Aid for Challenging Exploratory Research

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

Principal Investigator

SHIGA Hiroshige  東京工業大学, 理学院, 教授 (10154189)

Co-Investigator(Kenkyū-buntansha) 藤川 英華  千葉大学, 大学院理学研究科, 准教授 (80433788)
Project Period (FY) 2014-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords関数論 / トポロジー / Hyperbolic geometry / Teichmuller space / Riemann surface / 国際研究者交流 / 国際情報交換 / 多国籍
Outline of Final Research Achievements

On the deformation theory of Riemann surfaces, quasiconformal maps have played an important role. In this project, we succeeded in constructing a counterexample of Sullivan-Thurston’claim for quasiconformal motions. We also give a necessary and sufficient condition for a holomorphic motion of a subset of the Riemann sphere parametrized over a Riemann surface to be extended to a holomorphic motion of the Riemann sphere. The condition is given by the monodromy of the motion. This result is related to quasiconformal maps but it has also a topological aspect, the monodromy.
A new concept of Teichmuller space for Riemann surface with countably many points is considered. The research of those Teichmuller spaces will generalize holomorphic motions to those of subsets of Riemann surfaces. Therefore, it will give new interesting problems to related fields of mathematics.

Report

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

    (17 results)

All 2017 2016 2015 2014 Other

All Int'l Joint Research (1 results) Journal Article (5 results) (of which Peer Reviewed: 5 results,  Open Access: 3 results,  Acknowledgement Compliant: 1 results) Presentation (11 results) (of which Int'l Joint Research: 5 results,  Invited: 11 results)

  • [Int'l Joint Research] ニューヨーク市立大学(米国)

    • Related Report
      2015 Research-status Report
  • [Journal Article] The Teichmuller space of a countable set of points on a Riemann surface2017

    • Author(s)
      Ege Fujikawa and Masahiko Taniguchi
    • Journal Title

      Conformal Geometry and Dynamics

      Volume: 21 Issue: 2 Pages: 64-77

    • DOI

      10.1090/ecgd/301

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Conformal invariants defined by harmonic functions on Riemann surfaces2016

    • Author(s)
      H. Shiga
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 68 Issue: 1 Pages: 441-458

    • DOI

      10.2969/jmsj/06810441

    • NAID

      130005122700

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2016 Annual Research Report 2015 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] On analytic properties of deformation spaces of Kleinian groups2016

    • Author(s)
      H. Shiga
    • Journal Title

      Trans. Amer. Math. Soc.

      Volume: 368 Issue: 9 Pages: 6627-6642

    • DOI

      10.1090/tran/6563

    • Related Report
      2016 Annual Research Report 2015 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Holomorphic families of Riemann surfaces and monodromy2014

    • Author(s)
      H. Shiga
    • Journal Title

      Handbook of Teichmuller Theory, European Mathematical Society

      Volume: IV Pages: 439-460

    • DOI

      10.4171/117-1/12

    • ISBN
      9783037191170, 9783037196175
    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Convex hull of set in thick part of Teichmuller space2014

    • Author(s)
      L. Lui, H. Shiga and Z. Sun
    • Journal Title

      Science China Math.

      Volume: 57 Issue: 9 Pages: 1799-1810

    • DOI

      10.1007/s11425-014-4871-6

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Presentation] Holomorphic motions and the extension problem2017

    • Author(s)
      H. Shiga
    • Organizer
      「リーマン面・不連続群論」研究集会
    • Place of Presentation
      東北大学 片平キャンパス(仙台市青葉区)
    • Year and Date
      2017-01-09
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Holomorphic motions over Riemann surfaces and the extension problem2016

    • Author(s)
      H. Shiga
    • Organizer
      Topology related to Riemann surfaces
    • Place of Presentation
      The University of Tokyo, Meguro, Tokyo (Japan)
    • Year and Date
      2016-09-05
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Holomorphic motion over Riemann surface and monodromy2016

    • Author(s)
      H. Shiga
    • Organizer
      XVIII th Conference on Analytic Functions and related Topics
    • Place of Presentation
      Institute of Mathematics and Computer Science of the State School of Higher Education in Chelm (Poland)
    • Year and Date
      2016-06-28
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Tame qc-motions and the monodromy2016

    • Author(s)
      H. Shiga
    • Organizer
      AMS Spring Eastern Sectional Meeting, Special Session on Teichmuller Theory and Related Topics
    • Place of Presentation
      State University of New York at Stony Brook, Stony Brook, NY, USA
    • Year and Date
      2016-03-20
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Teichmuller spaces and holomorphic maps2015

    • Author(s)
      H. Shiga
    • Organizer
      Riemann surfaces and related topics
    • Place of Presentation
      Universidad Autonoma de Madrid, Madrid, Spain
    • Year and Date
      2015-06-02
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Klein群の変形空間について2015

    • Author(s)
      H. Shiga
    • Organizer
      日本数学会年会(特別講演)
    • Place of Presentation
      明治大学駿河台キャンパス(東京都千代田区)
    • Year and Date
      2015-03-23
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Complex analytic properties of deformation spaces of Kleinian groups I2014

    • Author(s)
      H. Shiga
    • Organizer
      低次元多様体モジュライ空間の幾何学
    • Place of Presentation
      京都大学数理科学研究所(京都府京都市左京区)
    • Year and Date
      2014-12-11
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Complex analytic properties of deformation spaces of Kleinian groups II2014

    • Author(s)
      H. Shiga
    • Organizer
      低次元多様体モジュライ空間の幾何学
    • Place of Presentation
      京都大学数理科学研究所(京都府京都市左京区)
    • Year and Date
      2014-12-11
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Conformal invariants defined by harmonic functions on Riemann surfaces2014

    • Author(s)
      H. Shiga
    • Organizer
      Conference of Potential theory
    • Place of Presentation
      福山大学宮地茂記念館(広島県福山市)
    • Year and Date
      2014-09-04
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Teichmuller curves and holomorphic maps on Riemann surfaces2014

    • Author(s)
      H. Shiga
    • Organizer
      Geometry on Groups and Spaces, ICM Satellite Conference on Geometric Group Theory & Geometric Structures
    • Place of Presentation
      KAIST, Daejeon, Korea
    • Year and Date
      2014-08-09
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] On analytic properties of deformation spaces of Kleinian groups2014

    • Author(s)
      H. Shiga
    • Organizer
      XVII th Conference on Analytic Functions and Related Topics
    • Place of Presentation
      Institute of Mathematics and Computer Science of the State School of Higher Education in Chelm, Poland
    • Year and Date
      2014-06-29
    • Related Report
      2014 Research-status Report
    • Invited

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Published: 2014-04-04   Modified: 2023-03-16  

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