Construction of geometric structure for infinite dimensional Teichmuller space

• Project/Area Number: 25400127
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tokyo Institute of Technology (2018-2019); Chiba University (2013-2017)
• Principal Investigator: Fujikawa Ege 東京工業大学, 理学院, 准教授 (80433788)
• Project Period (FY): 2013-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: リーマン面 / 擬等角写像 / 写像類群 / モジュライ空間 / 力学系理論 / 複素解析学 / タイヒミュラー空間 / モジュラー群 / 双曲幾何学 / 双曲幾何

Outline of Final Research Achievements

We consider the infinite dimensional Teichmuller space of a Riemann surface of general type. The asymptotic Teichmuller space is a certain quotient space of the Teichmuller space and there is a natural projection from the Teichmuller space to the asymptotic Teichmuller space. We consider the fibers of the projection over any point in the asymptotic Teichmuller space, and show a coherence of the discreteness on each fiber in the Teichmuller space. Furthermore, we formulate the concept of the Teichmuller space of a fractal structure and establish the fundamental theory on it. We introduce the Teichmuller space of a countable set of points associated with the fractal structure, and show that such a space admits a natural complex analytic structure if the fractal structure possesses standard bounded geometry.

Academic Significance and Societal Importance of the Research Achievements

タイヒミュラー空間は曲面の構造のパラメーター空間として，数学の諸分野や数理物理学の研究において重要な役割をもっている．有限型リーマン面に対する有限次元タイヒミュラー空間上とは異なり，無限型リーマン面に対する無限次元タイヒミュラー空間上では，タイヒミュラーモジュラー群の作用の様相は極めて複雑だが，その離散性，不連続性，安定性をはじめとする力学系理論を再構築し，タイヒミュラーモジュラー群の作用のカオス的様相とタイヒミュラー空間の非等質性をあわせて，無限次元タイヒミュラー空間および無限型リーマン面のモジュライ空間の構造理論を新しい研究対象としてとらえなおした.

Research Products

• [Journal Article] The Teichmuller space of a countable set of points on a Riemann surface (2017)
  • 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
  • Peer Reviewed / Open Access
• [Journal Article] Coherence of limit points in the fibers over the asymptotic Teichmuller space (2014)
  • Author(s): Ege Fujikawa
  • Journal Title: Computational Methods and Function Theory Volume: 14
  • Peer Reviewed
• [Journal Article] Topological characterization of the asymptotically trivial mapping class group (2014)
  • Author(s): Ege Fujikawa
  • Journal Title: Handbook of Group Actions, Advanced Lectures in Mathematics series Volume: 31
  • Peer Reviewed
• [Journal Article] Topological characterization of the asymptotically trivial mapping class group (2014)
  • Author(s): Ege Fujikawa
  • Journal Title: Handbook of Group Actions, Higher Education Press and International Press Volume: 未定
  • Peer Reviewed
• [Presentation] Coherence of limit points in the fibers over the asymptotic Teichmuller space
  • Author(s): Ege Fujikawa
  • Organizer: The 21st International Conference on Finite or Infinite Dimensional Complex Analysis and Applications
  • Place of Presentation: Nanjing University, China