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

• Project/Area Number: 18K18717
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 12:Analysis, applied mathematics, and related fields
• Research Institution: Kyoto 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: ¥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)
• Keywords: Holomorphic 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.

Research Products

• [Journal Article] On the quasiconformal equivalence of dynamical Cantor sets (2022)
  • Author(s): Shiga Hiroshige
  • Journal Title: Journal d'Analyse Mathematique Volume: 147 Issue: 1 Pages: 1-28
  • DOI: 10.1007/s11854-022-0214-7
  • Peer Reviewed / Open Access
• [Journal Article] A note on the extendability of holomorphic motions (2020)
  • 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
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] The quasiconformal equivalence of Riemann surfaces and the universal Schottky space (2019)
  • 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
  • Peer Reviewed
• [Journal Article] TEICHMÜLLER SPACES AND TAME QUASICONFORMAL MOTIONS (2018)
  • 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
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Presentation] Quasiconformal mappings and hyperbolic geometry (2023)
  • Author(s): 志賀 啓成
  • Organizer: Workshop: Quasiconformal mappings, hyperbolic geometry and Riemann surfaces
• [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
  • Invited
• [Presentation] Quasicircles and Dirichlet finite harmonic functions on open Riemann surfaces (2022)
  • Author(s): 志賀 啓成
  • Organizer: The POSTECH Conference 2022 on Complex Geometry
  • Int'l Joint Research / Invited
• [Presentation] Quasiconformal mappings and quasicircles on open Riemann surfaces (2022)
  • Author(s): 志賀 啓成
  • Organizer: Conference in honor of the 65th birthday of Athanase Papadopoulos
  • Int'l Joint Research / Invited
• [Presentation] Quasi-circles and Dirichlet finite harmonic functions on Riemann Surfaces (2021)
  • Author(s): 志賀啓成
  • Organizer: 静岡複素解析幾何セミナー
  • 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
  • Invited
• [Presentation] Dynamical Cantor sets and quasiconformal mappings (2020)
  • Author(s): 志賀 啓成
  • Organizer: Teichmuller Theory: Classical, Higher, Super and Quantum
  • Int'l Joint Research / Invited
• [Presentation] On dynamical Cantor sets and quasiconformal mappings (2019)
  • Author(s): 志賀 啓成
  • Organizer: First Analysis Mathematica International Conference
  • Int'l Joint Research / Invited
• [Presentation] On dynamical Cantor sets and quasiconformal mappings (2019)
  • Author(s): 志賀 啓成
  • Organizer: 函数論シンポジウム
  • 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