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タイヒミュラー空間論の複素解析的側面の深化と多角的視点からの新展開

Research Project

Project/Area Number 23K20211
Project/Area Number (Other) 20H01800 (2020-2023)
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeMulti-year Fund (2024)
Single-year Grants (2020-2023)
Section一般
Review Section Basic Section 11020:Geometry-related
Research InstitutionKanazawa University

Principal Investigator

宮地 秀樹  金沢大学, 数物科学系, 教授 (40385480)

Co-Investigator(Kenkyū-buntansha) 志賀 啓成  京都産業大学, 理学部, 教授 (10154189)
大鹿 健一  学習院大学, 理学部, 教授 (70183225)
山田 澄生  学習院大学, 理学部, 教授 (90396416)
Project Period (FY) 2020-04-01 – 2025-03-31
Project Status Granted (Fiscal Year 2024)
Budget Amount *help
¥14,170,000 (Direct Cost: ¥10,900,000、Indirect Cost: ¥3,270,000)
Fiscal Year 2024: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2023: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2022: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2021: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2020: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Keywordsタイヒミュラー空間 / 複素多様体 / モジュライ空間 / 多重ポテンシャル論 / 写像類群 / 無限小変形 / リーマン面 / 複素フィンスラー幾何学 / 小平-スペンサー理論 / 特異平坦構造 / ヒルベルト幾何学 / 擬等角写像 / 多重ポテンシャル理論 / ポアソン積分 / 双曲幾何学 / サーストン距離 / 凸幾何学 / ローレンツ軽量 / 正則運動 / 有界正則関数 / ベルグマン核 / 多重グリーン関数 / ローレンツ計量 / クライン群 / 位相幾何学
Outline of Research at the Start

タイヒミュラー空間は曲面上の複素構造(リーマン面)の集まり(変形空間)である。タイヒミュラー空間には複素構造が自然に入り,複素Euclid空間内の有界領域として複素解析的に実現される。理論的には有界領域は非常に取り扱いがよい一方で,その構造とリーマン面自身の情報との関係を見ることが難しく,いまだに整備されていない。本研究の目的は,その関係を解明することである。

Outline of Annual Research Achievements

Teichmuller距離の複素解析的幾何学の理解の促進のため,複素Finsler幾何学で重要な概念の2次の無限小構造の研究を行なった。Teichmuller計量は接束内の稠密な開集合においては十分滑らかであることが期待されており,複素Finsler幾何学の不変量がRiemann面のモジュライの視点からどのように記述されるのかは非常に興味あるところである。今年度は接束と余接束の接空間および余接空間と余接束の接束の接束に対して,小平-Spencer理論の視点からのモデル空間を得た。また,余接単位束は面積1の特異平坦構造の空間とも同一視され,余接束の射影によりこの関係により平坦構造の単位球と接束のTeichmuller計量の単位球の幾何学の関係が観測されつつある。大鹿健一とPapadopoulosと共に,Teichmuller-Randers計量の研究を行い,トーラスの場合の結果を一般の有限型の場合に拡張した。
志賀啓成は,擬円周がDirichlet有限な調和関数への有界作用素を誘導するという事実を一般のRiemann面に拡張した。さらに双曲距離の変分問題の関連から,擬等角写像に関する極値問題の新しい手法と評価を得た。大鹿健一は,Lecuireと共同で,40年来の懸案であったThurstonのbounded image theoremの証明を与えた。Papadopoulos, Yi Huangと共同で,Teichmuller空間のEarthquake距離について,非完備性,無限小剛性を証明することができた。山田澄生は,本研究期間においては、非ユークリッド幾何学としてのヒルベルト距離関数の幾何学を重点的に進めた。Papadopoulos氏と共同研究を行い、特に分担者(山田)の一般相対論の研究との関連もあることから、Timelike geometryという分野の定式化をおこなった。

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

タイヒミュラー空間論の複素解析的側面の研究において接束及び余接束の研究は多くあるが,それらの無限小構造の研究はHubbard-Masurの研究を除いて皆無であった。今回の研究では,タイヒミュラー空間に対して,複素フィンスラー幾何学の視点からの研究において必須となる接束,余接束の接構造及び余接構造のモデル空間を決定したため,複素微分幾何学ならびに複素フィンスラー幾何学によるタイヒミュラー空間の研究が可能となり,タイヒミュラー空間論の複素解析的側面の新展開を与えるものとなり得るものであると考えられる。

Strategy for Future Research Activity

2022年度は,2021年度の反省から,昨年度から所属する大学におけるセミナーの組織員を月一回のペースで行い,国内から3名を招聘,対面による研究集会の組織委員,勉強会の会場係,及び幾何構造に関する集中講義を企画した。そして研究交流をするために海外出張(フランス)を行いコミュニケーションによる研究を再開した。今年度も同様に活発に研究交流を行う。さらに,若い研究者の育成のため勉強会なども積極的に企画する。

Report

(3 results)
  • 2022 Annual Research Report
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • Research Products

    (43 results)

All 2022 2021 2020 Other

All Int'l Joint Research (2 results) Journal Article (15 results) (of which Int'l Joint Research: 9 results,  Peer Reviewed: 15 results,  Open Access: 9 results) Presentation (24 results) (of which Int'l Joint Research: 12 results,  Invited: 24 results) Remarks (1 results) Funded Workshop (1 results)

  • [Int'l Joint Research] ストラスブール大学(フランス)

    • Related Report
      2022 Annual Research Report
  • [Int'l Joint Research] Jinling Institute of Technology(中国)

    • Related Report
      2022 Annual Research Report
  • [Journal Article] Tangent spaces of the Teichmuller space of the torus with Thurston's weak metric2022

    • Author(s)
      Miyachi Hideki、Ohshika Ken'ichi、Papadopoulos Athanase
    • Journal Title

      Annales Fennici Mathematici

      Volume: 47 Issue: 1 Pages: 325-334

    • DOI

      10.54330/afm.113702

    • Related Report
      2022 Annual Research Report 2021 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Teichmuller Spaces and the Rigidity of Mapping Class Group Actions2022

    • Author(s)
      Ohshika Ken’ichi
    • Journal Title

      Surveys in geometry

      Volume: I Pages: 389-415

    • DOI

      10.1007/978-3-030-86695-2_10

    • ISBN
      9783030866945, 9783030866952
    • Related Report
      2022 Annual Research Report
    • Peer Reviewed
  • [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] Estimates of the Bergman kernel on Teichmuller space2022

    • Author(s)
      Hu Guangming、Miyachi Hideki
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: to appear Issue: 7 Pages: 2895-2905

    • DOI

      10.1090/proc/15580

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Universal commensurability augmented Teichm?ller space and moduli space2021

    • Author(s)
      Hu Guangming、Miyachi Hideki、Qi Yi
    • Journal Title

      Annales Fennici Mathematici

      Volume: 46 Issue: 2 Pages: 897-907

    • DOI

      10.5186/aasfm.2021.4660

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Geometry of Kleinian groups and its applications2021

    • Author(s)
      Ohshika Ken’ichi
    • Journal Title

      Sugaku Expositions

      Volume: 34 Issue: 2 Pages: 123-139

    • DOI

      10.1090/suga/460

    • NAID

      130007683915

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Shapes of hyperbolic triangles and once-punctured torus groups2021

    • Author(s)
      Kim Sang-hyun、Koberda Thomas、Lee Jaejeong、Ohshika Ken’ichi、Tan Ser Peow、Gao Xinghua
    • Journal Title

      Mathematische Zeitschrift

      Volume: 未定 Issue: 3-4 Pages: 2103-2130

    • DOI

      10.1007/s00209-021-02745-3

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Discontinuous motions of limit sets2021

    • Author(s)
      Mj Mahan、Ohshika Ken’ichi
    • Journal Title

      Algebraic and Geometric Topology

      Volume: 21 Issue: 7 Pages: 3401-3444

    • DOI

      10.2140/agt.2021.21.3401

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Balancing static vacuum black holes with signed masses in four and five dimensions2021

    • Author(s)
      Khuri Marcus、Weinstein Gilbert、Yamada Sumio
    • Journal Title

      Physical Review D

      Volume: 104 Issue: 4

    • DOI

      10.1103/physrevd.104.044063

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] What is a black hole? - A geometric introduction2021

    • Author(s)
      Sumio Yamada
    • Journal Title

      Topology and Geometry

      Volume: - Pages: 579-598

    • DOI

      10.4171/irma/33

    • ISBN
      9783985470013, 9783985475018
    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Classification of non-free Kleinian groups generated by two parabolic transformations2021

    • Author(s)
      Akiyoshi Hirotaka、Ohshika Ken’ichi、Parker John、Sakuma Makoto、Yoshida Han
    • Journal Title

      Transactions of the American Mathematical Society

      Volume: 374 Issue: 3 Pages: 1765-1814

    • DOI

      10.1090/tran/8246

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Variations of complex and hyperbolic structures on Riemann surfaces – a comparative viewpoint –2021

    • Author(s)
      Sumio Yamada
    • Journal Title

      Josai Mathematical Monographs

      Volume: 13 Pages: 173-192

    • DOI

      10.20566/13447777_13_173

    • NAID

      120007027883

    • URL

      https://libir.josai.ac.jp/il/meta_pub/G0000284repository_JOS-13447777-1311

    • Related Report
      2020 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 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Geometry and Topology of Geometric Limits I2020

    • Author(s)
      Ohshika Ken’ichi、Soma Teruhiko
    • Journal Title

      In the tradition of Thurston

      Volume: 1 Pages: 291-363

    • DOI

      10.1007/978-3-030-55928-1_9

    • ISBN
      9783030559274, 9783030559281
    • Related Report
      2020 Annual Research Report
    • Peer Reviewed
  • [Journal Article] 5-dimensional space-periodic solutions of the static vacuum Einstein equations2020

    • Author(s)
      Khuri Marcus、Weinstein Gilbert、Yamada Sumio
    • Journal Title

      Journal of High Energy Physics

      Volume: 2020 Issue: 12

    • DOI

      10.1007/jhep12(2020)002

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] Toward Complex analysis on Teichmuller space2022

    • Author(s)
      宮地秀樹
    • Organizer
      Indo-Japanese on-line workshop on surface groups and geometric structures
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Toward Complex analysis on Teichmuller space2022

    • Author(s)
      宮地秀樹
    • Organizer
      One day workshop on representations of groups, Teichmuller theory, and dynamics
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The Teichmuller-Randers metric2022

    • Author(s)
      宮地秀樹
    • Organizer
      Conference on Riemann surfaces and Discrete groups
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Quasiconformal mappings and hyperbolic geometry2022

    • Author(s)
      Hiroshige Shiga
    • Organizer
      Workshop: Quasiconformal mappings, hyperbolic geometry and Riemann surfaces
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Quasiconformal mappings and quasicircles on open Riemann surfaces2022

    • Author(s)
      Hiroshige Shiga
    • Organizer
      Conference in honor of the 65th birthday of Athanase Papadopoulos
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Quasicircles and Dirichlet finite harmonic functions on open Riemann surfaces2022

    • Author(s)
      Hiroshige Shiga
    • Organizer
      The POSTECH Conference 2022 on Complex Geometry
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Thurston's bounded image theorem revisited,2022

    • Author(s)
      Ken'ichi Ohshika
    • Organizer
      Geometrie, topologie et dynamique en basses dimensions
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometry of Teichmuller space with Thurston's asymmetric metric2022

    • Author(s)
      Ken'ichi Ohshika
    • Organizer
      Conference in Honor of 65th birthday of Athanase Papadopoulos
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Teichmuller Space from a Geometric Viewpoint2022

    • Author(s)
      Ken'ichi Ohshika
    • Organizer
      CIMPA SCHOOL ON GEOMETRIC STRUCTURES ON SURFACES, MODULI SPACES AND DYNAMICS
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Harmonics maps and Einstein spacetime2022

    • Author(s)
      山田澄生
    • Organizer
      Conference in Honor of 65th birthday of Athanase Papadopoulos
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Harmonics maps in general relativity2022

    • Author(s)
      山田澄生
    • Organizer
      Yamabe Memorial Symposium
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometry of the Einstein Equation --a concise introduction-- (連続講演4回)2022

    • Author(s)
      山田澄生
    • Organizer
      Geometric Structures on Surfaces, Moduli Spaces and Dynamics
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 位相幾何学との融和によるタイヒミュラー空間上の複素解析2022

    • Author(s)
      宮地秀樹
    • Organizer
      第 17 回代数・解析・幾何学セミナー(鹿児島大 Zoom meeting)
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] タイヒミュラー空間上の極値的長さの幾何学について2022

    • Author(s)
      宮地秀樹
    • Organizer
      OCAMI共同利用・共同研究種目(B)「双曲幾何に関連する幾何学的群論 」
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] 調和写像を介したアインシュタイン定常時空の構成2022

    • Author(s)
      山田澄生
    • Organizer
      第 17 回代数・解析・幾何学セミナー(鹿児島大 Zoom meeting)
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] タイヒミュラー空間論の問題2021

    • Author(s)
      宮地秀樹
    • Organizer
      複素幾何学の諸問題 II
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Realisation of measured laminations on the boundaries of convex cores2021

    • Author(s)
      大鹿健一
    • Organizer
      離散群と双曲空間の幾何学
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] The infinitesimal geometry of Teichmuller space with the Thurston metric2021

    • Author(s)
      大鹿健一
    • Organizer
      Riemann surfaces and related topics
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Evolution of geometric ideas, and the role of Relativity2021

    • Author(s)
      Sumio Yamada
    • Organizer
      Seminar on the History of Mathematics in St Petersburg (Zoom meeting)
    • Related Report
      2021 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 Annual Research Report
    • Invited
  • [Presentation] 楕円型変分問題としてのアインシュタイン方程式とその応用2021

    • Author(s)
      山田澄生
    • Organizer
      筑波大学微分幾何学セミナー
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Projective lamination spaceの凸体実現とThurston距離の無限小剛性2020

    • Author(s)
      大鹿健一
    • Organizer
      リーマン面に関する位相幾何学
    • Related Report
      2020 Annual Research 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 Annual Research Report
    • Invited
  • [Presentation] Complex analysis on Teichmuller space2020

    • Author(s)
      Hideki Miyachi
    • Organizer
      Theorie de Teichmuller : classique, supeieure, super et quantique Teichmuller Theory: Classical, Higher, Super and Quantum
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research / Invited
  • [Remarks] 宮地秀樹

    • URL

      http://hidekimyc.html.xdomain.jp

    • Related Report
      2022 Annual Research Report
  • [Funded Workshop] International coference:One day workshop on representations of groups, Teichmuller theory, and dynamics2022

    • Related Report
      2022 Annual Research Report

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Published: 2020-04-28   Modified: 2024-08-08  

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