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Analysis and Geometry of the Teichmuller Spaces

Research Project

Project/Area Number 13640164
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionShizuoka University

Principal Investigator

OKUMURA Yoshihide  Shizuoka University, Faculty of Science, Associate Professor, 理学部, 助教授 (90214080)

Co-Investigator(Kenkyū-buntansha) AKUTAGAWA Kazuo  Shizuoka University, Faculty of Science, Associate Professor, 理学部, 助教授 (80192920)
NAKANISHI Toshihiro  Shizuoka University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (00172354)
SATO Hiroki  Shizuoka University, Faculty of Science, Professor, 理学部, 教授 (40022222)
KUMURA Hisanori  Shizuoka University, Faculty of Science, Associate Professor, 理学部, 助教授 (30283336)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2002: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2001: ¥1,700,000 (Direct Cost: ¥1,700,000)
KeywordsTeichmuller space / Teichmuller modular group / discrete group / Riemann surface / Mobius transformation / simple dividing loop / hyperbolic manifold / real analytic manifold / 解散群
Research Abstract

My research during this term consists mainly of the following three branches :
1. Characterization of simple dividing loops on Riemann surfaces analytically.
2. Representation of the Teichmuller spaces global real analytically by angle parameters.
3. Representation of the Teichmuller modular groups (the mapping class groups) by angle parameters.
I Characterized the geometry of Mobius transformations by using the one-half powers of these transformations and these traces. Furthermore, I gave the necessary and sufficient condition of a simple loop L on a Riemann surface S to be dividing, by using the lifts of a Fuchsian group G representing S to the special linear group SL (2,C). For example, if S is a compact Riemann surface of genus p (>1), then the following is obtained :
The number of the lifts of G is 2 to the 2p-th power. Let g be an element of G corresponding to L. Then L is to be dividing if and only if for any lift of G, the matrix corresponding to g always has the negative trace.
I in … More troduced new angle parameters corresponding to the intersection angles between geodesics on the marked Riemann surface, in order to obtain global real analytic and simple representations of the Teichmuller spaces. I showed that the Teichmuller spaces are described by only angle parameters and it is easy to analyze such angle parameter spaces of the typical Teichmuller spaces of types (1,1), (2,0) and (3,0). Angle parameters correspond to the intersection angles between the axes of the generators and these products of the marked Fuchsian group. I found out the high symmetry of the arrangement of these axes. I investigated the relation among such geometry of Mobius transformations, traces and angle parameters. From these observations, the much relation and information of angle parameters were obtained.
Next, I considered the representations of the Teichmuller modular groups by only angle parameters. I especially studied the following :
I. Interpretation of the Teichmuller modular groups as the actions of some special hyperbolic polygons bounded by the axes to others.
II. Relation between angle parameters and length parameters representing these groups. Less

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (42 results)

All Other

All Publications (42 results)

  • [Publications] Yoshihide Okumura: "Lifting problem and its application to Riemann surfaces"Eighth International Conference on Complex Analysis. 173-179 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "Jorgensen groups and the Picard group"Proc. The Third ISAAC International Congress. (刊行予定). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "Jorgensen groups of parabolic type II (Countable infinite type)"RIMS kokyuroku, Kyoto Univ.. (刊行予定). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "Jorgensen groups of parabolic type I (Finite type)"RIMS kokyuroku, Kyoto Univ.. 1293. 65-77 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "The Jorgensen number of the Whitehead link"RIMS kokyuroku, Kyoto Univ.. 1270. 77-83 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "Jorgensen's inequality for classic Schottky groups of real type, II"J. Math. Soc. Japan. 53. 791-811 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "The Picard group, the figure-eight knot group and Jorgensen groups"RIMS kokyuroku, Kyoto Univ.. 1223. 37-42 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Toshihiro Nakanishi, M.Naatanen: "Areas of two-dimensional moduli spaces"Proc. Amer. Math. Soc.. 129. 3241-3252 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuo Akutagawa, Boris Botvinnik: "Yamabe metrics on cylindrical manifolds"Geom. Funct. Anal.. (刊行予定). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuo Akutagawa, Boris Botvinnik, Osamu Kobayashi, Harish Seshadri: "The Weyl functional near the Yamabe invariant"J. Geom. Anal.. 13. 1-20 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuo Akutagawa: "An obstruction to the positivity of relative Yamabe invariant"Math. Z. 243. 85-98 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuo Akutagawa, Boris Botvinnik: "Manifolds of positive scalar curvature and conformal cobordism theory"Math. Ann.. 324. 817-840 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuo Akutagawa, Boris Botvinnik: "The relative Yamabe invariant"Comm. Anal. Geom.. 10. 935-969 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hironori Kumura: "On the intrinsic ultracontractivity for compact manifolds with boundary"Kyushu J. Math.. 57. 29-50 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hironori Kumura: "A note on the absence of eigenvalues on negatively curved manifolds"Kyushu J. Math.. 56. 198-121 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Yoshihide Okumura: "Lifting problem and its application to Riemann surfaces"Eighth International Conference on Complex Analysis. 173-179 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "Jorgensen groups and the Picard group"Proc. The Third ISAAC International Congress. to appear. (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "Jorgensen groups of parabolic type II (Countable infinite type)"RIMS kokyuroku, Kyoto Univ.. to appear. (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "Jorgensen groups of parabolic type I (Finite type)"RIMS kokyuroku, Kyoto Univ.. 1293. 65-77 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "The Jorgensen number of the Whitehead link"RIMS kokyuroku, Kyoto Univ.. 1270. 77-83 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "Jorgensen's inequality for classical Schottky groups of real type, II"J. Math. Soc. Japan. 53. 791-811 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "The Picard group, the figure-eight knot group and Jorgensen groups"RIMS kokyuroku, Kyoto Univ.. 1223. 37-42 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Toshihiro Nakanishi and M. Naatanen: "Areas of two-dimensional module spaces"Proc. Amer. Math. Soc.. 129. 3241-3252 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuo Akutagawa and Boris Botvinnik: "Yamabe metrics on cylindrical manifolds"Geom. Funct. Anal.. to appear. (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuo Akutagawa, Boris Botvinnik, Osamu Kobayashi and Harish Seshadri: "The Weyl functional near the Yamabe invariant"J. Geom. Anal.. 13. 1-20 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuo Akutagawa: "An obstruction to the positivity of relative Yamabe invariants"Math. Z.. 243. 85-98 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuo Akutagawa and Boris Botvinnik: "Manifolds of positive scalar curvature and conformal cobordism theory"Math. Ann.. 324. 817-840 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kazuo Akutagawa and Boris Botvinnik: "The relative Yamabe invariant"Comm. Anal. Geom.. 10. 935-969 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hironori Kumura: "On the intrinsic ultracontractivity for compact manifolds with boundary"Kyusyu J. Math.. 57. 29-50 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hironori Kumura: "A note on the absence of eigenvalues on negatively curved manifolds"Kyushu J. Math.. 56. 109-121 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Hiroki Sato: "Jorgensen groups and the Picard group"Proc. The Third ISAAC International Congress. (刊行予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hiroki Sato: "Jorgensen groups of parabolic type I (Finite type)"RIMS kokyuroku, Kyoto Univ.. 1293. 65-77 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hiroki Sato: "The Jorgensen number of the Whitehead link"RIMS koukyoroku, Kyoto Univ.. 1270. 77-83 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kazuo Akutagawa: "Manifolds of projective scalar curvature and conformal cobordism theory"Math. Ann.. 324. 817-840 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kazuo Akutagawa: "An obstruction to the positivity of relative Yamabe invariant"Math. Z. 243. 85-98 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Hironori Kumura: "A note on the absence of eigenvzlues on negatively curved manifolds"Kyushu J. Math.. 56. 109-121 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Yoshihide Okumura: "Lifting problem and its application to Riemann surfaces"Eighth International Conference on Complex Analysis. 173-179 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 奥村 善英: "リーマン面上の単純閉曲線が分割曲線になるための必要十分条件"京都大学数理解析研究所講究録. 1163. 28-41 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hiroki Sato: "Jorgensen's inequality for classical Schottky groups of real type, II"J. Math. Soc. Japan. 53. 791-811 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Marjatta Naatanen: "Areas of two-dimensional moduli spaces"Proc. Amer. Math. Soc.. 129. 3241-3252 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kazuo Akutagawa: "An obstruction to the positivity of relative Yamabe invariants"Math. Z.. (印刷中). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hironori Kumura: "Nash inequalities for compact manifolds with boundary"Kodai Math. J.. 24. 352-378 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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