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The uniqueness and the degeneracy problems of meromorphic maps and the construction of meromorphic maps with deficient devisors.

Research Project

Project/Area Number 14540196
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKeio University (2004)
Numazu National College of Technology (2002-2003)

Principal Investigator

ATSUJI Atsushi (2004)  Keio University, Faculty of Economics, Professor, 経済学部, 教授 (00221044)

相原 義弘 (2002-2003)  沼津工業高等専門学校, 教養科, 助教授 (60175718)

Co-Investigator(Kenkyū-buntansha) MORI Seiki  Yamagata University, Faculty of Science, Professor, 理学部, 教授 (80004456)
KITAGAWA Yosihisa  Utsunomiya University, Faculty of Education, Professor, 教育学部, 教授 (20144917)
KAMADA Hiroyuki  Miyagi University of Education, Faculty of Education, Associate Professor, 教育学部, 助教授 (00249799)
AIHARA Yosihiro  Numazu National College of Technology, Division of Liberal Arts, Professor, 教養科, 教授 (60175718)
厚地 敦  慶應義塾大学, 経済学部, 助教授 (00221044)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2002: ¥1,500,000 (Direct Cost: ¥1,500,000)
Keywordsdeficiency / meromorphic map / flat tori / Nevanlinna theory / isometric deformation / indefinite Kaehler metric / self duality / delta-subharmonic function / 有理形写像 / 値分布論 / 対数微分の補題
Research Abstract

Our project was concerned with the value distribution of meromorphic maps,
in particular, uniqueness problem, degeneration and the construction of meromorphic maps with their defect to arbitrary divisors. From 2002-2003 the project leader Aihara and Mori studied the construction of meromorphic maps with positive defect on given hypersurfaces. In particular, we showed that for an arbitrary effective divisor on complex projective spaces there exists an interval of real numbers such that we can always construct the meromorphic maps whose defect equals to each value in the interval. These researches enabled us to improve the estimate on the defect more sharply. We also determined the divisors which defects cannot be evaluated. Aihara also considered some uniqueness problem of meromorphic maps between analytic covering spaces of complex Euclidean spaces and complex projective spaces. He specially discussed which kinds of constraints to the uniqueness are caused by some geometric conditions. From these works we achieved the first aim of our project. We pushed our project forward to some related problems. In 2004, the generalization of the domain of meromorphic maps was considered mainly by the project leader Atsuji. We showed that Nevanlinna theory, the main tool of the theory of value distribution of meromorphic maps, can be established for meromorphic functions on general complete Kaehler manifolds.
We also obtained the following results from 2002-2004 by the effort of the other members.
Kitagawa considered a conjecture : the volume of the domain surrounded by a closed surface in 3-sphere is invariant under any isometric deformations of the closed surface. He obtained the affirmative answer to the conjecture in the case when the closed surface is a flat torus.
Kamada showed that Hirzebruch surfaces which admit Kaehler metrics (may be indefinite) of constant scalar curvature, must be biholomorphic to a direct product of projective lines.

Report

(4 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (17 results)

All 2005 2004 2003 Other

All Journal Article (12 results) Publications (5 results)

  • [Journal Article] Deficiencies of meromorphic mappings for hypersurfaces2005

    • Author(s)
      Yoshihiro Aihara
    • Journal Title

      Jour.of Mathematical Society of Japan 57

      Pages: 233-258

    • NAID

      10016419813

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Compact scalar-flat indefinite Kahler surfaces with Hamiltonian S1-symmetry2005

    • Author(s)
      Hiroyuki Kamada
    • Journal Title

      Commun.Math.Phys. 254

      Pages: 23-44

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Compact scalar-flat indefinite Kaehler surfaces with Hamiltonian S1-symmetry2005

    • Author(s)
      Hiroyuki Kamada
    • Journal Title

      Commun.Math.Phys. 254

      Pages: 23-44

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Deficiencies of meromorphic mappings for hypersurfaces2005

    • Author(s)
      Aihara, Yoshihiro, Mori, Seiki
    • Journal Title

      Jour. of Mathematical Society of Japan 57

      Pages: 233-258

    • NAID

      10016419813

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Compact scalar-flat indefinite Kahler surfaces with Hamiltonian S1-symmetry2005

    • Author(s)
      Kamada, Hiroyuki
    • Journal Title

      Commun.Math.Phys. 254

      Pages: 23-44

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Parabolicity, projective volume and finiteness of total curvature of minimal surfaces2004

    • Author(s)
      Atsushi Atsuji
    • Journal Title

      Kodai Math.Jour. 27

      Pages: 227-236

    • NAID

      130003574462

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Meromorphic mappings and deficiencies2004

    • Author(s)
      Seiki Mori
    • Journal Title

      Advanced Studies in Pure Mathematics, Complex Analysis in Several Variables, Mathematical Society of Japan

      Pages: 237-242

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Parabolicity, projective volume and finiteness of total curvature of minimal surfaces2004

    • Author(s)
      Atsuji, Atsushi
    • Journal Title

      Kodai Math.Jour. 27・3

      Pages: 227-236

    • NAID

      130003574462

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Meromorphic mappings and deficiencies2004

    • Author(s)
      Mori, Seiki
    • Journal Title

      Advanced Studies in Pure Mathematics, Complex Analysis in Several Variables, Mathematical Society of Japan

      Pages: 237-242

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Algebraic dependence of meromorphic mappings in value distribution theory2003

    • Author(s)
      Yoshihiro Aihara
    • Journal Title

      Nagoya Math.Jour. 269

      Pages: 145-178

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Deformable flat tori in S^3 with constant mean curvature2003

    • Author(s)
      Yoshihisa Kitagawa
    • Journal Title

      Osaka Jour.Math. 40

      Pages: 103-119

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] 3次元球面内の平坦トーラス

    • Author(s)
      北川 義久
    • Journal Title

      「数学」日本数学会 57・2(掲載予定)

    • NAID

      10016419408

    • Related Report
      2004 Annual Research Report
  • [Publications] 相原義弘: "Uniqueness problem of meromorphic mappings on analytic covering spaces"Progress in Analysis (eds.H Begehr et al.). Vol.1. 279-286 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 北川義久: Osaka Journal of Mathematics. Vol.40. 103-119 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 相原義弘: "Algebraic dependence of meromorphic mappings in value distribution theory"Nagoya Mathematical Journal. Vol.169. 145-178 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 北川義久: Osaka Journal of Mathematics. Vol.40. 103-119 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 鎌田博行: "Self-dual Kaehler metrics of nutral signature on complex surfaces"Tohoku Mathematical Publications. Vol.24. 1-94 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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