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2007 Fiscal Year Final Research Report Summary

Existence Problem of Fxtremal Metran and Degeneration of Balanced Metrics

Research Project

Project/Area Number 16204006
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionOsaka University

Principal Investigator

MABUCHI Toshiki  Osaka University, Graduate School of Science, Professor (80116102)

Co-Investigator(Kenkyū-buntansha) NAKAJIMA Hiraku  Kyoto University, Graduate Schcol of Science, Professor (00201666)
ONO Kaoru  Hokkaido University, Graduate School of Science, Professor (20204232)
ITOH Mitsuhiro  University of Thukuba, Graduate Schcol of Pure and Applied Sciences, Professor (40015912)
UMEHARA Masaaki  Osaka University, Graduate Schcol of Science, Professor (90193945)
GOTO Ryushi  Osaka University, Graduate School of Science, Associate Professor (30252571)
Project Period (FY) 2004 – 2007
Keywordsgeometry / extremal metric / stability / balanced metric / degeneration / constant scalar curvature / test configuration / obstruction
Research Abstract

(1) Related to the existence problem of extremal metrics, we studied various kinds of stabilities for manifolds For instance, we succeeded in showing that Chow-Mumford stability and Hilbert-Mumford stability are asymptotically equivalent (Chow-Mumford stability implies HiItert-Mumford stability by the work of Fogarty, while nothing was known even asymptotically about its converse).
(2) Associated to the Monge-Ampere equation for the Kahler-Einstein metric on an Einstein toric surface, we have a hyperbolic affine sphere equation with Dirichlet condition defined on a bounded convex C^<2> domain in R^<2>, and for the solution of the equation, we obtain a very explicit asymptotic expansion along the boundary.
(3) It is known that the anticanonical bundle of a toric or Kahler Einstein Fano manifold admits a Ricci-flat Kahler metric inducing a Sasaki-Einstein metric on a suitable quotient. We studied analogous examples for anticanonical bundles of Fano manifolds with Kahler-Ricci solitons.

  • Research Products

    (14 results)

All 2008 2007 2006 2005 2004

All Journal Article (10 results) (of which Peer Reviewed: 5 results) Presentation (4 results)

  • [Journal Article] An affine sphere equation associated to Einstein toric surfaces2008

    • Author(s)
      T. Mabuchi
    • Journal Title

      4th ICCM 2007, Hangzhou, Higher Educ. Press & Internat. Press Vol. I

      Pages: 206-215

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] An affine sphere equation associated to Einstein toric surfaces2008

    • Author(s)
      T., Mabuchi
    • Journal Title

      4th ICCM 2007, Hangzhou, Higher educ. Press & Internat I(Press)

      Pages: 206-215

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Extremal metrics and stabilities on polarized manifolds2006

    • Author(s)
      T. Mabuchi
    • Journal Title

      ICM2606, Madrid, European Math. Soc. Vol. II

      Pages: 813-826

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Extremal metrics and stabilities on polarized manifolds2006

    • Author(s)
      T., Mabuchi
    • Journal Title

      ICM2006, Madrid, European Math. Soc., 2006 Vol.II

      Pages: 813-826

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, I2005

    • Author(s)
      T. Mabuchi
    • Journal Title

      Inventiones Math. 159

      Pages: 225-243

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds2005

    • Author(s)
      T., Mabuchi
    • Journal Title

      Inventiones Math 159

      Pages: 225-243

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Stability of extremal Kaehler manifolds2004

    • Author(s)
      T. Mabuchi
    • Journal Title

      Osaka J. Math. 41

      Pages: 563-582

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Uniqueness of extremal Kaehler metrics for an integral Kaehler class2004

    • Author(s)
      T. Mabuchi
    • Journal Title

      Internat. J. Math. 15

      Pages: 531-546

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Stability of extremal Kaehler manifolds2004

    • Author(s)
      T., Mabuchi
    • Journal Title

      Osaka J. Math 41

      Pages: 563-582

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Uniqueness of extremal Kaehler metrics for an integral Kaehler class2004

    • Author(s)
      T., Mabuchi
    • Journal Title

      Internat. J. Math 15

      Pages: 531-546

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] An affine sphere equation associated to Einstein toric surfaces2007

    • Author(s)
      T. Mabuchi
    • Organizer
      4th ICCM2007, Hangzhou
    • Place of Presentation
      Zhejiang University, Hangzhou
    • Year and Date
      2007-12-19
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] An affine sphere equation associated to Einstein toric surfaces2007

    • Author(s)
      T., Mabuchi
    • Organizer
      4th ICCM2006(The Fourth International Congress of Chinese Mathematicians)
    • Place of Presentation
      Zhejiang University, Hangzhou
    • Year and Date
      2007-12-09
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Extremal metrics and stabilities on polarized manifolds2006

    • Author(s)
      T. Mabuchi
    • Organizer
      ICM2006, Madrid(2006年度世界数学者会議)
    • Place of Presentation
      Palacio Municipal de Congresos, Madrid
    • Year and Date
      2006-08-29
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Extremal metrics and stabilities on polarized manifolds2006

    • Author(s)
      T., Mabuchi
    • Organizer
      ICM2006(International Congress of Mathematicians Madrid 2006)
    • Place of Presentation
      Palacio Municipal de Congress, Madrid
    • Year and Date
      2006-08-29
    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2010-02-04  

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