The Journal of Geometric Analysis

, Volume 10, Issue 4, pp 759–774 | Cite as

Kähler-Ricci soliton typed equations on compact complex manifolds withC 1(M) > 0

Article

Abstract

As a generalization of Calabi’s conjecture for Kähler-Ricci forms, which was solved by Yau in 1977, we discuss the existence of Kähler-Ricci soliton typed equation on a compact Kähler manifold (M, g) with positive first Chem C1 (M) > 0 as well as the uniqueness. For a given positively definite (1,1)-form Ω ∈ C1 (M) of M and a holomorphic vector field X on M, we prove that there is a Kähler form ω in the Kähler class [ωg] solving the Kähler-Ricci soliton typed equation if and only if, i) X is belonged to a reductive subalgebra of holomorphic vector fields and the imaginary part of X generates a compact one-parameter transformations subgroup of M; and ii) LX Ω is a real-valued (1,1)-form. Moreover, the solution ω is unique in the class [ωg].

Math Subject Classifications

53C25 32J15 53C55 58E11 

Key Words and Phrases

Kähler-Ricci solilon typed equations complex Monge-Ampére equations holomorphic vector field 

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References

  1. [1]
    Aubin, T. Réduction du cas positif de l’équation de Monge-Ampére sur les variétés Kählerinnes compactes à la démonstration d’un intégalité,J. Fund. Anal.,57, 143–153, (1984).CrossRefMathSciNetMATHGoogle Scholar
  2. [2]
    Bando, S. and Mabuchi, T. Uniqueness of Kähler-Einstein metrics modula connected group actions, Algebraic Geometry,Adv. Studies in Pure Math.,10, Sendai, (1987).Google Scholar
  3. [3]
    Cao. H.D. Existence of gradient Kähler-Ricci solitons. Elliptic and parabolic methods in geometry, Peters, A.K., Chow, B., Gulliver, R., Levy, S., and Sullivan, J., Eds., 1–16, 1994.Google Scholar
  4. [4]
    Cao, H.D. Limits of solutions of the Kähler-Ricci flow,J. Differ. Geom.,45, 257–272, (1997).MATHGoogle Scholar
  5. [5]
    Ding, W. and Tian, G. Kähler-Einstein metrics and the generalized Futaki invariants,Invent. Math.,110, 315–335, (1992).CrossRefMathSciNetMATHGoogle Scholar
  6. [6]
    Futaki, A. An obstruction to the existence of Kähler-Einstein metrics,Invent. Math.,73, 437–443, (1983).CrossRefMathSciNetMATHGoogle Scholar
  7. [7]
    Futaki, A. Kähler-Einstein metrics and integral invariants,Lect. Notes in Math.,1314, Springer-Verlag, Berlin, (1988).MATHGoogle Scholar
  8. [8]
    Futaki, A. and Mabuchi, T. Bilinear forms and extremal Kähler vector fields associated with Kähler classes,Math. Ann.,301, 199–210, (1995).CrossRefMathSciNetMATHGoogle Scholar
  9. [9]
    Hamilton, R.S. Eternal solutions to the Ricci-flow,J. Differ. Geom.,38, 1–11, (1993).MathSciNetMATHGoogle Scholar
  10. [10]
    Koiso, N. On rationally symmetric Hamilton’s equation for Kähler-Einstein metrics, Algebraic Geometry,Adv. Studies in Pure Math.,18-1, Sendai, (1990).Google Scholar
  11. [11]
    Siu, Y.T. The existence of Kähler-Einstein metrics on manifolds with positive anticanonnical line bundle and a suitable symmetry group,Ann. Math.,127, 585–627, (1988).CrossRefMathSciNetGoogle Scholar
  12. [12]
    Tian, G. On Calabi’s conjecture for complex surfaces with positive Chern class,Invent. Math.,101, 101–172, (1990).CrossRefMathSciNetMATHGoogle Scholar
  13. [13]
    Tian, G. Kähler-Einstein metrics on algebraic manifolds,Lect. Notes in Math.,1646, Springer-Verlag, Berlin, (1996).Google Scholar
  14. [14]
    Tian, G. Kähler-Einstein metrics with positive scalar curvature,Invent. Math.,130, 1–39, (1997).CrossRefMathSciNetMATHGoogle Scholar
  15. [15]
    Tian, G. and Zhu, X.H. Uniqueness of Kähler-Ricci solitons on compact complex manifolds withC 1 (M) > 0, to appear inActa Math. Google Scholar
  16. [16]
    Yau, S.T. On the Ricci curvature of a compact Kähler manifold and the Monge-Ampére equation,I *,Comm. Pure Appl. Math.,31, 339–441, (1978).CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2000

Authors and Affiliations

  1. 1.Department of MathematicsPeking University BeijingP.R. China

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