The Journal of Geometric Analysis

, Volume 10, Issue 4, pp 759–774

Kähler-Ricci soliton typed equations on compact complex manifolds withC1(M) > 0



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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Copyright information

© Mathematica Josephina, Inc. 2000

Authors and Affiliations

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

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