Abstract
In this paper, we consider the convergence of the generalized Kähler-Ricci flow with semi-positive twisted form \(\theta \) on Kähler manifold \(M\). We give detailed proofs of the uniform Sobolev inequality and some uniform estimates for the metric potential and the generalized Ricci potential along the flow. Then assuming that there exists a generalized Kähler-Einstein metric, if the twisting form \(\theta \) is strictly positive at a point or \(M\) admits no nontrivial Hamiltonian holomorphic vector field, we prove that the generalized Kähler-Ricci flow must converge in \(C^\infty \) topology to a generalized Kähler-Einstein metric exponentially fast, where we get the exponential decay without using the Futaki invariant.
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Acknowledgments
Both authors would like to thank professor X. Zhang for his useful discussion. We are also grateful to the referee for his or her careful reading and valuable suggestions. In particular, the referee points out that the exponential decay can be deduced on the basis of our arguments. The work was supported in part by NSF in China, No.11131007, the Hundred Talents Program of CAS and Zhejiang Provincial Natural Science Foundation of China, No.LY12A01028.
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Liu, J., Wang, Y. Convergence of the Generalized Kähler-Ricci Flow. Commun. Math. Stat. 3, 239–261 (2015). https://doi.org/10.1007/s40304-015-0058-x
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DOI: https://doi.org/10.1007/s40304-015-0058-x
Keywords
- Complex Monge-Ampère equation
- Generalized Kähler-Einstein metric
- Sobolev inequality
- Moser-Trudinger type inequality