Perelman’s entropy and Kähler-Ricci flow on a Fano manifold

Tian, Gang; Zhang, Shijin; Zhang, Zhenlei; Zhu, Xiaohua

doi:10.1090/S0002-9947-2013-06027-8

Perelman’s entropy and Kähler-Ricci flow on a Fano manifold
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by Gang Tian, Shijin Zhang, Zhenlei Zhang and Xiaohua Zhu PDF

Trans. Amer. Math. Soc. 365 (2013), 6669-6695 Request permission

Abstract:

In this paper, we extend the method in a recent paper of Tian and Zhu to study the energy level $L(\cdot )$ of Perelman’s entropy $\lambda (\cdot )$ for the Kähler-Ricci flow on a Fano manifold $M$. We prove that $L(\cdot )$ is independent of the initial metric of the Kähler-Ricci flow under an assumption that the modified Mabuchi’s K-energy is bounded from below on $M$. As an application of the above result, we give an alternative proof to the main theorem about the convergence of Kähler-Ricci flow found in a 2007 paper by Tian and Zhu.

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