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

In this paper, we extend the method in a recent paper of Tian and Zhu to study the energy level L(·) of Perelman's entropy λ(·) for the Kähler-Ricci flow on a Fano manifold M. We prove that L(·) 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.