Supremum of Perelman's entropy and K\"ahler-Ricci flow on a Fano manifold

Abstract

In this paper, we extend the method in [TZhu5] to study the energy level $L(\cdot)$ of Perelman's entropy $\lambda(\cdot)$ for K\"ahler-Ricci flow on a Fano manifold. Consequently, we first compute the supremum of $\lambda(\cdot)$ in K\"ahler class $2\pi c_1(M)$ under an assumption that the modified Mabuchi's K-energy $\mu(\cdot)$ defined in [TZhu2] is bounded from below. Secondly, we give an alternative proof to the main theorem about the convergence of K\"ahler-Ricci flow in [TZhu3].