Characterization of Einstein-Fano manifolds via the K\"ahler-Ricci flow Nefton Pali 32U05complex-Monge-Ampere equations We explain a characterization of Einstein-Fano manifolds in terms of the lower bound of the density of the volume of the K\"ahler-Ricci Flow. This is a consequence of Perelman's uniform estimate for the K\"ahler-Ricci Flow and a $C^0$ estimate of Tian and Zhu. We remark also a new monotonicity of Perelman's $\mathscr{W}$ functional along the K\"ahler-Ricci Flow with respect to the Ricci potential with scale factor $\frac{1}{2}$. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3426 10.1512/iumj.2008.57.3426 en Indiana Univ. Math. J. 57 (2008) 3241 - 3274 state-of-the-art mathematics http://iumj.org/access/