Remarks on Kähler Ricci Flow

We show the convergence of Kähler Ricci flow directly if the α-invariant of the canonical class is greater than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{n}{n+1}$\end{document}. Applying these convergence theorems, we can give a Kähler Ricci flow proof of Calabi conjecture on such Fano manifolds. In particular, the existence of KE metrics on a lot of Fano surfaces can be proved by flow method. Note that this geometric conclusion (based on the same assumption) was established earlier via elliptic method by Tian (Invent. Math. 89(2):225–246, 1987; Invent. Math. 101(1):101–172, 1990; Invent. Math. 130:1–39, 1997). However, a new proof based on Kähler Ricci flow should be still interesting in its own right.