  We study some estimates along the Kahler Ricci flow on Fano manifolds. Using
these estimates, we show the convergence of Kahler Ricci flow directly if the
$\alpha$-invariant of the canonical class is greater than $\frac{n}{n+1}$.
Applying these convergence theorems, we can give a flow proof of Calabi
conjecture on such Fano manifolds. In particular, the existence of Kahler
Einstein 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 G. Tian. However, a new proof based on Kahler
Ricci flow should be still interesting in its own right.
