Remarks on Kahler Ricci Flow

Abstract

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.

We note an overlap with the paper of Rubinstein [Ru1]. We add more reference

References in corpus (2)

• Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I
• On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow

Cited by in corpus (3)

• Kähler Ricci flow on Fano surfaces (I)
• Multiplier ideal sheaves and the Kähler-Ricci flow on toric Fano manifolds with large symmetry
• Multiplier ideal sheaves and geometric problems