Convergence of the Generalized Kähler-Ricci Flow

Abstract

In this paper, we consider the convergence of the generalized Kähler-Ricci flow with semi-positive twisted form $\theta $ on Kähler manifold $M$. We give detailed proofs of the uniform Sobolev inequality and some uniform estimates for the metric potential and the generalized Ricci potential along the flow. Then assuming that there exists a generalized Kähler-Einstein metric, if the twisting form $\theta $ is strictly positive at a point or $M$ admits no nontrivial Hamiltonian holomorphic vector field, we prove that the generalized Kähler-Ricci flow must converge in $C^\infty $ topology to a generalized Kähler-Einstein metric exponentially fast, where we get the exponential decay without using the Futaki invariant.

Keywords

Complex Monge-Ampère equation / Generalized Kähler-Einstein metric / Sobolev inequality / Moser-Trudinger type inequality