Title: On Modified Ricci Flow and Modified Calabi Flow

Zhuang-dan Daniel Guan (UCR)

Abstract: In this talk we shall explain the recent approach of using modified Calabi flow for searching the Calabi extremal metrics, which is a generalization of K\"ahler-Einstein metrics. We first use an approach of modified Ricci flow for the quasi-Einstein metrics and try to prove that the modified Ricci flow as a second order heat flow converges to the quasi-Einstein metrics for compact almost homogeneous manifolds with two ends. We also explain the weakness of this approach. Then we carry out the fourth order modified Calabi flow on almost homogeneous projective manifolds with two ends and prove that the modified Calabi flow converges to the extremal metrics. We shall explain why the fourth order modified Calabi flow is more natural.