The Kähler-Ricci flow and the [symbol] operator on vector fields

Abstract

The limiting behavior of the normalized Khler-Ricci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi Kenergy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabuchi Kenergy is bounded from below and if the lowest positive eigenvalue of the operator on smooth vector fields is bounded away from 0 along the flow, then the metrics converge exponentially fast in C to a Khler-Einstein metric.