THE KAHLER-RICCI FLOW AND THE (partial derivative)over-bar OPERATOR ON VECTOR FIELDS

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