The kähler-ricci flow and the ∂^¯ operator on vector fields

The limiting behavior of the normalized Kähler-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 ∂^¯†∂^¯ operator on smooth vector fields is bounded away from 0 along the flow, then the metrics converge exponentially fast in C∞ to a Kähler-Einstein metric.