The limiting behavior of the normalized KÄahler-Ricci °ow for manifolds with positive ¯rst 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 ¹@y ¹@ operator on smooth vector ¯elds is bounded away from0 along the °ow, then the metrics converge exponentially fast in
C1 to a KÄahler-Einstein metric.