Kahler-Ricci flow, Kahler-Einstein metric, and K-stability

We prove the existence of a Kahler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kahler-Ricci flows. The key ingredient is an algebrogeometric description of the asymptotic behavior of Kahler-Ricci flow on Fano manifolds. This is in turn based on a general finite-dimensional discussion, which is interesting on its own and could potentially apply to other problems. As one application, we relate the asymptotics of the Calabi flow on a polarized Kahler manifold to K-stability, assuming bounds on geometry.