(kähler-)ricci Flow on (kähler) Manifolds

One of the most interesting questions in Riemannian geometry is that of deciding whether a manifold admits curvatures of certain kinds. More specifically, one might want to know whether some given manifold admits a canonical metric, i.e. one with constant curvature of some form (sectional curvature, scalar curvature, etc.). (This will in fact have many topological implications.). One such problem is the production of Einstein metrics (metrics of constant Ricci curvature). The general procedure to find Einstein metrics on a manifold requires the solution of Einstein equations which is in general very difficult. For more on Einstein metrics and manifolds, one can check [B]. In this paper we will be interested in a specific method, Ricci flow, used in a similar problem, the production of Einstein metrics of positive scalar curvature and constant sectional curvature. Ricci flow arises naturally from the observation that Einstein metrics on a compact manifold of dimension n ≥ 3 may be viewed as the critical points of the normalized total scalar curvature functional on the space of all (Riemannnian) metrics on the manifold. See [B] for a description of this functional and Einstein metrics as its critical points. See the introduction to [Y] for an explicit derivation of the normalized Ricci flow equation via this approach. The main idea is to start with an initial metric on the given manifold and deform it along its Ricci tensor. The corresponding flow equation is: