Some parabolic and elliptic problems in complex Riemannian geometry

This dissertation consists of three parts, the first one is on the blow-up behavior of K"ahler Ricci flow on $cp^n$ blown-up at one point, and the second one on the convergence of K"ahler Ricci flow on minimal projective manifolds of general type, and the last one is on the existence of canonical conical K"ahler metrics on toric manifolds. In the first part, we consider the Ricci flow on $cp^n$ blown-up at one point starting with any rotationally symmetric K"ahler metric. We show that if the total volume does not go to zero at the singular time, then any parabolic blow-up limit of the Ricci flow along the exceptional divisor is a non-compact complete shrinking K"ahler Ricci soliton with rotational symmetry on $mathbb C^n$ blown-up at one point, hence the FIK soliton constructed in cite{FIK}. In the second part, we consider the K"ahler Ricci flow on a smooth minimal model of general type, following the ideas of Song (cite{S1,S2}), we show that if the Ricci curvature is uniformly bounded below along the K"ahler-Ricci flow, then the diameter is uniformly bounded. As a corollary we show that under the Ricci curvature lower bound assumption, the Gromov-Hausdorff limit of the flow is homeomorphic to the canonical model of the manifold. Moreover, we will give a purely analytic proof of a recent result of Tosatti-Zhang (cite{TZ}) that if the canonical line bundle $K_X$ is big and nef, but not ample, then the Ricci flow is of Type IIb. In the last part, we give criterion for the existence of toric conical K"ahler-Einstein and K"ahler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci lower bound and Bakry-Emery-Ricci lower bound. It is shown that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with conical K"ahler-Einstein metrics in the Gromov-Hausdorff topology.Ph.D.Includes bibliographical referencesby Bin Gu