SOME PARABOLIC AND ELLIPTIC PROBLEMS IN COMPLEX RIEMANNIAN GEOMETRY

Abstract

OF THE DISSERTATION Some parabolic and elliptic problems in complex Riemannian geometry by Bin Guo Dissertation Director: Professor Jian Song This dissertation consists of three parts, the first one is on the blow-up behavior of Kahler Ricci flow on CP blown-up at one point, and the second one on the convergence of Kahler Ricci flow on minimal projective manifolds of general type, and the last one is on the existence of canonical conical Kahler metrics on toric manifolds. In the first part, we consider the Ricci flow on CP blown-up at one point starting with any rotationally symmetric Kahler 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 Kahler Ricci soliton with rotational symmetry on Cn blown-up at one point, hence the FIK soliton constructed in [36]. In the second part, we consider the Kahler Ricci flow on a smooth minimal model of general type, following the ideas of Song ([76, 77]), we show that if the Ricci curvature is uniformly bounded below along the Kahler-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 ([102]) that if the canonical line bundle KX is big and nef, but not