Curvature Estimates for Geometric Deformations in Kähler Geometry

Abstract

We study curvature properties of two geometric deformation equations in Kahler geometry: the Kahler-Ricci flow and the continuity method. These equations are central to the analytic minimal model program proposed by J. Song and G. Tian. This thesis contains three main works. In the first work, we confirm a conjecture by V. Tosatti and show that the singularity type of long-time solutions to the Kahler Ricci flow does not depend on the initial metric. This extends a previous work by Y. Zhang to the general numerically effective case. The second work pertains to the continuity method, in which we show that curvature bounds for one long-time solution can be transferred to curvature bounds on a second long-time solution to the continuity method. Finally, the third work analyses the curvature blow-up of the continuity method on a manifold exhibiting Calabi symmetry. These are important examples of the less understood finite time singularities. Furthermore, we show that finite-time solutions to the continuity method must have unbounded scalar curvature, which parallels the same results for the Kahler-Ricci flow by Z Zhang.