Kahler-Einstein metrics and K-stability

Abstract

In this thesis, we study several problems related to the existence problem of Kahler-Einstein metric on Fano manifold. After introduction in the first chapter, in the second chapter, we review the basic theory both from PDE and variational point of view. Tian's program using finite dimensional approximation is then explained. Futaki invariant is discussed in detail for both its definition and calculation. K-stability is introduced following Tian and Donaldson. In the third chapter, we extend the basic theory to the twisted setting. As an important case, the analytic and algebraic theory are both extended to the conic setting. In the third chapter, we study the continuity method on toric Fano manifolds. We calculate the maximal value of parameter for solvability and study the limit behavior of the solution metrics. As a corollary, we prove Tian's partial C0-estimate on toric Fano manifolds. The log-Futaki invariant is calculated on toric Fano manifolds too. In the fourth chapter, we discuss the recent joint work with Dr. Chenyang Xu. We use Minimal Model Program (MMP) to simplify the degeneration and prove Tian's conjecture which reduce the test for K-stability to special degenerations. In the final chapter, we construct examples of rotationally symmetric solitons. These solitons are local models of special singularities of Kahler-Ricci flow.