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.