Gang Tian

Abstract:   This is a general talk on recent progresses on the problem of the existence of Kahler-Einstein metrics. This problem was raised by E. Calabi in the 50s for compact Kahler manifolds with definite first Chern class. In 70s, the existence of Kahler-Einstein metrics was established by Yau in the case of vanishing first Chern class and independently by Aubin, Yau in the case of negative first Chern class. It has been a difficult problem to study the existence of a Kahler-Einstein metrics on Fano manifolds, i.e., those algebraic manifolds with positive first Chern class, because of known obstructions, such as, the Futaki invariant and certain geometric stability condition. There are two approaches to this existence problem: One is to show that the existence is equivalent to the K-stability, a geometric stability condition on the underlying Fano manifold, the other is to study asymptotic limiting behavior of the Kahler-Ricci flow. In this talk, I will first explain why the K-stability implies the existence of Kahler-Einstein metrics on Fano manifolds. I will discuss main techniques used in establishing the existence of Kahler-Einstein metrics. In the end, I will discuss more recent progresses on studying asymptotic behaviors of the Kahler-Ricci flow on Fano manifolds. This talk is based on my own work as well as a joint work with Zhenlei Zhang.