ICM邀请报告——K?hler-Ricci flow on Fano manifolds
报告人：Xiaohua Zhu (Peking University)
时间：2022-07-07 20:15-21:00
地点：Room 1560, Sciences Building No. 1
Abstract: The K?hler-Ricci flow is simply the Ricci flow restricted to K?hler metrics on a K?hler manifold M. If M is a Fano manifold, we usually consider the following normalized flow,
(0.1)
?ω(t) / ?t = ?Ric(ω(t)) + ω(t), ω(0) = ω_{0},
where ω(t) denote the solutions of K?hler-Ricci flow with initial metric ω_{0} in 2πc_{1}(M). Then the flow preserves the K?hler class, i.e., [ω(t)] = 2πc_{1}(M) for all t. In particular, the flow preserves the volume of ω(t). It is well-know that the solutions of (0.1) exist for any times t > 0 and their smooth limits (if exists) are K?hler-Ricci solitons. Because of obstructions, a Fano manifold may not admit any K?hler-Ricci soliton in general. Thus, the flow (0.1) may develop singularity. It makes the investigation more complicated, when studying the limit behavior of the flow. In this talk, we will introduce some basic tools as well as some recent developments of the K?hler-Ricci flow, including Perelman’s fundamental estimates in K?hler-Ricci flow, the smooth convergence of K?hler-Ricci flow, the progress on Hamilton-Tian conjecture and the K?hler-Ricci flow on G-manifolds with singular limits.
