ICM邀请报告——Kähler-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 Kähler-Ricci flow is simply the Ricci flow restricted to Kähler metrics on a Kähler 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 Kähler-Ricci flow with initial metric ω_{0} in 2πc_{1}(M). Then the flow preserves the Kähler 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 Kähler-Ricci solitons. Because of obstructions, a Fano manifold may not admit any Kähler-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 Kähler-Ricci flow, including Perelman’s fundamental estimates in Kähler-Ricci flow, the smooth convergence of Kähler-Ricci flow, the progress on Hamilton-Tian conjecture and the Kähler-Ricci flow on G-manifolds with singular limits.
