The Kähler–Ricci flow and optimal degenerations

Abstract

We prove that on Fano manifolds, the Kähler–Ricci flow produces a “most destabilising” degeneration, with respect to a new stability notion related to the $H$-functional. This answers questions of Chen–Sun–Wang and He.

We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman’s $\mu$‑functional on Fano manifolds, resolving a conjecture of Tian–Zhang-Zhang-Zhu as a special case. Secondly, we use this to prove that if a Fano manifold admits a Kähler–Ricci soliton, then the Kähler–Ricci flow converges to it modulo the action of automorphisms, with any initial metric. This extends work of Tian–Zhu and Tian–Zhang–Zhang–Zhu, where either the manifold was assumed to admit a Kähler–Einstein metric, or the initial metric of the flow was assumed to be invariant under a maximal compact group of automorphism.