We prove that on Fano manifolds, the Kahler-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 Kahler-Ricci soliton, then the Kahler-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 Kahler-Einstein metric, or the initial metric of the flow was assumed to be invariant under a maximal compact group of automorphism.

中文翻译：

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Kähler-Ricci 流和最佳退化

我们证明，在 Fano 流形上，Kahler-Ricci 流会产生“最不稳定”的退化，相对于与 H 泛函相关的新稳定性概念。这回答了陈孙旺和何的问题。我们给出了这个结果的两个应用。首先，我们给出了法诺流形上 Perelman 的 {\mu}-泛函的上限值的纯代数几何公式，解决了作为特例的 Tian-Zhang-Zhang-Zhu 猜想。其次，我们用它来证明如果 Fano 流形允许 Kahler-Ricci 孤子，那么 Kahler-Ricci 流以自同构的作用为模收敛到它，具有任何初始度量。这扩展了 Tian-Zhu 和 Tian-Zhang-Zhang-Zhu 的工作，其中假设流形接纳 Kahler-Einstein 度量，