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arXiv · 1612.07299

The K\"ahler-Ricci flow and optimal degenerations

Ruadhaí Dervan·Gábor Székelyhidi
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Abstract

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

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Ruadhaí Dervan, Gábor Székelyhidi. 2016-12-21. The K\"ahler-Ricci flow and optimal degenerations. https://arxiv.org/abs/1612.07299

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