Kähler–Ricci soliton and $H$-functional

We consider Kähler–Ricci soliton on a Fano manifold $M$. We introduce an $H$-functional on $M$; we show that its critical point has to be a Kähler–Ricci soliton and the Kähler–Ricci flow can be viewed as its reduced gradient flow. We then obtain a natural lower bound of $H$-functional in terms of an invariant of holomorphic vector fields on $M$. As an application, we prove that a Kähler–Ricci soliton, if exists, maximizes Perelman’s $\mu$-functional. Second we consider a conjecture proposed by S. K. Donaldson regarding the existence of Kähler metrics with constant scalar curvature in terms of $\mathcal{K}$-energy; a simple observation is that on Fano manifolds, one can consider Donaldson’s conjecture in terms of Ding’s $\mathcal{F}$-functional. We then state geodesic stability conjecture on Fano manifolds in terms of $\mathcal{F}$-functional. Similar pictures can be naturally extended to a Kähler–Ricci soliton and modified $\mathcal{F}$-functional.

Keywords

Kähler–Ricci soliton, $H$-functional, geodesic stability

2010 Mathematics Subject Classification

Primary 53C55. Secondary 32Q15, 58E11.

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Published 1 November 2016