$\cF$-functional and geodesic stability

Abstract

We consider canonical metrics on Fano manifolds. First we introduce a norm-type functional on Fano manifolds, which has Kahler-Einstein or Kahler-Ricci soliton as its critical point and the Kahler-Ricci flow can be viewed as its (reduced) gradient flow. We then obtain a natural lower bound of this functional. As an application, we prove that Kahler-Ricci soliton, if exists, maximizes Perelman's $\mu$-functional without extra assumptions. Second we consider a conjecture proposed by S.K. Donaldson in terms of $\cK$-energy. Our simple observation is that $\cF$-functional, as $\cK$-energy, also integrates Futaki invariant. We then restate geodesic stability conjecture on Fano manifolds in terms of $\cF$-functional. Similar pictures can also be extended to Kahler-Ricci soliton and modified $\cF$-functional.

Comments are welcome; published version by Asian J. of Math

References in corpus (4)

• K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
• Space of Kähler metrics (IV)--On the lower bound of the K-energy
• Probability measures related to geodesics in the space of Kähler metrics
• Supremum of Perelman's entropy and Kähler-Ricci flow on a Fano manifold

Cited by in corpus (6)

• K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
• Kähler-Einstein metrics along the smooth continuity method
• Geodesic Rays and Kähler-Ricci Trajectories on Fano Manifolds
• On the space of Kahler potentials
• Boltzmann's Entropy and Kähler-Ricci Solitons
• On the Spectrum of weighted Laplacian operator and its application to uniqueness of Kähler Einstein metrics