3 on Asymptotics for the Mabuchi Energy Functional

Many canonical geometric structures have been found to be closely related to stability in the sense of geometric invariant theory. For the existence of Kähler-Einstein metrics, this is the classical conjecture of Yau [14, 15, 16]. The necessity of several notions of stability has been established in this case by Tian [12] and by Donaldson [3]. In the variational approach, the existence of Kähler-Einstein metrics can be reduced to the behavior of energy functionals [12]. Of particular importance is the Futaki energy functional F 0 ω0(φ) [4, 2] F 0 ω0(φ) = − 1 (n + 1)V ∫