Vector field energies and critical metrics on Kähler manifolds

Abstract

Associated with a Hamiltonian holomorphic vector field on a compact Kähler manifold, a nice functional on a space of Kähler metrics will be constructed as an integration of the bilinear pairing in [FM] contracted with the Hamiltonian holomorphic vector field. As applications, we have functionals $\hat{\mu}$, $\hat{\nu}$ whose critical points are extremal Kähler metrics or "Kähler-Einstein metrics" in the sense of [M4], respectively. Finally, the same method as used by [G1] allows us to obtain, from the convexity of $\hat{\nu}$, the uniqueness of "Kähler-Einstein metrics" on nonsingular toric Fano varieties possibly with nonvanishing Futaki character.