Extremal Kähler Metrics II

Abstract

Given a compact, complex manifold M with a Kähler metric, we fix the deRham cohomology class Ω of the Kahler metric, and consider the function space ℊ_Ω of all Kahler metrics in M in that class. To each (g) ∈ G_Ω we assign the non-negative real number $\Phi (g) = \int\limits_{M} {R_{g}^{2}d{V_{g}}}$ (R _g = scalar curvature, d V _g = volume element).

Aiming to find a (g) ∈ ℊ_Ω that minimizes the function Φ, we study the geometric properties in M of any (g) ∈ ℊ_Ω that is a critical point of Φ, with the following results.

1) Any metric (g) that is a critical point of Φ is necessarily invariant under a maximal compact subgroup of the identity component ℌ₀(M) of the complex Lie group of all holomorphic automorphisms of M.

2) Any critical metric (g) ∈ ℊ_Ω of Φ achieves a local minimum value of Φ in ℊ_Ω; the component of (g) in the critical set of Φ coincides with the orbit of Φ under the action of the group ℌ₀(M), it is diffeomorphic to an open euclidean ball, and the critical set is always non-degenerate in the sense of ℌ₀(M)-equivariant Morse theory.

3) If there exists a (g) ∈ ℊ_Ω with constant scalar curvature R, then it achieves an absolute minimum value of Φ; furthermore every critical metric in ℊ_Ω has constant R, and achieves the same value of Φ.

4) Whenever the existence of a critical Kahler metric (g) can be guaranteed (i.e., always, according to a conjecture ²), then Futaki’s obstruction determines a necessary and sufficient condition for the existence of a (g) ∈ ℊ_Ω with constant scalar curvature.

Research supported by NSF Grant No. MCS 81–15107 and by the Institute for Advanced Study, Princeton, NJ.