Extremal Kähler Metrics II

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 ℊΩ of all Kahler metrics in M in that class. To each (g) ∈ GΩ 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) ∈ ℊΩ that minimizes the function Φ, we study the geometric properties in M of any (g) ∈ ℊΩ 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) ∈ ℊΩ of Φ achieves a local minimum value of Φ in ℊΩ; 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) ∈ ℊΩ with constant scalar curvature R, then it achieves an absolute minimum value of Φ; furthermore every critical metric in ℊΩ 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) ∈ ℊΩ with constant scalar curvature.