1985 | OriginalPaper | Chapter
Extremal Kähler Metrics II
Author : Eugenio Calabi
Published in: Differential Geometry and Complex Analysis
Publisher: Springer Berlin Heidelberg
Included in: Professional Book Archive
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
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 $$ \Phi (g) = \int\limits_{M} {R_{g}^{2}d{V_{g}}}$$ (Rg = 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.