Let (M, J) be a compact complex manifold of Kähler type. To each Kähler metric g is associated the Calabi energy, which is defined to be the integral of the square of the scalar curvature of g, computed with respect to the volume form of g itself. This energy functional is the simplest non-trivial energy functional on the set of Kähler forms representing a fixed deRham class, and a critical metric, if any, may be regarded as a `distinguished' representative of its class. A metric that is critical for the Calabi energy among metrics whose Kähler forms represent a fixed (1,1) class is an extremal metric (in the sense of Calabi).
Let N be a compact complex manifold, and assume a compact group of automorphisms acts on N with real hypersurface orbits and disconnected exceptional set. Then N admits an extremal Kähler metric in each Kähler class; this metric is unique up to the action of the connected automorphism group. Further, if the connected automorphism group is reductive, then N admits a Kähler metric of constant (positive) scalar curvature. In fact, the set of Kähler classes containing such a metric is a real-algebraic hypersurface which separates the Kähler cone.
The main results of this paper are of interest for two reasons:
Let N be a compact complex manifold, and let K be a Kähler class. Suppose there exists an extremal metric representing the class K. Then this metric minimizes the Calabi energy on K, and the minimum value can be computed from the class K; in particular, two extremal metrics representing K have the same Calabi energy. This answers affirmatively two questions of Calabi.
The main theorem of this paper justifies the terminology `extremal Kähler metric', and shows that the critical value of the energy varies smoothly as the Kähler class varies. It also puts some results of Calabi and Futaki into a broader conceptual picture: The norm-squared of the Futaki character is exactly the amount by which the energy of an extremal metric fails to achieve the bound given by the Schwarz inequality (the latter is saturated exactly when the metric has constant scalar curvature).
Let F be a Hirzebruch surface, that is, a ruled surface over the projective line. We prove that a Kähler metric on F is locally conformally Einstein (i.e is distinguished) if and only if it is either a global minimum for the Calabi energy over the entire Kähler cone, or else it represents the degenerate Kähler class, for which the negative section has zero area. The conformality is global if and only if F is Fano. On F_2, there is only one distinguished metric (up to homothety and automorphisms). This metric is conformal, away from the negative section, to the Eguchi-Hansen metric with the opposite orientation.
The use of ODE techniques in construction of Kähler metrics goes back at least two decades, and has a number of successes, including: the Eguchi-Hansen graviton; the complete, Ricci-flat metrics constructed by Calabi on the tangent bundle of complex projective spaces; the almost-homogeneous Einstein-Kähler metrics of Sakane, Mabuchi, and Koiso-Sakane; the complete metrics of constant scalar curvature constructed by LeBrun, and by Pedersen-Poon; the extremal Kähler metrics of Calabi, Hwang, and Guan. These metrics are of interest to mathematicians, and to physicists working in cosmology (the positive mass theorem) and gauge theory (gravitational instantons). Using a well-known ansatz of Calabi with a change of coordinate inspired by the work of Koiso and Sakane, Singer and I develop a general framework for the method, including sufficient (and very likely necessary) curvature hypotheses for the ansatz to yield metrics of constant scalar curvature. Up to possible existence of `sporadic' metrics (which we believe do not exist), we find all complete Einstein-Kähler and constant scalar curvature metrics that arise from the Calabi ansatz. The examples we construct are rather more general than previously-known metrics, and include an apparently new complete, scalar-flat Kähler metric on the complex plane C2 (this metric is anti-selfdual in the sense of 4-dimensional conformal geometry), as well as new families of complete Kähler metrics of constant scalar curvature (including some complete negative Einstein-Kähler metrics) on total spaces of certain disk bundles.