The energy of a Kähler class on admissible manifolds

Abstract

On a compact complex manifold of Kahler type, the energy E(Ω) of a Kahler class Ω is given by the squared L 2-norm of the projection onto the space of holomorphic potentials of the scalar curvature of any Kahler metric representing the said class, and any one such metric whose scalar curvature has squared L 2-norm equal to E(Ω) must be an extremal representative of Ω. A strongly extremal metric is an extremal metric representing a critical point of E(Ω) when restricted to the set of Kahler classes of fixed positive top cup product. We study the existence of strongly extremal metrics and critical points of E(Ω) on certain admissible manifolds, producing a number of nontrivial examples of manifolds that carry this type of metrics, and where in many of the cases, the class that they represent is one other than the first Chern class, and some examples of manifolds where these special metrics and classes do not exist. We also provide a detailed analysis of the gradient flow of E(Ω) on admissible ruled surfaces, show that this dynamical system can be extended to one beyond the Kahler cone, and analyze the convergence of solution paths at infinity in terms of conditions on the initial data, in particular proving that for any initial data in the Kahler cone, the corresponding path is defined for all t, and converges to a unique critical class of E(Ω) as time approaches infinity.