Uniqueness of Extremal Kähler Metrics for an Integral Kähler Class

Abstract

For an integral Kähler class on a compact connected complex manifold, an extremal Kähler metric, if any, in the class is unique up to the action of Aut 0(M). This generalizes a recent result of Donaldson (see [4] for cases of metrics of constant scalar curvature) and that of Chen [3] for c1(M)≤0.

Blowing up and desingularizing constant scalar curvature Kähler manifolds

“…blowing up and desingularizing kähler manifolds 209 Using (40) and (41), and increasingĉ § if necessary, one can check that…”

Calabi energies of extremal toric surfaces

“…It was later shown [13] that any extremal Kähler metric on a compact complex manifold actually minimizes the Calabi energy (1.1) in its Kähler class. Moreover, when such a minimizer exists, it is actually unique in its Kähler class, modulo automorphisms of the complex manifold [14,20,43]. Our knowledge of existence remains imperfect, but considerable progress [2,16,21] has recently been made in the toric case that is focus of the present paper.…”

Canonical metrics on generalized Cartan-Hartogs domains

“…The existence and uniqueness of the extremal metrics in some given Kähler classes have been studied (see [7,13,18]). The important relationship between the existence of extremal metrics and various stability notions of the corresponding polarized manifolds has also been deeply investigated (see [2, 11-12, 19-22, 24]).…”