A criterion for the properness of the $$K$$ K -energy in a general Kähler class

Abstract

In this paper, we give a criterion for the properness of the \(K\)-energy in a general Kahler class of a compact Kahler manifold by using Song–Weinkove’s result in (Commun Pure Appl Math 61(2):210–229, 2008). As applications, we give some Kahler classes on \(\mathbb {C}\mathbb {P}^2\#3\overline{\mathbb {C}\mathbb {P}^2}\) and \(\mathbb {C}\mathbb {P}^2\#8\overline{\mathbb {C}\mathbb {P}^2}\) in which the \(K\)-energy is proper. Finally, we prove Song-Weinkove’s result on the existence of critical points of \(\hat{J}\) functional by the continuity method.