A criterion for the properness of the \(K\)-energy in a general Kähler class. II (Q2828648)

In a previous paper [Math. Ann. 361, No. 1--2, 135--156 (2015; Zbl 1329.32010)], by using the \(J\)-flow introduced by \textit{S. K. Donaldson} [Asian J. Math. 3, No. 1, 1--15 (1999; Zbl 0999.53053)] and \textit{X. Chen} [Int. Math. Res. Not. 2000, No. 12, 607--623 (2000; Zbl 0980.58007)], the authors et al. gave a sufficient condition of the \(K\)-energy in a general Kähler class of a compact Kähler manifold. This paper is a continuation of this paper, namely the authors extend previous results to the case of extremal \(K\)-energy associated to extremal Kähler metrics. The proof is based on the study of the extremal \(J\)-flow, an extremal version of the J-flow defined by Donaldson and Chen. Moreover they answer a question of \textit{J. Song} and \textit{B. Weinkove} [Commun. Pure Appl. Math. 61, No. 2, 210--229 (2008; Zbl 1135.53047)] on the properness of the K-energy in any dimension.