On the constant scalar curvature Kähler metrics (II)—Existence results

In this paper, we apply our previous estimates in Chen and Cheng [On the constant scalar curvature Kähler metrics (I): a priori estimates, Preprint] to study the existence of cscK metrics on compact Kähler manifolds. First we prove that the properness of K K -energy in terms of L 1 L^1 geodesic distance d 1 d_1 in the space of Kähler potentials implies the existence of cscK metrics. We also show that the weak minimizers of the K K -energy in ( E 1 , d 1 ) (\mathcal {E}^1, d_1) are smooth cscK potentials. Finally we show that the non-existence of cscK metric implies the existence of a destabilized L 1 L^1 geodesic ray where the K K -energy is non-increasing, which is a weak version of a conjecture by Donaldson. The continuity path proposed by Xiuxiong Chen [Ann. Math. Qué. 42 (2018), pp. 69–189] is instrumental in the above proofs.