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.