  We derive some consequences of the Liouville theorem for plurisubharmonic
functions of L.-F. Tam and the author. The first result provides a nonlinear
version of the complex splitting theorem (which splits off a factor of
$\mathbb{C}$ isometrically from the simply-connected K\"ahler manifold with
nonnegative bisectional curvature and a linear growth holomorphic function) of
L.-F. Tam and the author. The second set of results concerns the so-called
$k$-hyperbolicity and its connection with the negativity of the $k$-scalar
curvature (when $k=1$ they are the negativity of holomorphic sectional
curvature and Kobayashi hyperbolicity) introduced recently by F. Zheng and the
author. We lastly prove a new Schwarz Lemma type estimate in terms of {\it only
the holomorphic sectional curvatures of both domain and target manifolds}.
