Liouville Theorems and a Schwarz Lemma for Holomorphic Mappings Between Kähler Manifolds

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 ℂ isometrically from the simply connected Kähler 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 in [33] by F. Zheng and the author. We lastly prove a new Schwarz‐lemma‐type estimate in terms of only the holomorphic sectional curvatures of both domain and target manifolds .