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. © 2020 Wiley Periodicals LLC.

中文翻译：

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Liouville定理和Kähler流形之间全纯映射的Schwarz引理

我们得出Liouville定理对L.F的多次谐波函数的一些结果。谭和作者。第一个结果提供了L.-F的复分解定理的非线性形式（它从具有简单非负二等分曲率和线性增长全同函数的简单连接的Kähler流形中等距分解了ℂ个因子）。谭和作者。第二组结果的涉及所谓ķ -hyperbolicity及其与的消极连接ķ -scalar曲率（当ķ  = 1它们是正三态截面曲率和小林双曲率的负性（最近由F. Zheng和作者在[33]中介绍）。最后，我们仅根据域和目标流形的全纯截面曲率证明了新的Schwarz-lemma型估计。©2020 Wiley Periodicals LLC。