Liouville theorems and a Schwarz Lemma for holomorphic mappings between Kähler manifolds

Abstract

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 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 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}.

A correction was made

References in corpus (2)

• Negative Holomorphic curvature and positive canonical bundle
• An extension of a theorem of Wu-Yau

Cited by in corpus (4)

• RC-positive metrics on rationally connected manifolds
• General Schwarz Lemmata and their applications
• Integral inequalities for holomorphic maps and applications
• RC-positivity and rigidity of harmonic maps into Riemannian manifolds