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 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.

Schwarz lemma: the case of equality and an extension

“…H is the unique hyperbolic metric on S. This corollary generalizes the equality case of Theorem 2 in [18]. Theorem 1.4 may be generalized to the case of non-smooth functions and then to other forms of Schwarz lemmata in [16] .…”

“…Recently, by applying a viscosity consideration from PDE theory, Ni proves a Schwarz lemma on Kähler manifolds as follows (see also related work in [4]): any holomorphic map from a complete Kähler manifold with holomorphic sectional curvature bounded from below by K 1 ≤ 0 into a Kähler manifold with holomorphic sectional curvature bounded from above by K 2 < 0 is distance decreasing up to a constant K 1 K 2 , provided that the bisectional curvature of the domain is bounded from below. There are also several other types of Schwarz lemmata obtained in [15] and [16].…”

“…Namely, by using the known bound of df 2 and analyzing the differential inequalities, we can invoke the classical strong maximum principle of linear elliptic operators to get the global equality. Then we obtain the geometric implications (see also the appendix in [16]). A special consequence is the following.…”

Tamed exhaustion functions and Schwarz type lemmas for almost Hermitian manifolds

“…Schwarz lemma is extremely useful in complex analysis and differential geometry, and has been extended to several cases, such as holomorphic maps between higher dimensional complex manifolds (cf. [Ch,Lu,Ni], etc. ), conformal immersions and harmonic maps between Riemannian manifolds (cf.…”

Liouville Theorems for holomorphic maps on pseudo-Hermitian manifolds

“…Since then, there have been a lot of developments on the generalizations and applications of Schwarz lemma. We refer to [9] [17] [18] [19] and especially to [14] [15] for the recent advances.…”