On the Schwarz Lemma for Complete Hermitian Manifolds

Abstract

This paper generalizes a result from Chen, Cheng and Lu [1] and improves a result from Yau [2]. Let M be a complete Hermitian manifold whose holomorphic sectional curvature is bounded from below by k₁ (or whose second Ricci curvature is bounded from below by R ^T₁ ). Let N be a Hermitian manifold whose holomorphic sectional curvature is bounded from above by k₂ < 0. We shall prove that if f: M → N is a holomorphic mapping and some conditions on the curvature and torsion of M and N are given, then

$$ \begin{gathered} f^* ds_N^2 \leqslant \,\frac{{k_1 }} {{k_2 }}ds_M^2 \hfill \\ ({\text{or}}\quad \,{\text{f}}^{\text{*}} {\text{ds}}_{\text{N}}^{\text{2}} \leqslant \,\frac{{{\text{R}}_{\text{1}}^{\text{T}} }} {{{\text{k}}_{\text{2}} }}{\text{ds}}_{\text{M}}^{\text{2}} ). \hfill \\ \end{gathered} $$