Weighted Yosida Mappings of Several Complex Variables

TL;DR

This paper introduces weighted Yosida mappings for holomorphic functions from complex Euclidean spaces to complete Hermitian manifolds, providing criteria for identifying such mappings based on boundedness conditions involving the metric and a weight function.

Contribution

It defines weighted Yosida mappings in several complex variables and establishes new criteria to characterize these mappings, extending the classical concept to a broader geometric setting.

Findings

Criteria for holomorphic mappings to be weighted Yosida mappings.

Conditions involving the weight function and metric boundedness.

Extension of Yosida mappings concept to complex manifolds.

Abstract

Let $M$ be a complete complex Hermitian manifold with metric $E_{M}$ and let $\varphi: [0,\infty)\rightarrow (0,\infty)$ be positive function such that $$\gamma_r=\sup\limits_{r\leq a<b}\left|(\varphi(a)-\varphi(b))/(a-b)\right|\leq C,~r\in (0,\infty),$$ for some $C\in (0,1],$ and $\lim_{r\rightarrow\infty}\gamma_r=0.$ A holomorphic mapping $f:\mathbb{C}^{m}\rightarrow M$ is said to be a weighted Yosida mapping if for any $z,~\xi\in\mathbb{C}^{m}$ with $\|\xi\|=1,$ the quantity $\varphi(\|z\|)E_{M}(f(z), df(z)(\xi))$ remains bounded above, where $df(z)$ is the map from $T_z(\mathbb{C}^{m})$ to $T_{f(z)}(M)$ induced by $f.$ We present several criteria of holomorphic mappings belonging to the class of all weighted Yosida mappings.