Weighted Yosida Mappings of Several Complex Variables

Abstract

Let $M$ be a complete complex Hermitian manifold with metric $E_M$ and let $\ddot{I}†:[0,\infty )\to (0,\infty )$ be positive function such that $${\hat{I}}_r^3=\underset{r\le a<b}{\sup }\left|(\ddot{I}†(a)-\ddot{I}†(b))/(a-b)\right|\le C,r\in (0,\infty ),$$ for some $C\in (0,1],$ and ${\lim }_{r\to \infty }{\hat{I}}_r^3=0.$ A holomorphic mapping $f:{\mathbb{C}}^m\to M$ is said to be a weighted Yosida mapping if for any $z,\hat{I}¾\in {\mathbb{C}}^m$ with $\Vert \hat{I}¾\Vert =1,$ the quantity $\ddot{I}†(\Vert z\Vert )E_M(f(z),df(z)(\hat{I}¾))$ 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.

15 pages