Characterization of Holomorphic Invariant Complex Finsler Metrics and Schwarz Lemma on the Polyball

A domain B subset of CN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pmb {B}\subset \mathbb {C}<^>N$$\end{document} is called a polyball if it is given as a direct product of open unit balls. We prove that every holomorphic invariant strongly pseudoconvex complex Finsler metric F:T1,0B ->[0,+infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F:T<^>{1,0}\pmb {B}\rightarrow [0,+\infty )$$\end{document} on a polyball B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pmb {B}$$\end{document} is necessary a K & auml;hler-Berwald metric with the holomorphic sectional curvature bounded between two negative constants, and its holomorphic bisectional curvature is nonpositive and bounded from below by a negative constant. These important curvature properties make it possible for us to establish a Schwarz lemma for holomorphic mappings f from a polyball B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pmb {B}_1$$\end{document} into another polyball B2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pmb {B}_2$$\end{document} whenever they are endowed with arbitrary holomorphic invariant K & auml;hler-Berwald metrics which are not necessary Hermitian quadratic.