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

A domain \(\pmb {B}\subset \mathbb {C}^N\) $\mathit{B}\subset {\mathbb{C}}^N$ 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 \(F:T^{1,0}\pmb {B}\rightarrow [0,+\infty )\) $F:T^{1,0}\mathit{B}\to [0,+\infty )$ on a polyball \(\pmb {B}\) $\mathit{B}$ is necessary a Kähler-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 \(\pmb {B}_1\) ${\mathit{B}}_1$ into another polyball \(\pmb {B}_2\) ${\mathit{B}}_2$ whenever they are endowed with arbitrary holomorphic invariant Kähler-Berwald metrics which are not necessary Hermitian quadratic.