Curvature of left-invariant complex Finsler metric on Lie groups

Abstract

Let $ G $ be a connected Lie group with real Lie algebra $ \mathfrak{g}$. Suppose $G$ is also a complex manifold. We obtain explicit holomorphic sectional and bisectional curvature formulas of left-invariant strongly pseudoconvex complex Finsler metrics $F$ on $G$ in terms of the complex Lie algebra $\mathfrak{g}^{1,0}$; we also obtain a necessary and sufficient condition for $F$ to be a K\"ahler-Finsler metric and a weakly K\"ahler-Finsler metric, respectively. As an application, we obtain the rigidity result: if $F$ is a left-invariant strongly pseudoconvex complex Finsler metric on a complex Lie group $G$, then $F$ must be a complex Berwald metric with vanishing holomorphic bisectional curvature; moreover, $F$ is a K\"ahler-Berwald metric iff $G$ is an Abelian complex Lie group.