Rigidity of Kähler-Finsler Metrics with Constant Radial Curvature

Abstract

In this paper, we prove that a smooth strongly pseudoconvex Kähler–Finsler metric $F$ satisfying the constant radial curvature condition $Ω(χ,\barχ)χ=cF^2χ, c\in\mathbb R\setminus\{0\},$ is induced by a Hermitian Kähler metric of holomorphic sectional curvature $2c$. Under the Abate–Patrizio curvature symmetry $\langleΩ(\cdot,\barχ)χ,χ\rangle =\langleΩ(χ,\barχ)\cdot,χ\rangle,$ this result gives Hermitian rigidity for nonzero constant holomorphic sectional curvature and the classification of complete simply connected metrics of negative curvature. As an application, we show that the Kobayashi metric of a bounded strongly convex domain with smooth boundary is Kähler–Finsler if and only if the domain is biholomorphic to the unit ball. We also prove that the universal cover of a bounded strongly pseudoconvex domain $D$ with smooth boundary is biholomorphic to the unit ball if and only if its Kobayashi metric is a smooth strongly pseudoconvex Kähler–Finsler metric. Furthermore, we give two explicit examples to illustrate the distinction between the weakly Kähler–Finsler and Kähler–Finsler conditions.