Characterization of strongly convex Kähler-Berwald metrics

TL;DR

The paper characterizes strongly convex complex Finsler metrics that are Kähler-Berwald by examining the parallelism of the canonical complex structure on $T^{1,0}M$ with the Cartan connection. It proves that horizontal parallelism $\nabla J=0$ characterizes metrics where the Cartan connection coincides with the Chern-Finsler connection, and shows that this condition, together with constant holomorphic sectional curvature, forces the metric to be Kähler-Einstein with explicit global models in negative, zero, and positive curvature cases. A rigidity theorem follows: complete, simply connected spaces with $\nabla J=0$ and constant curvature are holomorphically isometric to the Bergman ball, a complex Minkowski space, or $\mathbb{CP}^n$ with scaled standard metrics, respectively. The work thus extends classical Hermitian geometry insights to complex Finsler geometry, highlighting Kähler-Berwald metrics as the natural rigidity class in this setting.

Abstract

Let $F: T^{1,0}M\rightarrow[0,+\infty)$ be a strongly convex complex Finsler metric on a complex manifold $M$ and $\pmb{J}$ the canonical complex structure on the complex manifold $T^{1,0}M$. We give a geometric characterization of strongly convex Kähler-Berwald metrics. In particular, we prove that $\pmb{J}$ is horizontally parallel with respect to the Cartan connection iff $F$ is a Kähler-Berwald metric. We also prove that the Cartan connection and the Chern-Finsler connection associated to $F$ coincide iff $\pmb{J}$ is both horizontal and vertical parallel with respect to the Cartan connection. Based on these results, we give a rigidity theorem of strongly convex Kähler-Berwald metrics with constant holomorphic sectional curvatures.

Theorems & Definitions (64)

• Definition 1.1
• Proposition 1.2: Mok
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Definition 2.2
• Theorem 2.3