On $U(n)$-invariant strongly convex complex Finsler metrics

TL;DR

This paper characterizes $U(n)$-invariant strongly convex complex Finsler metrics, establishing conditions for convexity, relationships with Hermitian metrics, and properties of geodesics and curvature in complex geometry.

Contribution

It provides a necessary and sufficient condition for strong convexity, a rigidity theorem linking complex Finsler and Hermitian metrics, and explicit curvature formulas for $U(n)$-invariant metrics.

Findings

Characterization of strong convexity for $U(n)$-invariant complex Finsler metrics.

Rigidity theorem relating strongly convex metrics to Hermitian metrics.

Explicit formulas for holomorphic curvature and properties of geodesics.

Abstract

In this paper, we obtain a necessary and sufficient condition for a $U(n)$-invariant complex Finsler metric $F$ on domains in $\mathbb{C}^n$ to be strongly convex, which also makes it possible to investigate relationship between real and complex Finsler geometry via concrete and computable examples. We prove a rigid theorem which states that a $U(n)$-invariant strongly convex complex Finsler metric $F$ is a real Berwald metric if and only if $F$ comes from a $U(n)$-invariant Hermitian metric. We give a characterization of $U(n)$-invariant weakly complex Berwald metrics with vanishing holomorphic sectional curvature and obtain an explicit formula for holomorphic curvature of $U(n)$-invariant strongly pseudoconvex complex Finsler metric. Finally, we prove that the real geodesics of some $U(n)$-invariant complex Finsler metric restricted on the unit sphere $\pmb{S}^{2n-1}\subset\mathbb{C}^n$ share a specific property as that of the complex Wrona metric on $\mathbb{C}^n$.cc