  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
