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

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 $S^{2n-1}\subset {\mathbb{C}}^n$ share a specific property as that of the complex Wrona metric on ${\mathbb{C}}^n$.cc

22 pages, has been accepted for publication in SCIENCE CHINA Mathematics