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Hongchuan Xia

Publications and source records attributed to Hongchuan Xia.

3 recordsLinked to original sources
arXiv2023

Characterizations of complex Finsler Metrics

Munteanu defined the canonical connection associated to a strongly pseudoconvex complex Finsler manifold $(M,F)$. We first prove that the holomorphic sectional curvature tensors of the canonical connection coincide with those of the Chern-Finsler connection associated to $F$ if and only if $F$ is a Kähler-Finsler metric. We also investigate the relationship of the Ricci curvatures (resp. scalar curvatures) of these two connections when $M$ is compact. As an application, two characterizations of balanced complex Finsler metrics are given. Next, we obtain a sufficient and necessary condition for a balanced complex Finsler metric to be Kähler-Finsler. Finally, we investigate conformal transformations of a balanced complex Finsler metric.

math.DG↗
arXiv2020

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

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

math.DG↗