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Chunping Zhong

Publications and source records attributed to Chunping Zhong.

10 recordsLinked to original sources
arXiv2026

Characterization of strongly convex K\"ahler-Berwald metrics

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 K\"ahler-Berwald metrics. In particular, we prove that $\pmb{J}$ is horizontally parallel with respect to the Cartan connection iff $F$ is a K\"ahler-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 K\"ahler-Berwald metrics with constant holomorphic sectional curvatures.

math.DG↗
arXiv2025

Curvature of left-invariant complex Finsler metric on Lie groups

Let $ G $ be a connected Lie group with real Lie algebra $ \mathfrak{g}$. Suppose $G$ is also a complex manifold. We obtain explicit holomorphic sectional and bisectional curvature formulas of left-invariant strongly pseudoconvex complex Finsler metrics $F$ on $G$ in terms of the complex Lie algebra $\mathfrak{g}^{1,0}$; we also obtain a necessary and sufficient condition for $F$ to be a K\"ahler-Finsler metric and a weakly K\"ahler-Finsler metric, respectively. As an application, we obtain the rigidity result: if $F$ is a left-invariant strongly pseudoconvex complex Finsler metric on a complex Lie group $G$, then $F$ must be a complex Berwald metric with vanishing holomorphic bisectional curvature; moreover, $F$ is a K\"ahler-Berwald metric iff $G$ is an Abelian complex Lie group.

math.DG↗
arXiv2023

Characterization of invariant complex Finsler metrics and Schwarz lemma on the classical domains

Our goal of this paper is to give a complete characterization of all holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains and establish a corresponding Schwarz lemma for holomorphic mappings with respect to these invariant metrics. We prove that every $\mbox{Aut}(\mathfrak{D})$-invariant strongly pseudoconvex complex Finsler metric $F$ on a classical domain $\mathfrak{D}$ is a K\"ahler-Berwald metric which is not necessary Hermitian quadratic, but it enjoys very similar curvature property as that of the Bergman metric on $\mathfrak{D}$. In particular, if $F$ is Hermitian quadratic, then $F$ must be a constant multiple of the Bergman metric on $\mathfrak{D}$. This actually answers the $4$-th open problem posed by Bland and Kalka (Variations of holomorphic curvature for K\"ahler Finsler metrics, American Mathematical Society, 1996).We also obtain a general Schwarz lemma for holomorphic mappings from a classical domain $\mathfrak{D}_1$ into another classical domain $\mathfrak{D}_2$ whenever $\mathfrak{D}_1$ and $\mathfrak{D}_2$ are endowed with arbitrary holomorphic invariant K\"ahler-Berwald metrics $F_1$ and $F_2$, respectively. The method used to prove the Schwarz lemma is purely geometric. Our results show that the Lu constant of $(\mathfrak{D},F)$ is both an analytic invariant and a geometric invariant. This can be better understood in the complex Finsler setting.

math.CV↗
arXiv2023

Geometry of holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains

In this paper, a class of holomorphic invariant metrics is introduced on the irreducible classical domains of type I-IV, which are strongly pseudoconvex complex Finsler metrics in the strict sense of M. Abate and G. Patrizio[2]. These metrics are of particular interest in several complex variables since they are holomorphic invariant complex Finsler metrics found so far in literature which enjoy good regularity as well as strong pseudoconvexity and can be explicitly expressed so as to admit differential geometric studies. They are, however, not necessarily Hermitian quadratic as that of the Bergman metrics. These metrics are explicitly constructed via deformation of the corresponding Bergman metric on the irreducible classical domains of type I-IV, respectively, and they are all proved to be complete Kahler-Berwald metrics. They enjoy very similar curvature properties as that of the Bergman metric on the irreducible classical domains, namely their holomorphic sectional curvatures are bounded between two negative constants, and their holomorphic bisectional curvatures are always non positive and bounded below by negative constants, respectively. From the viewpoint of complex analysis, these metrics are analogues of Bergman metrics in complex Finsler geometry which do not necessarily have Hermitian quadratic restrictions in the viewpoint of S.-S. Chern[7].

math.DG↗
arXiv2023

Schwarz lemma on polydiscs endowed with holomorphic invariant Kähler-Berwald metrics

In this paper, we obtain a Schwarz lemma for holomorphic mappings from the unit polydisc $P_m$ into the unit polydisc $P_n$, here $P_m$ and $P_n$ are endowed with $\mbox{Aut}(P_m)$-invariant Kähelr-Berwald metric $F_{t,k}$ and $\mbox{Aut}(P_n)$-invariant Kähler-Berwald metric $\tilde{F}_{\tilde{t},\tilde{k}}$ respectively. Our result generalizes the Schwarz lemma for holomorphic mappings from $P_m$ into $P_n$ whenever $P_m$ and $P_n$ are endowed with the Bergman metrics respectively. We also obtain a distortion theorem on the unit polydisc $P_m$, where $P_m$ is endowd with an $\mbox{Aut}(P_m)$-invariant Kähler-Berwald metric $F_{t,k}$, and show that for each fixed $t\in[0,+\infty)$ and integer $k\geq 2$, $F_{t,k}$ is actually a Kähler Finsler-Einstein metric in the sense of T. Aikou.

math.CV↗
arXiv2022

Holomorphic invariant strongly pseudoconvex complex Finsler metrics

Let $B_n$ and $P_n$ be the unit ball and the unit polydisk in $\mathbb{C}^n$ with $n\geq 2$ respectively. Denote $\mbox{Aut}(B_n)$ and $\mbox{Aut}(P_n)$ the holomorphic automorphism group of $B_n$ and $P_n$ respectively. In this paper, we prove that $B_n$ admits no $\mbox{Aut}(B_n)$-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Poincar$\acute{\mbox{e}}$-Bergman metric, while $P_n$ admits infinite many $\mbox{Aut}(P_n)$-invariant complete strongly convex complex Finsler metrics other than the Bergman metric. The $\mbox{Aut}(P_n)$-invariant complex Finsler metrics are explicitly constructed which depend on a real parameter $t\in [0,+\infty)$ and integer $k\geq 2$. These metrics are proved to be strongly convex Kähler-Berwald metrics, and they posses very similar properties as that of the Bergman metric on $P_n$. As applications, the existence of $\mbox{Aut}(M)$-invariant strongly convex complex Finsler metrics is also investigated on some Siegel domains of the first and the second kind which are biholomorphic equivalently to the unit polydisc in $\mathbb{C}^n$. We also give a characterization of strongly convex Kähler-Berwald spaces and give a de Rahm type decomposition theorem for strongly convex Kähler-Berwald spaces.

math.CV↗
arXiv2022

A Schwarz lemma for weakly Kähler-Finsler manifolds

In this paper, we first establish several theorems about the estimation of distance function on real and strongly convex complex Finsler manifolds and then obtain a Schwarz lemma from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold. As applications, we prove that a holomorphic mapping from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold is necessary constant under an extra condition. In particular, we prove that a holomorphic mapping from a complex Minkowski space into a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant is necessary constant.

math.DG↗
arXiv2021

Schwarz lemma from a Kähler manifold into a complex Finsler manifold

Suppose that $M$ is a Kähler manifold with a pole such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is also bounded from below. Suppose that $N$ is a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant. In this paper, we establish a Schwarz lemma for holomorphic mappings $f$ form $M$ into $N$. As applications, we obtain a Liouville type rigidity result for holomorphic mappings $f$ from $M$ into $N$, as well as a rigidity theorem for bimeromorphic mappings from a compact complex manifold into a compact complex Finsler manifold.

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↗