arXiv subjects

Enchao Bi

Publications and source records attributed to Enchao Bi.

6 recordsLinked to original sources

arXiv2025

Rigidity of proper holomorphic self-mappings of the hexablock

The hexablock $\mathbb{H}$, introduced by Biswas-Pal-Tomar \cite{Hexablock}, is a Hartogs domain in $\mathbb{C}^4$ fibered over the tetrablock $\mathbb{E}$ in $\mathbb{C}^3$, arising in the context of $\mu$-synthesis problems. In this paper, we prove that every proper holomorphic self-map of $\mathbb{H}$ is necessarily an automorphism. Consequently, we resolve the conjecture $G(\mathbb{H}) = \mathrm{Aut}(\mathbb{H})$ on the automorphism group structure, originally posed by Biswas-Pal-Tomar in \cite{Hexablock}.

math.CV↗

arXiv2018

Rigidity of proper holomorphic mappings between generalized Fock-Bargmann-Hartogs domains

A generalized Fock-Bargmann-Hartogs domain $D_n^{\mathbf{m},\mathbf{p}}$ is defined as a domain fibered over $\mathbb{C}^{n}$ with the fiber over $z\in \mathbb{C}^{n}$ being a generalized complex ellipsoid $Σ_z({\mathbf{m},\mathbf{p}})$. In general, a generalized Fock-Bargmann-Hartogs domain is an unbounded non-hyperbolic domains without smooth boundary. The main contribution of this paper is as follows. By using the explicit formula of Bergman kernels of the generalized Fock-Bargmann-Hartogs domains, we obtain the rigidity results of proper holomorphic mappings between two equidimensional generalized Fock-Bargmann-Hartogs domains. We therefore exhibit an example of unbounded weakly pseudoconvex domains on which the rigidity results of proper holomorphic mappings can be built.

math.CV↗

arXiv2018

The Kobayashi pseudometric for the Fock-Bargmann-Hartogs domain and its application

The Fock-Bargmann-Hartogs domain $D_{n,m}$ in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. This paper mainly consists of three parts. Firstly, we give the explicit expression of geodesics of $D_{n,1}$ in the sense of Kobayashi pseudometric; Secondly, using the formula of geodesics, we calculate explicitly the Kobayashi pseudometric on $D_{1,1}$; Lastly, we establish the Schwarz lemma at the boundary for holomorphic mappings between the nonequidimensional Fock-Bargmann-Hartogs domains by using the formula for the Kobayashi pseudometric on $D_{1,1}$.

math.CV↗

arXiv2018

Rawnsley's $\varepsilon$-function on some Hartogs type domains over bounded symmetric domains and its applications

The purpose of this paper is twofold. Firstly, we will compute the explicit expression of the Rawnsley's $\varepsilon$-function $\varepsilon_{(α,g(μ;ν))}$ of $\big(\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ),g(μ;ν)\big)$, where $g(μ;ν)$ is a Kähler metric associated with the Kähler potential $-\sum_{j=1}^kν_j\ln N_{Ω_j}(z_j,\overline{z_j})^{μ_j}-\ln(\prod_{j=1}^kN_{Ω_j}(z_j,\overline{z_j})^{μ_j}-\|w\|^2)$ on the generalized Cartan-Hartogs domain $\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ)$ and obtain necessary and sufficient conditions for $\varepsilon_{(α,g(μ;ν))}$ to become a polynomial in $1-\|\widetilde{w}\|^2$. Secondly, we study the Berezin quantization on $\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ)$ with the metric $ g(μ;ν)$.

math.CV↗

arXiv2017

Remarks on the canonical metrics on the Cartan-Hartogs domains

The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. For a Cartan-Hartogs domain $Ω^{B}(μ)$ endowed with the natural Kähler metric $g(μ),$ Zedda conjectured that the coefficient $a_2$ of the Rawnsley's $\varepsilon$-function expansion for the Cartan-Hartogs domain $(Ω^{B}(μ), g(μ))$ is constant on $Ω^{B}(μ)$ if and only if $(Ω^{B}(μ), g(μ))$ is biholomorphically isometric to the complex hyperbolic space. In this paper, following Zedda's argument, we give a geometric proof of the Zedda's conjecture by computing the curvature tensors of the Cartan-Hartogs domain $(Ω^{B}(μ), g(μ))$.

math.CV↗

arXiv2015

Balanced metrics on the Fock-Bargmann-Hartogs domains

The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2 0)$ on $D_{n,m}(μ)$, where $g(μ;ν)$ is the Kähler metric associated with the Kähler potential $Φ(z,w):=μν{\Vert z\Vert}^{2}-\ln(e^{-μ{\Vert z\Vert}^{2}}-\Vert w\Vert^2)$ ($ν>-1$) on $D_{n,m}(μ)$. The purpose of this paper is twofold. Firstly, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space of square integrable holomorphic functions on $(D_{n,m}(μ), g(μ;ν))$ with the weight $\exp\{-αΦ\}$ for $α>0$. Secondly, using the explicit expression of the Bergman kernel, we obtain the necessary and sufficient condition for the metric $αg(μ;ν)$ $(α>0)$ on the domain $D_{n,m}(μ)$ to be a balanced metric. So we obtain the existence of balanced metrics for a class of Fock-Bargmann-Hartogs domains.

math.CV↗