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Zhenhan Tu

Publications and source records attributed to Zhenhan Tu.

16 recordsLinked to original sources
arXiv2019

Growth and Distortion Results for a Class of Biholomorphic Mapping and Extremal Problem with Parametric Representation in $\mathbb{C}^n$

Let $\widehat{\mathcal {S}}_g^{α, β}(\mathbb{B}^n)$ be a subclass of normalized biholomorphic mappings defined on the unit ball in $\mathbb{C}^n,$ which is closely related to the starlike mappings. Firstly, we obtain the growth theorem for $\widehat{\mathcal {S}}_g^{α, β}(\mathbb{B}^n)$. Secondly, we apply the growth theorem and a new type of the boundary Schwarz lemma to establish the distortion theorems of the Fréchet-derivative type and the Jacobi-determinant type for this subclass, and the distortion theorems with $g$-starlike mapping (resp. starlike mapping) are partly established also. At last, we study the Kirwan and Pell type results for the compact set of mappings which have $g$-parametric representation associated with a modified Roper-Suffridge extension operator, which extend some earlier related results.

math.CV↗
arXiv2019

$L^{p}$ regularity of weighted Bergman projection on Fock-Bargmann-Hartogs domain

The Fock-Bargmann-Hartogs domain $D_{n, m}(μ)$ is defined by $$ D_{n, m}(μ):=\{(z, w)\in\mathbb{C}^{n}\times\mathbb{C}^m:\Vert w \Vert^2 0.$ The Fock-Bargmann-Hartogs domain $D_{n, m}(μ)$ is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. In this paper, we first compute the weighted Bergman kernel of $D_{n, m}(μ)$ with respect to the weight $(-ρ)^α$, where $ρ(z,w):=\|w\|^2-e^{-μ\|z\|^2}$ is a defining function for $D_{n, m}(μ)$ and $α>-1$. Then, for $p\in [1,\infty),$ we show that the corresponding weighted Bergman projection $P_{D_{n, m}(μ), (-ρ)^α}$ is unbounded on $L^p(D_{n, m}(μ), (-ρ)^α)$, except for the trivial case $p=2$. In particular, this paper gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is $L^p$ irregular when $p\in [1,\infty)\setminus\{2\}$. This result turns out to be completely different from the well-known positive $L^p$ regularity result on bounded strongly pseudoconvex domain.

math.CV↗
arXiv2019

Special Toeplitz operators on a class of bounded Hartogs domains

We introduce a wider class of bounded Hartogs domains, which contains some generalizations of the classical Hartogs triangle. A sharp criteria for the $L^p-L^q$ boundedness of the Toeplitz operator with symbol $K^{-t}$ is obtained on these domains, where $K$ is the Bergman kernel on diagonal and $t\geq 0$. It generalizes the results by Chen and Beberok in the case $1<p<\infty$.

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↗
arXiv2018

Classification of proper holomorphic mappings between certain unbounded non-hyperbolic domains

The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. Recently, Tu-Wang obtained the rigidity result that proper holomorphic self-mappings of $D_{n,m}(μ)$ are automorphisms for $m\geq 2$, and found a counter-example to show that the rigidity result isn't true for $D_{n,1}(μ)$. In this article, we obtain a classification of proper holomorphic mappings between $D_{n,1}(μ)$ and $D_{N,1}(μ)$ with $N<2n$.

math.CV↗
arXiv2017

The Schwarz Lemma at the Boundary of the Symmetrized Bidisc

The symmetrized bidisc ${\textbf{G}}_{2}$ is defined by $${\textbf{G}}_{2}:=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: |z_1|<1,|z_2|<1,\; z_1,z_2\in\mathbb{C}\}.$$ It is a bounded inhomogeneous pseudoconvex domain without $\mathcal{C}^1$ boundary, and especially the symmetrized bidisc hasn't any strongly pseudoconvex boundary point and the boundary behavior of both Carathéodory and Kobayashi metrics over the symmetrized bidisc is hard to describe precisely. In this paper, we study the boundary Schwarz lemma for holomorphic self-mappings of the symmetrized bidisc ${\textbf{G}}_2$, and our boundary Schwarz lemma in the paper differs greatly from the earlier related results.

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↗
arXiv2014

Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains

The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbf{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbf{C}^n\times \mathbf{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbf{C}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock-Bargmann-Hartogs domains in terms of the polylogarithm functions and Kim-Ninh-Yamamori determined the automorphism group of the domain $D_{n,m}(μ)$. In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock-Bargmann-Hartogs domains. Our rigidity result implies that any proper holomorphic self-mapping on the Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ with $m\geq 2$ must be an automorphism.

math.CV↗
arXiv2014

Balanced metrics on some Hartogs type domains over bounded symmetric domains

The definition of balanced metrics was originally given by Donaldson in the case of a compact polarized Kähler manifold in 2001, who also established the existence of such metrics on any compact projective Kähler manifold with constant scalar curvature. Currently, the only noncompact manifolds on which balanced metrics are known to exist are homogeneous domains. The generalized Cartan-Hartogs domain $\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ)$ is defined as the Hartogs type domain constructed over the product $\prod_{j=1}^kΩ_j$ of irreducible bounded symmetric domains $Ω_j$ $(1\leq j \leq k)$, with the fiber over each point $(z_1,...,z_k)\in \prod_{j=1}^kΩ_j$ being a ball in $\mathbb{C}^{d_0}$ of the radius $\prod_{j=1}^kN_{Ω_j}(z_j,\bar{z_j})^{\frac{μ_j}{2}}$ of the product of positive powers of their generic norms. Any such domain $\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ)$ $(k\geq 2)$ is a bounded nonhomogeneous domain. The purpose of this paper is to obtain necessary and sufficient conditions for the metric $αg(μ)$ $(α>0)$ on the domain $\big(\prod_{j=1}^kΩ_j\big)^{\mathbb{B}^{d_0}}(μ)$ to be a balanced metric, where $g(μ)$ is its canonical metric. As the main contribution of this paper, we obtain the existence of balanced metrics for a class of such bounded nonhomogeneous domains.

math.CV↗
arXiv2014

Rigidity of proper holomorphic mappings between equidimensional Hua domains

Hua domain, named after Chinese mathematician Loo-Keng Hua, is defined as a domain in $\mathbb{C}^{n}$ fibered over an irreducible bounded symmetric domain $Ω\subset \mathbb{C}^{d}\;(d<n)$ with the fiber over $z\in Ω$ being a $(n-d)$-dimensional generalized complex ellipsoid $Σ(z)$. In general, a Hua domain is a nonhomogeneous domain without smooth boundary. The purpose of this paper is twofold. Firstly, we obtain what seems to be the first rigidity results on proper holomorphic mappings between two equidimensional Hua domains. Secondly, we determine the explicit form of the biholomorphisms between two equidimensional Hua domains. As a special conclusion of this paper, we completely describe the group of holomorphic automorphisms of the Hua domain.

math.CV↗
arXiv2014

Rigidity of Proper Holomorphic Self-mappings of the Pentablock

The pentablock is a Hartogs domain over the symmetrized bidisc. The domain is a bounded inhomogeneous pseudoconvex domain, and does not have a $\mathcal{C}^{1}$ boundary. Recently, Agler-Lykova-Young constructed a special subgroup of the group of holomorphic automorphisms of the pentablock, and Kosiński completely described the group of holomorphic automorphisms of the pentablock. The purpose of this paper is to prove that any proper holomorphic self-mapping of the pentablock must be an automorphism.

math.CV↗
arXiv2014

On canonical metrics on Cartan-Hartogs domains

The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan-Hartogs domain $Ω^{B^{d_0}}(μ)$ endowed with the canonical metric $g(μ)$, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space $\mathcal{H}_α$ of square integrable holomorphic functions on $(Ω^{B^{d_0}}(μ), g(μ))$ with the weight $\exp\{-αφ\}$ (where $φ$ is a globally defined Kähler potential for $g(μ)$) for $α>0$, and, furthermore, we give an explicit expression of the Rawnsley's $\varepsilon$-function expansion for $(Ω^{B^{d_0}}(μ), g(μ)).$ Secondly, using the explicit expression of the Rawnsley's $\varepsilon$-function expansion, we show that the coefficient $a_2$ of the Rawnsley's $\varepsilon$-function expansion for the Cartan-Hartogs domain $(Ω^{B^{d_0}}(μ), g(μ))$ is constant on $Ω^{B^{d_0}}(μ)$ if and only if $(Ω^{B^{d_0}}(μ), g(μ))$ is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.

math.CV↗