Cartan-Hartogs Domain Research Articles

Weighted Composition Operators between Bers-Type Spaces on Generalized Hua–Cartan–Hartogs Domains

We address weighted composition operators between Bers-type spaces on generalized Hua–Cartan–Hartogs domains and provide the necessary and sufficient conditions for their boundedness and compactness. We then apply our results to study the boundedness and the compactness of weighted composition operators between Bers-type spaces on four different domains: generalized Hua domains, generalized Cartan–Hartogs domains, generalized Cartan–Hartogs domains over different Cartan domains and generalized ellipsoidal-type domains, by proving that the above four different domains are special cases of the generalized Hua–Cartan–Hartogs domains.

Zero problems of the Bergman kernel function on the second type of Cartan-Hartogs domain

We give the condition that the Bergman kernel function of the second type of Cartan-Hartogs domain has zeros. When the Bergman kernel function of this type of domain has zeros, the topological properties of the zero cluster are proved, and the conditions are given to judge whether a boundary point on the definition domain of the Bergman kernel function is an aggregation point of the zero cluster.

A Cartan–Hartogs version of the polydisk theorem

We extend the Polydisk Theorem for symmetric bounded domains to Cartan–Hartogs domains, and apply it to prove that a Cartan–Hartogs domain inherits totally geodesic submanifolds from the bounded symmetric domain which is based on, and to give a characterization of Cartan–Hartogs’s geodesics with linear support.

Submanifolds of some Hartogs domain and the complex Euclidean space

Two Kähler manifolds are called relatives if they admit a common Kähler submanifold with the same induced metrics. In this paper, we show that a Hartogs domain over an irreducible bounded symmetric domain equipped with the Bergman metric is not a relative to the complex Euclidean space. This generalizes the results in Cheng and Hao [On the non-existence of common submanifolds of Kähler manifolds and complex space forms. Ann Glob Anal Geom. 2021;60(1):167–180] and Cheng X, Niu Y. Submanifolds of Cartan–Hartogs domains and complex Euclidean spaces. J Math Anal Appl 2017;452(2):1262–1268.] and the novelty here is that the Bergman kernel of the Hartogs domain is not necessarily Nash algebraic.

Weighted Composition Operators from H ∞ to α , m -Bloch Space on Cartan-Hartogs Domain of the First Type

Let Y I be nonhomogeneous Cartan-Hartogs domain of the first type, ϕ a holomorphic self-map, and ψ a fixed holomorphic function on Y I . We study the weighted composition operator ψ C ϕ f = ψ f ∘ ϕ for a function f holomorphic on Y I . Our main results generalize both cases of the unit ploydisc and the unit ball obtained by Li and Stević (Li 2007 and Li 2008). Firstly, we obtain two crucial inequalities on Y I ; furthermore, the boundedness and compactness of operator ψ C ϕ from the space H ∞ of all bounded holomorphic functions to the α , m -Bloch space B α , m on Y I are investigated.

Symplectic geometry of Cartan–Hartogs domains

This paper studies the geometry of Cartan–Hartogs domains from the symplectic point of view. Inspired by duality between compact and noncompact Hermitian symmetric spaces, we construct a dual counterpart of Cartan–Hartogs domains and give explicit expression of global Darboux coordinates for both Cartan–Hartogs domains and their dual. Further, we compute their symplectic capacity and show that a Cartan–Hartogs domain admits a symplectic duality if and only if it reduces to be a complex hyperbolic space.

Zero Problems of the Bergman Kernel Function on the First Type of Cartan-Hartogs Domain

Zero Problems of the Bergman Kernel Function on the First Type of Cartan-Hartogs Domain

On the rigidity of proper holomorphic mappings for the Bergman-Hartogs domains

In this paper, we mainly discuss the proper holomorphic mappings of the Bergman-Hartogs domains raised by Roos, which can also be regarded as a natural generalization of Cartan-Hartogs domains. We show that any proper holomorphic mapping between two equidimensional Bergman-Hartogs domains over bounded circular homogeneous domains is a biholomorphism. As an application of our result, we can firstly obtain the rigidity of the proper holomorphic mappings for the Cartan-Hartogs domains. Secondly, we are able to describe the biholomorphism between two Bergman-Hartogs domains over any bounded circular homogeneous domains, and thus we can completely determine its automorphism group without fibre's restriction.

Formula omitted]-cohomology vanishing theorem on a type of generalized Cartan–Hartogs domain

Based on computing the Bergman kernel function of a type of generalized Cartan–Hartogs domain, we prove that the Kähler form of this type of domain with respect to Bergman metric is d-bounded, so the L2-cohomology vanishing theorem holds on this type of domain.

The Holomorphic Equivalence of Two Equidimensional Hartogs Domains over Bounded Symmetric Domains

This paper is concerned with the biholomorphism of two equidimensional Hartogs type domains over irreducible bounded symmetric domains $H_{\Omega}(p)$ (see (1.1)) which is a Hua construction of Cartan–Hartogs domain. We develop a new simple methods to give an sufficient and necessary condition for the two Hua domains to be biholomorphic equivalent by using the function $\mathcal{L}_{\Omega}(z,\omega)$ (see (1.2)). Furthermore as an application, we can also give an equivalent description for the automorphism of Hua domain.

A vanishing theorem on generalized Cartan-Hartogs domain of the second type

In this paper, we consider the vanishing theorem of square integrable harmonic forms with respect to the Bergman metric on the second type of generalized Cartan-Hartogs domain by using three main tools: the properties of Bergman kernel function, holomorphic invariant and holomorphic automorphism group.

Области Бергмана–Гартогса и их автоморфизмы

For Cartan–Hartogs domains and also for Bergman–Hartogs domains, the determination of their automorphism groups is given for the cases when the base is any bounded symmetric domain and a general bounded homogeneous domain respectively.

The first two coefficients of the Bergman function expansions for Cartan–Hartogs domains

Let [Formula: see text] be a globally defined real Kähler potential on a domain [Formula: see text], and [Formula: see text] be a Kähler metric on the Hartogs domain [Formula: see text] associated with the Kähler potential [Formula: see text]. First, we obtain explicit formulas of the coefficients [Formula: see text] of the Bergman function expansion for the Hartogs domain [Formula: see text] in a momentum profile [Formula: see text]. Second, using explicit expressions of [Formula: see text], we obtain necessary and sufficient conditions for the coefficients [Formula: see text] to be constants. Finally, we obtain all the invariant complete Kähler metrics on Cartan–Hartogs domains such that their the coefficients [Formula: see text] of the Bergman function expansions are constants.

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 ΩB(μ) endowed with the natural Kähler metric g(μ), Zedda conjectured that the coefficient a2 of the Rawnsley's ε-function expansion for the Cartan–Hartogs domain (ΩB(μ),g(μ)) is constant on ΩB(μ) if and only if (ΩB(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 (ΩB(μ),g(μ)).

Submanifolds of Cartan–Hartogs domains and complex Euclidean spaces

We study the non-existence of common submanifolds of a complex Euclidean space and a Cartan–Hartogs domain equipped with their canonical metrics.

Canonical metrics on generalized Cartan-Hartogs domains

In this paper, the author considers a class of bounded pseudoconvex domains, i.e., the generalized Cartan-Hartogs domains Ω(μ, m). The first result is that the natural Kahler metric gΩ(μ, m) of Ω(μ, m) is extremal if and only if its scalar curvature is a constant. The second result is that the Bergman metric, the Kahler-Einstein metric, the Caratheodary metric, and the Koboyashi metric are equivalent for Ω(μ, m).

Berezin–Engliš’ quantization of Cartan–Hartogs domains

We prove the existence of a Berezin–Engliš quantization for Cartan–Hartogs domains.

Composition Operators from p-Bloch Space to q-Bloch Space on the Fourth Cartan-Hartogs Domains

We obtain new generalized Hua’s inequality corresponding to YIV(N,n;K), where YIV(N,n;K) denotes the fourth Cartan-Hartogs domain in CN+n. Furthermore, we introduce the weighted Bloch spaces on YIV(N,n;K) and apply our inequality to study the boundedness and compactness of composition operator Cϕ from βp(YIV(N,n;K)) to βq(YIV(N,n;K)) for p≥0 and q≥0.

On the Szegö kernel of Cartan–Hartogs domains

Inspired by the work of Z. Lu and G. Tian [21] in the compact setting, in this paper we address the problem of studying the Szeg\"o kernel of the disk bundle over a noncompact K\"ahler manifold. In particular we compute the Szeg\"o kernel of the disk bundle over a Cartan-Hartogs domain based on a bounded symmetric domain. The main ingredients in our analysis are the fact that every Cartan-Hartogs domain can be viewed as an "iterated" disk bundle over its base and the the ideas given in [4] for the computation of the Szeg\"o kernel of the disk bundle over an Herimitian symmetric space of compact type.

Properties of squeezing functions and global transformations of bounded domains

The central purpose of the present paper is to study boundary behaviors of squeezing functions on some bounded domains. We prove that the squeezing function of any strongly pseudoconvex domain tends to 1 near the boundary. In fact, such an estimate is proved for the squeezing function on any bounded domain near its globally strongly convex boundary points. We also study the stability properties of squeezing functions on a sequence of bounded domains, and give some comparisons of intrinsic measures and metrics on bounded domains in terms of squeezing functions. As applications, we give new proofs of several well-known results about geometry of strongly pseudoconvex domains, and prove that all Cartan-Hartogs domains are homogenous regular. Finally, some related problems for further study are proposed.