Non-hyperbolic Domains Research Articles

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

The Fock-Bargmann-Hartogs domain Dn,m(μ)(μ>0) in Cn+m is defined by the inequality ‖w‖2<e−μ‖z‖2, where (z,w)∈Cn×Cm, which is an unbounded non-hyperbolic domain in Cn+m. In this article, we prove that any proper holomorphic mapping from Dn,m(μ) to Dn,m+1(μ)(m≥3) is necessarily the standard linear embedding up to their automorphisms and therefore is a totally geodesic isometric embedding with respect to their Bergman metrics.

Generating Shilnikov chaos in 3D piecewise linear systems

We propose a construction algorithm that takes advantage of the geometric features of linear eigenspaces with focus-saddle and center-node equilibrium points to construct piecewise linear systems with Shilnikov chaos. Unlike previously published algorithms to generate chaos, we construct homoclinic (heteroclinic) orbits combining hyperbolic and nonhyperbolic domains by appropriately choosing locations where the trajectories move between linear domains through the switching surfaces at their crossing sections. We present two variants of our construction algorithm: On the first, a homoclinic orbit is constructed locating the exit and return points on the crossing section of the switching surface are connected by a nonhyperbolic center-node domain. In the second variant, we extend the construction method to construct heteroclinic loops with two switching surfaces between them. For the resulting systems a wide range of parameter values results on Shilnikov chaos. These chaotic systems are illustrated with numerical simulations.

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 $D_{1,1}$.

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 $\Sigma_z({\mathbf{m},\mathbf{p}})$. 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.

Remarks on Rawnsley’s $$\varvec{\varepsilon }$$ε-function on the Fock–Bargmann–Hartogs domains

In this paper, we mainly study a family of unbounded non-hyperbolic domains in $$\mathbb {C}^{n+m}$$, called Fock–Bargmann–Hartogs domains $$D_{n,m}(\mu )$$ ($$\mu >0$$) which are defined as a Hartogs type domains with the fiber over each $$z\in \mathbb {C}^{n}$$ being a ball of radius $$e^{-\frac{\mu }{2} {\Vert z\Vert }^{2}}$$. The purpose of this paper is twofold. Firstly, we obtain necessary and sufficient conditions for Rawnsley’s $$\varepsilon $$-function $$\varepsilon _{(\alpha ,g)}(\widetilde{w})$$ of $$\big (D_{n,m}(\mu ), g(\mu ;\nu )\big )$$ to be a polynomial in $$\Vert \widetilde{w}\Vert ^2$$, where $$g(\mu ;\nu )$$ is a Kahler metric associated with the Kahler potential $$\nu \mu {\Vert z\Vert }^{2} -\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$. Secondly, using above results, we study the Berezin quantization on $$D_{n,m}(\mu )$$ with the metric $$\beta g(\mu ;\nu )$$$$(\beta >0)$$.

English

We consider a certain class of unbounded nonhyperbolic Reinhardt domains which is called the twisted Fock-Bargmann-Hartogs domains. By showing Cartan’s linearity theorem for our unbounded nonhyperbolic domains, we give a complete description of the automorphism groups of twisted Fock-Bargmann-Hartogs domains.

Classification of Proper Holomorphic Mappings Between Certain Unbounded Non-hyperbolic Domains

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

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

The Fock–Bargmann–Hartogs domain Dn,m(μ) (μ>0) in Cn+m is defined by the inequality ‖w‖2<e−μ‖z‖2, where (z,w)∈Cn×Cm, which is an unbounded non-hyperbolic domain in Cn+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 Dn,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 Dn,m(μ) with m≥2 must be an automorphism.

The automorphism group of a certain unbounded non-hyperbolic domain

In this paper we determine the automorphism group of the Fock–Bargmann–Hartogs domain Dn,m in Cn×Cm which is defined by the inequality ‖ζ‖2<e−μ‖z‖2.

A VERY SIMPLE CHARACTERIZATION OF GROMOV HYPERBOLICITY FOR A SPECIAL KIND OF DENJOY DOMAINS

In this paper we provide characterizations for the Gromov hyperbolicity of some particular Denjoy domains and besides some sufficient conditions to guarantee or discard the hyperbolicity of some others. The conditions obtained involve just the lengths of some special simple closed geodesics in the domain. These results, on the one hand, show the extraordinary complexity of determining the hyperbolicity of a domain and, on the other hand, allow us to construct easily a large variety of both hyperbolic and non-hyperbolic domains.