Non-hyperbolic unbounded Reinhardt domains: non-compact automorphism group, Cartan's linearity theorem and explicit Bergman kernel

Abstract

In the study of the holomorphic automorphism groups, many researches have been carried out inside the category of bounded or hyperbolic domains. On the contrary to these cases, for unbounded non-hyperbolic cases, only a few results are known about the structure of the holomorphic automorphism groups. Main result of the present paper gives a class of unbounded non-hyperbolic Reinhardt domains with non-compact automorphism groups, Cartan's linearity theorem and explicit Bergman kernels. Moreover, a reformulation of Cartan's linearity theorem for finite volume Reinhardt domains is also given. Introduction.In several complex variables, complex domains with properties boundedness or hyperbolicity are fundamental research objects. On the other hand, one may not expect some useful function-theoretic properties for complex domains without the two properties. Our primary motivation is to investigate some useful properties for unbounded non-hyperbolic domains.1.1. Background. One of the most important problem in several complex variables is to classify all complex domains in C n . If n = 1, then the Riemann mapping theorem tells us that every simply connected proper subdomain of C is biholomorphically equivalent to the unit disk. By showing the inequivalence of the unit ball and the polydisk in C 2 , Poincaré found that the Riemann mapping theorem does not hold even simply connected domains in C 2 . Thus the purely topological condition "simply connectedness" is not enough for the holomorphic equivalence problem in C n . Because of this background, in several complex variables, it is important to investigate biholomorphic invariant objects of complex domains.The purpose of this paper is to study two objects (the holomorphic automorphism group and the Bergman kernel) for a certain class of non-hyperbolic unbounded Reinhardt domains. The holomorphic automorphism group is a biholomorphic invariant object. Moreover, from the Bergman kernel, one can construct an invariant metric which is so-called the Bergman metric. In the theory of the automorphism groups, one important problem is to understand how the information of the holomorphic automorphism group characterizes complex domains. The next theorem is a notable result concerning this problem (cf.

On Proper Holomorphic Mappings Between Two Equidimensional FBH-Type Domains

“…Our next task is to show that f : D p → D q is linear, that is, f is a non-singular linear transformation of C N such that f (D p ) = D q . This can be shown along the same lines as in the proof of Yamamori [25,Theorem 4.1]. Indeed, the inclusion M ⊂ A 2 (D) in (2.2) guarantees that K D s (0, 0) > 0 and T D s (0, 0) is positive definite for s = p, q, where K D s is the Bergman kernel for D s and T D s is an N × N matrix defined by…”

“…This together with f ( D p ) = D q from Theorem 1 yields that f (0) = 0. Thus, by the same method as in the proof of Yamamori[25, Theorem 4.1] based on the results in Ishi and Kai [7, Propositions 2.1 and 2.6] it can be shown that f is linear, as desired. Finally, express f (z, w) = (Az, Bw) by some A ∈ GL(n; C) and B ∈ GL(m; C).…”

“…Indeed, we know by (2.2) that the set M of all monomials ζ α is contained in A 2 (D). Therefore, once it is shown that f (0) = 0, the linearity of f is a direct consequence of Yamamori [25,Theorem 4.1]. Now, in order to check that f (0) = 0, notice that f induces a CR-diffeomorphism of the pseudoconvex real analytic hypersurface ∂ D, because f as well as f −1 extends holomorphically beyond ∂ D by Fact A.…”

The holomorphic automorphism groups of twisted Fock-Bargmann-Hartogs domains

“…, θ n ∈ R. For Reinhardt domains, the following theorem is known. This theorem follows from Proposition 3.2 after checking that K D (0, 0) > 0 and that T D (0, 0) is positive definite as in [28].…”

“…For works related to the Hua domain, see [17], [23]. In [28], the Bergman kernel and the automorphism group of Ω m,p are studied when Ω = C n and Φ(z) = e −µ z 2 . Let us consider the domain…”