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

Abstract

The Fock-Bargmann-Hartogs domain Dn,m(µ) (µ > 0) in C n+m is defined by the inequality w 2 < e −µ z 2 , where (z, w) ∈ C n × C m , which is an unbounded nonhyperbolic domain in 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 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.

On the linearity of origin-preserving automorphisms of quasi-circular domains in Cn

“…• an explicit description of the automorphism group of D n,m [7], • rigidity properties of proper holomorphic mappings for D n,m [13].…”

Proper holomorphic mappings between generalized Hartogs triangles

“…Remark 8 Corollary 7 generalizes the main result of [6], where it is proved that for n ≥ 2, m ≥ 2, p ∈ N n , and q ∈ N m any proper holomorphic self-mapping in F p,q is an automorphism. For more information on rigidity of proper holomorphic mappings between special kind of domains in C n , such as Cartan domains, Hua domains, etc., we refer the reader to [14], [15], [16], [17], and [18].…”

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

“…Recently, Tu-Wang [20] obtained the rigidity result on proper holomorphic mappings between two equidimensional Fock-Bargmann-Hartogs domains. Theorem 1.D (Tu-Wang [20]) If D n,m (µ) and D n ′ ,m ′ (µ ′ ) are two equidimensional Fock-Bargmann-Hartogs domains with m ≥ 2 and f is a proper holomorphic mapping from D n,m (µ) into D n ′ ,m ′ (µ ′ ), then f is a biholomorphism between D n,m (µ) and D n ′ ,m ′ (µ ′ ).…”

“…Theorem 2.5 in Tu-Wang[20]) If D n,m (µ) and D n ′ ,m ′ (µ ′ ) are two equidimensional Fock-Bargmann-Hartogs domains and f is a proper holomorphic mapping from D n,m (µ) into D n ′ ,m ′ (µ ′ ), then f extends to be holomorphic in a neighborhood of D n,m (µ).…”