Classification of Proper Holomorphic Mappings between Hartogs Domains over Homogeneous Siegel Domains

The Hartogs domain over homogeneous Siegel domain D_(N,s)(s>0)is defined by the inequality■,where D is a homogeneous Siegel domain of typeⅡ,(z,ζ)∈D×C~N and KD(z,z)is the Bergman kernel of D.Recently,Seo obtained the rigidity result that proper holomorphic mappings between two equidimensional domains D_(N,s)and D'_(N',s')are biholomorphisms for N≥2.In this article,we find a counter-example to show that the rigidity result is not true for D_(1,s)and obtain a classification of proper holomorphic mappings between D_(1,s)and D'_(1,s').