Rigidity of Proper Self-Mapping on Some Kinds of Generalized Hartogs Triangle

The classification of proper holomorphic mappings between special Hartogs triangles of different dimensions

“…Hence, there exist constant matrices A ∈ U(n 1 ) and B ∈ U(n 2 ) such that (A(w), 0)B(w) = (A, 0)B. Thus By the main theorem in [7], any proper holomorphic self-mapping of Ω(n 1 , m 1 ) or Ω(n 2 , m 2 ) is automorphism. Without loss of generality, let σ ∈ Aut(Ω(n 1 , m 1 )), τ ∈ Aut(Ω(n 2 , m 2 )), such that σ : (z, w) → (zA, wA ),…”

“…Proper holomorphic mapping theory dates from 1950s, and there are many good results on it (see [1][2][3][4][5][6][7][8][9]). The classification of proper holomorphic mappings (see the definition in [1]) is an important and difficult problem, especially between bounded domains of different dimensions (see [2][3][4]).…”

Proper holomorphic self-maps of smooth bounded Reinhardt domains in ℂ2

“…Besides the obvious examples, Berteloot and Loeb [10] proved, based on the complex dynamics in C 2 , that there exists a complete circular domain in C 2 with the real analytic strictly pseudoconvex boundary outside of the union of three circles (where the boundary is not smooth) such that there is a proper holomorphic self-map of the domain which is not biholomorphic. However for some other non-smooth domains, certain generalized Hartogs triangles, Chen and Xu [11] proved that the self-maps are biholomorphic. We also point out that the proper self-maps of smooth bounded domains in a complex manifold may not be the automorphisms by the examples of Burns and Schnider.…”

Proper holomorphic mappings between generalized Hartogs triangles

“…and the group Aut(F p,q ) of holomorphic automorphisms of F p,q has been investigated in many papers (see, e.g., [12], [5], [6], [2], [3] for the equidimensional case and [4] for the non-equidimensional one). It was Landucci who considered the mappings (1) in 1989 as examples of proper holomorphic mappings between non-smooth pseudoconvex Reinhardt domains, with the origin on the boundary, which do not satisfy a regularity property for the Bergman projection (the so-called R-condition).…”

“…The next step was made one year later, when the same authors fully described proper holomorphic self-mappings of F p,q for n > 1, m > 1, p ∈ N n , and q ∈ N m (cf. [6]). In the same year, Chen in [2] characterized the existence of mappings (1) in the case n > 1, m > 1, p,p ∈ R n >0 , and q,q ∈ R m >0 .…”