Proper holomorphic mappings between generalized pseudoellipsoids

Proper holomorphic mappings between symmetrized ellipsoids

“…A theorem of that type was obtained in the case of B n in [3] and its generalization on complex ellipsoids was done in [8] and [5]. Recently, a similar result was obtained in [7] for the tetrablock, which is a (1, 1, 2)-balanced domain in C 3 .…”

“…φ III ) is the automorphism of E 1,2 (resp. E 1/2,2 ) defined in Corollary 4. Similar classification for the class of generalized complex ellipsoids (with not necessarily equal exponents on each coordinate) was done in [8] (the case of positive integer exponents) and [5] (case of positive real exponents).…”

Holomorphic vector fields and proper holomorphic self-maps of Reinhardt domains

“…S. Bell proved that mappings between circular domains are algebraic as soon as they preserve the origin [6]. Mappings between particular classes of Reinhardt domains were studied by G. Dini and A. Selvaggi [13], M. Landucci and G. Patrizio [15] and M. Landueei and S. Pinehuk [16]. In [9], F. Berteloot and S. Pinchuk classified the proper maps between bounded complete Reinhardt domains in C 2 and characterized the bidisc as being the only domain in this class which admits non-injective proper holomorphie self-maps (see also [17]).…”

Classification of proper holomorphic maps between Reinhardt domains in ${\Bbb C}^2$

“…Now, in caseD is a generalized pseudoellipsoid and D / =D, then there exists an annulus A = {(z 1 , 0) : 0 < a 1 < |z 1 | < a 2 } or B = {(0, z 2 ) : 0 < b 1 < |z 2 | < b 2 }, which is in ∂D and which is necessarily mapped by F onto another annulus lying inD ∩ I . By direct inspection of the proper maps between generalized pseudoellipsoids (see [5], [4]), this implies that F splits. Proof.…”