Geometry of the symmetrized polydisc

Abstract

Abstract. We describe all proper holomorphic mappings of the symmetrized polydisc and study its geometric properties. We also apply the obtained results to the study of the spectral unit ball in Mn(C n ).

Explicit Formulas for the Bergman Kernel on Certain Forelli-Rudin Construction

“…In 1998, Roos and Yin [20] introduced so called Cartan-Hartogs domains on which Bergman kernels were obtained in close form by Yin in 1999. In [8], the Bergman kernel for the symmetrized polydisc G n was obtained by Edigarian and Zwonek. Recently, Jong-Do Park computed the explicit formula of the Bergman kernel for a nonhomogeneous domain {z ∈ C 2 : |z 1 | 4 + |z 2 | 4 < 1} and the kernel is not algebraic [17].…”

Structure of the group of automorphisms of the spectral $$2$$ -ball

“…a domain in C 2 given by the formula G 2 := {(tr x, det x) : x ∈ } (see [3] for its basic properties).…”

“…Composing it, if necessary, with a Möbius map one may assume that ϕ(0) = 0 (see [3,9]). Then ϕ (0) is also an automorphism of the spectral ball.…”

“…[3]) that for any ϕ ∈ Aut( ) there is a Möbius map m such that σ (ϕ(x)) = σ (m(x)). Therefore, composing ϕ with a Möbius map, a transposition and a linear automorphism of we may always assume that ϕ is normalized, i.e.…”

Proper holomorphic mappings between symmetrized ellipsoids

“…Recently, a similar result was obtained in [7] for the tetrablock, which is a (1, 1, 2)-balanced domain in C 3 . Characterization of proper holomorphic self-mappings of symmetrized polydisc is done in [6].…”