Function Theory of Several Complex Variables

A new invariant metric and applications

“…We cannot prove this result here, but see [KRA1]. Another key fact for us, that is proved in [GRK3], is the following:…”

“…Among smoothly bounded domains, the only domain with transitive automorphism group is the unit ball (up to biholomorphic equivalence). See the discussion in the next section as well as [KRA1], [ROS], [WON].…”

“…See [KRA1] for definitions and background on such domains. A fundamental result for these types of domains is due to Bun Wong [WON] and Rosay [ROS]:…”

“…In fact write Ω 0 = {z ∈ C n : ρ 0 (z) < 0}, where ρ 0 is a smooth defining function for Ω 0 (see [KRA1] for this concept). There is a positive integer k and an ǫ > 0 such that if Ω = {z ∈ C n : ρ(z) < 0} and ρ − ρ 0 C k < ǫ then the automorphism group Aut (Ω) is a subgroup of Aut (Ω 0 ).…”

Hyper-Conjugate Harmonic Function of Conic Regular Functions in Conic Quaternions

“…From Krantz [9], the δ-closed form ψ of z 1 and z 2 are δ-exact form on γ 1 (Ω). Since Ω is a pseudoconvex domain, there exists the 1st conic conjugate harmonic function g 2 of class C ∞ in Ω, where ∂-closed form γ −1 1 ψ = ∂g 2 on Ω of z 1 and z 2 are ∂-exact (0, 1)-forms on Ω such that g(Z) is the 1st conic regular function in Ω.…”