Explicit description for the automorphism group of the Fornæss domain

Abstract

Motivated by the Kohn-Nirenberg domain, J.E. Fornaess considered the germ of a domain near the origin in C 2 such that Ω t = (z, w) ∈ C 2 : Re w + |zw| 2 + |z| 6 + t|z| 2 Re z 4 < 0 to study the holomorphic peak function that is smooth up to the boundary (Fornaess, 1977 [6]). J.E. Fornaess proved that for 1 < t < 9/5 the domain Ω t does not admit a holomorphic function on Ω t that is C 1 up to the boundary and that peaks at the origin. We define Π(z, w) = (e i π 2 z, w). In this paper, we prove that for 1 < t < 9/5, the automorphism group of Ω t is equal to the set {Π k : k = 1, 2, 3, 4}.

New and Old Results of Computations of Automorphism Group of Domains in the Complex Space

“…Fornaess proved that for 1 < t < 9/5 the domain Ω t does not admit a holomorphic function on Ω t that is C 1 up to the boundary and that peaks at the origin. In [3], they compute all automorphism of the Fornaess domain. Moreover, they generalize domain Ω n,t defined by …”

A Bounded Kohn Nirenberg Domain

“…In [4] the explicit description of the group of holomorphic automorphisms of the Kohn Nirenberg domain D KN is given; in particular, Aut (D KN ) = Z 6 , and so it is compact (as a Lie group). Moreover, in [2] it is proved that the space Aut (D KN ) is sequences-wise compact; but J. Byun could not infer from this result that the group Aut (D KN ) is compact since it is not known whether the Kohn Nirenberg domain is φ-bounded.…”

“…Until recently, a great deal of work was done, among the others, by M. Kolar (see [7], [8]), by J. Byun (see [2], [3]), and by J. Byun and H. R. Cho (see [4]) in order to understand the properties and peculiarities of convexifiability and of the domain D KN .…”

“…Moreover, in [4], Byun and Cho were able to explicitly compute the automorphism group. We will observe that the automorphism group of the Kohn Nirenberg domain is contained in our modification.…”

“…We will observe that the automorphism group of the Kohn Nirenberg domain is contained in our modification. It would be interesting to study whether the methods used in [4] can be generalized to our domain.…”