Möbius automorphisms of plane domains

Quasiconformally Bi-Homogeneous Compacta in the Complex Plane

“…To be speci®c, let D n h n H and D hH, where H fz: Im z > 0g. Then D is a component of ker n31 D n 1 n1 int 1 mn D m and h À1 n 3 h À1 pointwise in D. We shall produce a contradiction by demonstrating that the domain D is conformally homogeneous, and hence, in view of Theorem 3 in [7] (see also [13]), that ¶D is a circle in C. Since the points 0, 1 and 1 lie on ¶D, ¶D R È f1g. Thus hI, a subarc of ¶D that has endpoints 0 and 1 but does not contain 1, must coincide with I.…”

“…Because KE c < KE´E c , the second assertion follows from Theorem A and from Theorem 5.5 in [9], which prevents E from being a ®nite set of points. The ®nal assertion in the lemma is a consequence of Theorem 2.3 in [18] or of Theorem 3 in [7] (see [13] as well).…”

Ambient quasiconformal homogeneity of planar domains

“…One may combine Theorem 8.1 of Gehring-Palka [12] with the main result of Erkama [11] to obtain a complete characterization of ambiently conformally homogeneous domains. (1) If C ⊂ Ω is compact, then there exists N such that C ⊂ Ω n if n ≥ N , and (2) if an open set U is contained in Ω n for infinitely many values of n, then U ⊂ Ω.…”

“…Notice that any K-quasidisk (which is not round) is ambiently Kquasiconformally homogeneous, but not ambiently 1-quasiconformally homogeneous (see Gehring-Palka [12] and Erkama [11]) so one cannot bound the constant away from 1 in the simply connected case. Theorem 1.1.…”