On the scaling methods by Pinchuk and Frankel

Abstract

Abstract. The main purpose of this paper is to study two scaling methods developed respectively by Pinchuk and Frankel. We introduce first a continuouslyvarying global coordinate system, and give an alternative proof to the convergence of Pinchuk's scaling sequence (and of our modification) on bounded domains with finite type boundaries in C 2 . Using this, we discuss the modification of the Frankel scaling sequence on nonconvex domains. We also observe that two modified scalings are equivalent.

On Boundary Points at Which the Squeezing Function Tends to One

“…Proof. The proof is almost the same as those for Propositions 2.8 and 2.10 of [10]. However, the surjectivity part of σ requires a few, simple but perhaps subtle adjustments.…”

“…Note that the convergences P j → P , R j → 0 and Q j → 0 are uniform on compact subsets of C 2 , while P is a nonzero real-valued subharmonic polynomial of degree 2k. This is proved in detail in Lemma 2.4 of [10]. Consequently, ρ j converges uniformly to ρ := Re w+ P (z,z) on compact…”

“…Taking a subsequence if necessary, we may assume that the uniform convergence holds for σ −1 j → σ −1 and (σ q j ) −1 → σ q −1 on compact subsets of σ( Ω) and σ q ( Ω q ) respectively. (See Lemma 2.9 in [10]).…”

“…The next step is to use the dilation sequence {α j : Ω → C 2 } introduced in [10], Section 2.2. (We remark that this is a mild but necessary modification of Pinchuk's stretching sequence [1,16]).…”