On the scaling methods by Pinchuk and Frankel

TL;DR

This paper analyzes and compares two scaling methods in complex analysis, providing new proofs of convergence and demonstrating the equivalence of modified scalings on certain domains.

Contribution

It introduces a global coordinate system and offers an alternative convergence proof for Pinchuk's scaling, also discussing modifications and equivalences of Frankel's scaling.

Findings

Convergence of Pinchuk's scaling sequence is established on bounded domains with finite type boundaries.

Modified scalings for nonconvex domains are shown to be equivalent.

A new global coordinate system simplifies the analysis of scaling methods.

Abstract

The main purpose of this paper is to study two scaling methods developed respectively by Pinchuk and Frankel. We introduce first a continuously-varying 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 $\mathbb{C}^2$. Using this, we discuss the modification of the Frankel scaling sequence on nonconvex domains. We also observe that two modified scalings are equivalent.