Scaling methods in Several Complex Variables and some Holomorphic Invariants

Abstract

The scaling methods were developed in the 1980s by Pinchuk and Frankel independently as a technique to study bounded domains with non-compact automorphism group. This thesis observes that how to apply the scaling methods under various boundary conditions. In particular, we give a detailed proof to the convergence of Pinchuk's scaling sequence (forward sequence in particular) on bounded domains with finite type boundaries in C2. This is the first main result of this thesis. We also discuss the modification of the Frankel scaling sequence on non-convex domains and observe that two scalings are equivalent. Using this technique, we show that a point at which the squeezing function tends to one is strongly pseudoconvex if the given domain is bounded in C2 with finite type boundary in the sense of D'Angelo. This is the second main result of this thesis.