A method of potential scaling in the study of pseudoconvex domains with noncompact automorphism group

TL;DR

This paper introduces the potential scaling method as an alternative to affine scaling for analyzing pseudoconvex domains with noncompact automorphism groups, focusing on constructing potential functions for Kähler-Einstein metrics.

Contribution

It presents a new potential scaling approach and proves its connection to automorphism groups in pseudoconvex domains with Kähler-Einstein metrics.

Findings

Potential functions with constant differential length imply a 1-parameter automorphism family.

The method offers an alternative to affine scaling in complex domain analysis.

New insights into the structure of automorphism groups in pseudoconvex domains.

Abstract

The affine scaling method has been a typical approach to study complex domains with noncompact automorphism group. In this article, we will introduce an alternative approach, so called, the method of potential scaling to construct a certain class of potential functions of the K\"ahler-Einstein metric. We will also prove that if a bounded pseudoconvex domain admits a potential function of the K\"ahler-Einstein metric whose differential has constant length, then there is an $1$-parameter family of automorphisms.