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

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