Fridman's invariant, squeezing functions, and exhausting domains

TL;DR

The paper investigates conditions under which bounded domains in complex analysis are equivalent to the unit ball, focusing on Fridman's invariant and exhaustion by strictly pseudoconvex domains, revealing new characterizations of such domains.

Contribution

It establishes that domains exhausted by strictly pseudoconvex domains are equivalent to the ball or a similar domain, and links Fridman's invariant growth to domain equivalence.

Findings

Domains exhausted by strictly pseudoconvex domains are equivalent to the ball or the domain they exhaust.

Growth conditions on Fridman's invariant characterize when a domain is equivalent to the ball.

The results unify concepts of exhaustion, squeezing functions, and domain equivalence in complex analysis.

Abstract

We show that if a bounded domain $\Omega$ is exhausted by a bounded strictly pseudoconvex domain $D$ with $C^2$ boundary, then $\Omega$ is holomorphically equivalent to $D$ or the unit ball, and show that a bounded domain has to be holomorphically equivalent to the unit ball if its Fridman's invariant has certain growth condition near the boundary.