On the comparison of the Fridman invariant and the squeezing function

TL;DR

This paper investigates the relationship between the Fridman invariant and the squeezing function on bounded domains in complex space, showing that their quotient does not necessarily characterize the domain as the unit ball.

Contribution

It provides counterexamples to the conjecture that the quotient of these invariants being 1 implies the domain is biholomorphic to the unit ball.

Findings

The quotient invariant equals 1 at some point does not imply the domain is the ball.

The quotient invariant being constantly 1 does not hold for all domains.

Counterexamples show the invariants' quotient does not characterize the domain.

Abstract

Let $D$ be a bounded domain in $\mathbb{C}^n$, $n\ge 1$. In this paper, we study two biholomorphic invariants on $D$, the Fridman invariant $e_D(z)$ and the squeezing function $s_D(z)$. More specifically, we study the following two questions about the \textit{quotient invariant} $m_D(z)=s_D(z)/e_D(z)$: 1) If $m_D(z_0)=1$ for some $z_0\in D$, is $D$ biholomorphic to the unit ball? 2) Is $m_D(z)$ constantly equal to 1? We answer both questions negatively.