On the comparison of the Fridman invariant and the squeezing function

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.