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 $e_D\left(z\right)$ and the squeezing function $s_D\left(z\right)$. More specifically, we study two questions about the quotient invariant $m_D\left(z\right)=s_D\left(z\right)/e_D\left(z\right)$: (1) If $m_D\left(z_0\right)=1$ for some $z_0\in D$, is D biholomorphic to the unit ball? (2) Is $m_D\left(z\right)$ constantly equal to 1? We answer both questions negatively.

COMMUNICATED BY:

• S. Krantz

Keywords:

• Fridman invariant
• squeezing function
• quotient invariant

AMS SUBJECT CLASSIFICATIONS:

• 32H02
• 32F45

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors are partially supported by the National Natural Science Foundation of China (grant no. 11871333).