Fridman Function, Injectivity Radius Function and Squeezing Function

TL;DR

This paper investigates the Fridman function's properties on hyperbolic complex manifolds, establishing bounds, explicit formulas for specific domains, and analyzing boundary behaviors, thus advancing understanding of complex geometric functions.

Contribution

It proves the Fridman function is bounded by the injectivity radius, derives explicit formulas for certain domains, and explores boundary behaviors, extending the concept of uniform thickness.

Findings

Fridman function is bounded above by the injectivity radius function.

Explicit formulas for Fridman functions on annulus and punctured disk.

Boundary behavior analysis of Fridman and squeezing functions.

Abstract

Very recently, the Fridman function of a complex manifold $X$ has been identified as a dual of the squeezing function of $X$. In this paper, we prove that the Fridman function for certain hyperbolic complex manifold $X$ is bounded above by the injectivity radius function of $X$. This result also suggests us to use the Fridman function to extend the definition of uniform thickness to higher-dimensional hyperbolic complex manifolds. We also establish an expression for the Fridman function (with respect to the Kobayashi metric) when $X = \mathbb{D} \diagup \Gamma$ and $\Gamma$ is a torsion-free discrete subgroup of isometries on the standard open unit disk $\mathbb{D}$. Hence, explicit formulae of the Fridman functions for the annulus $A_r$ and the punctured disk $\mathbb{D}^*$ are derived. These are the first explicit non-constant Fridman functions. Finally, we explore the boundary behaviour of the Fridman functions (with respect to the Kobayashi metric) and the squeezing functions for regular type hyperbolic Riemann surfaces and planar domains respectively.