On the squeezing function for finitely connected planar domains

TL;DR

This paper investigates the squeezing function for finitely connected planar domains, disproves a conjecture about extremal mappings, and provides a new potential-theoretic proof for the annulus case.

Contribution

It disproves a conjecture on extremal conformal mappings for finitely connected domains and offers a simple potential-theoretic proof for the annulus case.

Findings

Disproved the conjecture that extremal mappings are circularly slit disks for all finitely connected domains.

Provided a simple potential-theoretic proof for the explicit squeezing function of an annulus.

Identified all extremal functions for the annulus case.

Abstract

In a recent paper, Ng, Tang and Tsai (Math. Ann. 2020) have found an explicit formula for the squeezing function of an annulus via the Loewner differential equation. Their result has led them to conjecture a corresponding formula for planar domains of any finite connectivity stating that the extremum in the squeezing function problem is achieved for a suitably chosen conformal mapping onto a circularly slit disk. In this paper we disprove this conjecture. We also give a conceptually simple potential-theoretic proof of the explicit formula for the squeezing function of an annulus which has the added advantage of identifying all extremal functions.