The squeezing function on doubly-connected domains via the Loewner differential equation

Abstract

Inspired by the work of Liu, Sun and Yau (2004) on holomorphic homogeneous regular (HHR) domains and Yeung (2009)’s work on domains with uniform squeezing property (another name for HHR domains), Deng, Guan and Zhang (2012) introduced a new biholomorphic invariant, namely, the squeezing function for bounded domains in the n-dimensional complex Euclidean space. Since then it has been one of the most active area in several complex variables in recent years. On the other hand, until now, there is only one explicit example of non-constant squeezing functions, namely the squeezing function of the punctured ball. In this talk, we will establish an explicit formula for the squeezing functions of annuli and hence (up to biholomorphisms) for any doubly connected planar domain. The main tools used to prove this result are the Schottky-Klein prime function (following the work of Crowdy) and a version of the Loewner differential equation on annuli due to Komatu. We will also show that these results can be used to obtain lower bounds on the squeezing function for certain product domains in the n-dimensional complex Euclidean space.