Some properties of squeezing functions on bounded domains

Abstract

This paper introduces the notion of squeezing functions on bounded domains and studies some of their properties. The relation to geometric and analytic structures of bounded domains will be investigated. Existence of related extremal maps and continuity of squeezing functions are proved. Holomorphic homogeneous regular domains introduced by Liu, Sun and Yau are exactly domains whose squeezing functions have positive lower bounds. Completeness of certain intrinsic metrics and pseudoconvexity of holomorphic homogeneous regular domains are proved by alternative method. In the dimension one case, we get a neat description of boundary behavior of squeezing functions of finitely connected planar domains. This leads to necessary and sufficient conditions for a finitely connected planar domain to be a holomorphic homogeneous regular domain. Consequently, we can recover some important results in complex analysis. For annuli, we obtain some interesting properties of their squeezing functions. Finally, we present some examples of bounded domains whose squeezing functions can be given explicitly.