Geometry of domains with the uniform squeezing property

TL;DR

This paper introduces the uniform squeezing property for domains, exploring their geometric and analytic features, and demonstrating their relevance to complex spaces like Teichmuller spaces and Hermitian symmetric spaces.

Contribution

It defines the uniform squeezing property, studies its implications, and shows its applicability to various complex geometric spaces and problems.

Findings

Domains with the uniform squeezing property are pseudoconvex and hyperconvex.

Such domains exhibit Kaehler-hyperbolicity and vanishing cohomology groups.

The property provides criteria for Steinness in holomorphic fiber bundles.

Abstract

We introduce the notion of domains with uniform squeezing property, study various analytic and geometric properties of such domains and show that they cover many interesting examples, including Teichmuller spaces and Hermitian symmetric spaces of non-compact type. The properties supported by such manifolds include pseudoconvexity, hyperconvexity, Kaehler-hyperbolicity, vanishing of cohomology groups and quasi-isometry of various invariant metrics. It also leads to nice geometric properties for manifolds covered by bounded domains and a simple criterion to provide positive examples to a problem of Serre about Stein properties of holomorphic fiber bundles.