GEOMETRY OF DOMAINS WITH THE UNIFORM SQUEEZING PROPERTY

Abstract We introduce the notion of domains with (a, b)-uniform squeezing prop-erty, study various analytic and geometric properties of such domains and show that they cover many interesting examples, including Teichmüller spaces and Her-mitian symmetric spaces of non-compact type. The properties supported by such manifolds include pseudoconvexity, hyperconvexity, Kähler-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. The purpose of this article is to introduce a class of bounded domains in Cn which on one hand is sufficiently general to include interesting classes of examples and on the other hand leads to interesting analytic and geometric properties. Definition 1. Denote by Br(x) a ball of radius r in Cn. Let 0 < a < b <