Tests for equivalence of hyperbolic manifolds

Boundary behaviour of the Carathéodory and Kobayashi-Eisenman volume elements

“…This was first proved by Wong [26] with the hypothesis that D is bounded and complete hyperbolic, which was relaxed by Rosay [24] to D being any bounded domain. Dektyarev [9] further relaxed this condition to D being only hyperbolic and later Graham and Wu [12] showed that no assumption on D is required for the result to be true, in fact, it is true for any complex manifold. Thus q D measures the extent to which the Riemann mapping theorem fails for D. This fact is a fundamental step in the proof of the Wong-Rosay theorem and several other applications can be found in [13][14][15].…”

Intrinsic measures and holomorphic retracts

“…In this paper we examine the consequences of the equality of the Eisenman and Caratheodory norms on k~vectors, 2 < k < n -1, at a point in an -dimensional complex manifold M. (The Eisenman norm is the analog of the infinitesimal Kobayashi metric and is the object of a recent study by H. Wu and the author [9]; see also the papers of Eisenman [7,15].) H. Wu and the author considered the top-dimensional case in [10] (for earlier results see [6,17,22]) obtaining a criterion for biholomorphic equivalence with the unit ball in C". We also investigate the consequences of the existence of a large number of two-dimensional holomorphic retracts of a complex manifold-one tangent to each 2-vector at a given point.…”