Loewner chains and biholomorphic mappings in Cn and reflexive complex Banach spaces

This paper is a survey of very recent results about biholomorphic mappings of the ball in Cn and in reflexive complex Banach spaces. After recalling existence and regularity results in Cn, we present certain applications including univalence criteria and quasiconformal extension results. We also consider nonuniqueness phenomena for solutions of the Loewner differential equation, and a geometric characterization of Loewner chains which satisfy a growth condition in t based on a generalization of the Caratheodory convergence theorem. Finally we describe some properties of Loewner chains and the Loewner equation on the unit ball of a reflexive complex Banach space.