An abstract approach to Loewner chains

Abstract

We present a new geometric construction of Loewner chains in one and several complex variables which holds on a complete hyperbolic complex manifold M and prove that there is essentially a one-to-one correspondence between evolution families of order d and Loewner chains of the same order. As a consequence we obtain a solution for any Loewner-Kufarev PDE, given by univalent mappings (f_t) from M to a complex manifold N. The problem of finding solutions given by univalent mappings with range in C^n is reduced to investigating whether the union of the images f_t(M) is biholomorphic to a domain in C^n. We apply such results to the study of univalent mappings from the unit ball B^n to C^n.