Evolution Families and the Loewner Equation II: complex hyperbolic manifolds

TL;DR

This paper establishes a correspondence between evolution families and specific holomorphic vector fields on complex hyperbolic manifolds, solving a general Loewner type differential equation in this setting.

Contribution

It extends the Loewner theory to complex hyperbolic manifolds by linking evolution families with semicomplete non-autonomous holomorphic vector fields.

Findings

Established a one-to-one correspondence between evolution families and holomorphic vector fields.

Provided a solution framework for a general Loewner type differential equation on manifolds.

Extended classical Loewner theory to complex hyperbolic manifolds.

Abstract

We prove that evolution families on complex complete hyperbolic manifolds are in one to one correspondence with certain semicomplete non-autonomous holomorphic vector fields, providing the solution to a very general Loewner type differential equation on manifolds.