An abstract approach to Loewner chains

2010/02/23 by Arosio, Leandro, Bracci, Filippo, Hamada, Hidetaka +1
#30C45 #32H02 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1002.4262

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 Cⁿ is reduced to investigating whether the union of the images f_t(M) is biholomorphic to a domain in Cⁿ. We apply such results to the study of univalent mappings from the unit ball Bⁿ to Cⁿ.

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• Evolution Families and the Loewner Equation II: complex hyperbolic manifolds
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