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An abstract approach to Loewner chains

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Abstract

We present a new geometric construction of Loewner chains in one and several complex variables which holds on complete hyperbolic complex manifolds 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 univalent solution (f t : MN) of any Loewner-Kufarev PDE. The problem of finding solutions given by univalent mappings (f t : M → ℂn) is reduced to investigating whether the complex manifold ∪ t≥0 f t (M) is biholomorphic to a domain in ℂn. We apply such results to the study of univalent mappings f: B n → ℂn.

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Correspondence to Leandro Arosio.

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Titolare di una Borsa della Fondazione Roma — Terzo Settore bandita dall’Istituto Nazionale di Alta Matematica.

Partially supported by Grant-in-Aid for Scientific Research (C) No. 22540213 from Japan Society for the Promotion of Science, 2011.

Partially supported by the Romanian Ministry of Education and Research, UEFISCSU-CNCSIS Grant PN-II-ID 524/2007.

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Arosio, L., Bracci, F., Hamada, H. et al. An abstract approach to Loewner chains. JAMA 119, 89–114 (2013). https://doi.org/10.1007/s11854-013-0003-4

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