An abstract approach to Loewner chains

By Leandro Arosio, Filippo Bracci, H. Hamada, G. Kohr. 2010

78 citations recorded by the source corpus.

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 (ft: M → N) of any Loewner-Kufarev PDE. The problem of finding solutions given by univalent mappings (ft: M → ℂn) is reduced to investigating whether the complex manifold ∪t≥0ft (M) is biholomorphic to a domain in ℂn. We apply such results to the study of univalent mappings f: Bn → ℂn.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/25e1966adc0a073026da6e0d0b2d53ce60fe9a3d

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Solving the Loewner PDE in complete hyperbolic starlike domains of $\C^N$ (2012, background): “ogether with M. D. Contreras and S. D´ıaz-Madrigal [8], [9] generalized and solved the Loewner ODE on complete hyperbolic manifolds, and later the ﬁrst two authors together with H. Hamada and G. Kohr [7] showed that the Loewner PDE on complete hyperbolic manifolds always admits an (essentially uniqu”
Variation of Loewner Chains, Extreme and Support Points in the Class $$S^0$$S0 in Higher Dimensions (2014, background): “lent. Deﬁnition 2.6. A normalized Loewner chain (ft)t≥0 on Bn is called a normal Loewner chain if the family {e−tf t(·)}t≥0 is normal. Putting together [17, Chapter 8], [19], [3, Prop. 2.6] (see also [2], [9]), we have the following result: Theorem 2.7. (1) If (ft)t≥0 is any normalized Loewner chain”