Loewner chains in the unit disk

By Manuel D. Contreras, S. Díaz-Madrigal, P. Gumenyuk. 2009

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In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [arXiv:0807.1594], of the radial and chordal variant of the Loewner differential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general setting, we establish a deep correspondence between these chains and the evolution families introduced in [arXiv:0807.1594]. Among other things, we show that, up to a Riemann map, such a correspondence is one-to-one. In a similar way as in the classical Loewner theory, we also prove that these chains are solutions of a certain partial differential equation which resembles (and includes as a very particular case) the classical Loewner-Kufarev PDE.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2f31f43dfa0113dfd1a64d91aacb190bc352ca4b

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

The evolution of Loewner's differential equations (2010, background): “An intrinsic study of evolution families and of their relationship with other aspects of the theory has not been carried out until recently; let us describe the approach proposed in [11] and [14].”