In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. To appear in J. Reine Angew. Math.] 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 [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. To appear in J. Reine Angew. Math.]. 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.
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Manuel D. Contreras, Santiago Díaz-Madrigal, Pavel Gumenyuk, Loewner chains in the unit disk. Rev. Mat. Iberoam. 26 (2010), no. 3, pp. 975–1012DOI 10.4171/RMI/624