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• W1907414511 abstract "The work at hand studies problems from Loewner theory and is divided into two parts:In part 1 (chapter 2) we present the basic notions of Loewner theory. Here we use a modern form which was developed by F. Bracci, M. Contreras, S. Diaz-Madrigal et al. and which can be applied to certain higher dimensional complex manifolds.We look at two domains in more detail: the Euclidean unit ball and the polydisc. Here we consider two classes of biholomorphic mappings which were introduced by T. Poreda and G. Kohr as generalizations of the class S.We prove a conjecture of G. Kohr about support points of these classes. The proof relies on the observation that the classes describe so called Runge domains, which follows from a result by L. Arosio, F. Bracci and E. F. Wold.Furthermore, we prove a conjecture of G. Kohr about support points of a class of biholomorphic mappings that comes from applying the Roper-Suffridge extension operator to the class S.In part 2 (chapter 3) we consider one special Loewner equation: the chordal multiple-slit equation in the upper half-plane.After describing basic properties of this equation we look at the problem, whether one can choose the coefficient functions in this equation to be constant. D. Prokhorov proved this statement under the assumption that the slits are piecewise analytic. We use a completely different idea to solve the problem in its general form.As the Loewner equation with constant coefficients holds everywhere (and not just almost everywhere), this result generalizes Loewner’s original idea to the multiple-slit case.Moreover, we consider the following problems: • The “simple-curve problem” asks which driving functions describe the growth of simple curves (in contrast to curves that touch itself). We discuss necessary and sufficient conditions, generalize a theorem of J. Lind, D. Marshall and S. Rohde to the multiple-slit equation and we give an example of a set of driving functions which generate simple curves because of a certain self-similarity property. • We discuss properties of driving functions that generate slits which enclose a given angle with the real axis. • A theorem by O. Roth gives an explicit description of the reachable set of one point in the radial Loewner equation. We prove the analog for the chordal equation." @default.
• W1907414511 created "2016-06-24" @default.
• W1907414511 creator A5050408343 @default.
• W1907414511 date "2013-01-01" @default.
• W1907414511 modified "2023-09-27" @default.
• W1907414511 title "Embedding Problems in Loewner Theory" @default.
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