Embedding Problems in Loewner Theory

Abstract

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. Díaz-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 and we prove a conjecture about support points of these classes. 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. 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.