In this paper we present explicit solutions to the radial and chordal Loewner PDEs and we make an extensive study of their geometry. Specifically, we study multi-slit Loewner flows, driven by the time-dependent point masses in the radial case and in the chordal case, where all the above parameters are chosen arbitrarily. Furthermore, we investigate their close connection to the semigroup theory of holomorphic functions, which also allows us to map the chordal case to the radial one.

We study parabolic semigroups of finite shift in the unit disk with regard to their geometric properties and the rate of convergence of their orbits to the Denjoy-Wolff point. We examine this rate in terms of Euclidean distance, hyperbolic distance and harmonic measure. We further discuss the corresponding rates of convergence for parabolic semigroups of positive hyperbolic step and infinite shift. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data

We study the chordal Loewner equation associated with certain driving functions that produce infinitely many slits. Specifically, for a choice of a sequence of positive numbers (b_n)_n≥1 and points of the real line (k_n)_n≥1, we explicitily solve the Loewner PDE

in H×[0,1). Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as t→1⁻.

This thesis deals with two topics in complex analysis that are related to families of Riemann maps that depend on some parameter function called the driving function of the family. One is Loewner theory and the other is the theory of semigroups of holomorphic self-maps of the unit disc. This thesis consists of four works, three of which lie in the intersection of the two theories and the other one refers solely on semigroups.

In Paper I, we deal with the Loewner equation both in the unit disc (the radial  case) and in the upper half-plane (the chordal case). The solutions to these equations, which depend on the space and time variables, are called (radial or chordal) Loewner chains. Its main purpose is to present explicitly solutions to certain choices of driving functions and additionally visualize their geometry as time evolves. In particular, we deal with conformal maps with finitely many slits, for both cases. Thus, the aforementioned evolution involves the growth of multiple curves either in the unit disc or in the upper half-plane. Secondly, we discover the semigroup nature of these families, which we utilize in order to connect the radial with the chordal case through a Möbius transform, although in the general theory this is not always possible.

The second paper is a continuation of Paper I, where we extend the study of the chordal Loewner chains of Paper I to chains with infinitely many slits. Again, we study the geometry of the chains as time evolves and we find the same geometric behaviour as in Paper I. However, this study is more complicated and requires a different approach that involves techniques from classical complex analysis and the use of the harmonic measure.

In Paper III we are concentrated in a specific type of semigroups. We call those semigroups of finite shift. In the general theory of semigroups, several authors have studied the rate of convergence of a semigroup to the Denjoy-Wolff point, in terms of the Euclidean distance. In this direction, we also examine the rate of convergence for this case, in terms of the Euclidean distance, the hyperbolic distance and also in terms of the harmonic measure. 

In Paper IV, we present some computational examples of Loewner chains. Some of them are related to those appearing in Papers I and II. We work similarly in the sense that we solve the Loewner equation for some certain driving functions. In addition, we have collected some Loewner chains that do not appear in the literature and we recover their driving functions. Our intention is to visualize these elementary examples in an effort to compare the geometry of the chains with their driving functions.