Conformal Maps, Bergman Spaces, and Random Growth Models

Sammanfattning: This thesis consists of an introduction and five research papers on topics related to conformal mapping, the Loewner equation and its applications, and Bergman-type spaces of holomorphic functions. The first two papers are devoted to the study of integral means of derivatives of conformal mappings. In Paper I, we present improved upper estimates of the universal means spectrum of conformal mappingsof the unit disk. These estimates rely on inequalities  obtained by Hedenmalm and Shimorin using Bergman space techniques, and on computer calculations. Paper II is a survey of recent results on the universal means spectrum, with particular emphasis on Bergman spacetechniques.Paper III concerns Bergman-type spaces of holomorphic functions in subsets of $\textbf{C}^d$ and their reproducing kernel functions. By expanding the norm of a function in a Bergman space along the zero variety of a polynomial, we obtain a series expansion of reproducing kernel functions in terms of kernels associated with lower-dimensionalspaces of holomorphic functions. We show how this general approach can be used to explicitly compute kernel functions for certain weighted Bergman and Bargmann-Fock spaces defined in domains in $\textbf{C}^2$.The last two papers contribute to the theory of Loewner chains and theirapplications in the analysis of planar random growth model defined in terms of compositions of conformal maps.In Paper IV, we study Loewner chains generated by unimodular L\'evy processes.We first establish the existence of a capacity scaling limit for the associated growing hulls in terms of whole-plane Loewner chains driven by a time-reversed process. We then analyze the properties of Loewner chains associated with a class of two-parameter compound Poisson processes, and we describe the dependence of the geometric properties of the hulls on the parameters of the driving process. In Paper V, we consider a variation of the Hastings-Levitov growth model, with anisotropic growth. We again establish results concerning scaling limits, when the number of compositions increases and the basic conformal mappings tends to the identity. We show that the resulting limit sets can be associated with solutions to the Loewner equation.We also prove that, in the limit, the evolution of harmonic measure on the boundary is deterministic and is determined by the flow associated with an ordinary differential equation, and we give a description of the fluctuations around this deterministic limit flow.