Loewner Theory in Several Complex Variables and Related Problems.

摘要

The first part of the thesis deals with aspects of Loewner theory in several complex variables. First we show that a Loewner chain with minimal regularity assumptions (Df (0, ˙) of local bounded variation) satisfies an associated Loewner equation. Next we give a way of renormalizing a general Loewner chain so that it corresponds to the same increasing family of domains. To do this we will prove a generalization of the converse of Caratheodory's kernel convergence theorem. Next we address the problem of finding a Loewner chain solution to a given Loewner chain equation. The main result is a complete solution in the case when the infinitesimal generator satisfies Dh (0, t) = A where inf {Re ⟨Az, z⟩ : ||z|| = 1} > 0. We will see that the existence of a bounded solution depends on the real resonances of A, but there always exists a polynomially bounded solution. Finally we discuss some properties of classes of biholomorphic mappings associated to A-normalized Loewner chains. In particular we give a characterization of the compactness of the class of spirallike mappings in terms of the resonance of A.;The second part of the thesis deals with the problem of finding examples of extreme points for some classes of mappings. We see that straightforward generalizations of one dimensional extreme functions give examples of extreme Caratheodory mappings and extreme starlike mappings on the polydisc, but not on the ball. We also find examples of extreme Caratheodory mappings on the ball starting from a known example of extreme Caratheodory function in higher dimensions.