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Quasiregular Mappings extend quasiconformal theory to the noninjective case.They give a natural and beautiful generalization of the geometric aspects ofthe theory of analytic functions of one complex variable to Euclidean n-space or, more generally, to Riemannian n-manifolds. This book is a self-contained exposition of the subject. A braod spectrum of results of both analytic and geometric character are presented, and the methods vary accordingly. The main tools are the variational integral method and the extremal length method, both of which are thoroughly developed here. Reshetnyak's basic theorem on discreteness and openness is used from the beginning, but the proof by means of variational integrals is postponed until near the end. Thus, the method of extremal length is being used at an early stage and leads, among other things, to geometric proofs of Picard-type theorems and a defect relation, which are some of the high points of the present book.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
Quasiregular mappings are defined in the same way as quasieonformal mappings, absent the homeomorphism requirement. They were first introduced and studied by Yu.G. Reshetnyak in a series of articles that began to appear in 1966. His discoveries were enhanced and furthered by the group O. Martio, S. Rickman, and J. Väisälä a few years later. The theory of quasiregular mappings gives a natural and beautiful generalization of the geometric aspects of the theory of (complex) analytic functions in the plane to Euclidean n-space ℝ n , or more generally, to Riemannian n-manifolds. Quasiregular mappings are interesting not only because of the results obtained about them, but also because of the many new ideas generated in the course of the development of their theory. In addition, a part of classical complex function theory is enriched by its exposure to a different point of view.
Seppo Rickman

Chapter I. Basic Properties of Quasiregular Mappings

Abstract
We shall define quasiregular mappings analytically following Yu.G. Reshetnyak. At an early stage we take full advantage of Reshetnyak’s important discovery that a nonconstant quasiregular mapping is discrete and open, but put off the proof to Chapter VI. This way we are able to have a coherent geometric treatment without introducing an excessive amount of machinery at the outset. Section 1 on ACLP mappings contains fairly standard preliminary results. Discrete open mappings are considered in Section 4 as a separate topic, mostly without proofs. The material of the first chapter is primarily concerned with various aspects of the definition of quasiregularity.
Seppo Rickman

Chapter II. Inequalities for Moduli of Path Families

Abstract
In this chapter we shall derive the important inequalities for moduli of path families which form the basis for the geometric part of qr theory. The main reference for path families in connection with qc theory is the book [V4] by J. Väisälä. When convenient, we shall refer to [V4] rather than repeat proofs. The main results, namely Poletskiĭ’s and Väisälä ‘s inequalities, involve a somewhat technical measure theoretic step concerning path families whose modulus is neglible. We shall call this step Poletskiĭ’s lemma. In qc theory a corresponding result is known as Fuglede’s theorem. Inequalities for path families are more general and more effective than inequalities for capacities of condensers, which historically came first [MRV1]. Here we shall obtain the capacity inequalities as corollaries in Section 10.
Seppo Rickman

Chapter III. Applications of Modulus Inequalities

Abstract
In this chapter we shall give our first applications of the inequalities for moduli of path families proved in the preceding chapter. Further applications will be given in Chapters IV, V, and VII. We start with some global distortion results and continue by proving, among other things, that a nonconstant qr mapping of ℝ n into itself omits at most a set of zero capacity. A local form of the latter result will be used in the proof of a Picard-type theorem in Chapter IV. Next, we shall establish a generalization of the theorem of V.A. Zorich which was mentioned in the introduction. In all these results Poletskiĭ’s KI-inequality is used. The rest of this chapter is devoted to local questions. There the sharper Väisälä’s inequality, or its capacity variant due to O. Martio, and the K O -inequality play essential roles for the problems treated.
Seppo Rickman

Chapter IV. Mappings into the n-Sphere with Punctures

Abstract
The main results of this chapter are a Picard-type theorem (Theorem 2.1) on omitted values and its variants. In dimension three it is known that this result is qualitatively best possible (Theorem 2.2). The proof of 2.2 is very technical and will not be presented here. We merely refer the reader to the article [R11]. In Section 3 we will give a quantitative growth estimate for mappings of the unit ball into the n-sphere with punctures, which delivers as a special case a counterpart to the Picard-Schottky theorem of classical function theory.
Seppo Rickman

Chapter V. Value Distribution

Abstract
Value distribution theory is concerned with how evenly a given mapping covers points. For example, the discussion in the preceding chapter of how many points in the target space can be omitted by a mapping falls under this heading.
Seppo Rickman

Chapter VI. Variational Integrals and Quasiregular Mappings

Abstract
Extremals of certain variational integrals, like the one appearing in the definition II.(10.1) of the capacity, serve in connection with the theory of qr mappings as counterparts for harmonic functions in the plane. Nonlinearity enters in the theory for dimensions n ≥ 3: the Euler—Lagrange equations for such variational integrals are not linear, but only quasilinear partial differential equations. For that reason methods familiar from the classical theory are for the most part not applicable to this nonlinear potential theory.
Seppo Rickman

Chapter VII. Boundary Behavior

Abstract
Many results of the boundary behavior of the theory of planar analytic functions have their counterparts for qr mappings for all dimensions n ≥ 2. In the early stage of the development of the theory some of these counterparts were established by O. Martio and S. Rickman [MR1]. Later M. Vuorinen continued this line of research in a number of articles, see 2.3 and 7.4. The tool in [MR1] and mostly in Vuorinen’s articles is the method of extremal length. S. Granlund, P. Lindqvist, and O. Martio introduced for a given variational kernel F in [GLM2] a substitute for the harmonic measure, called F-harmonic measure. This potential theoretic notion has turned out to be a useful tool when studying the properties of qr mappings near the boundary.
Seppo Rickman

Backmatter

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