Quasiregular Mappings (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 26)

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. Ch. I. Basic Properties Of Quasiregular Mappings -- 1. Acl[superscript P] Mappings -- 2. Quasiregular Mappings -- 3. Examples -- 4. Discrete Open Mappings -- Ch. Ii. Inequalities For Moduli Of Path Families -- 1. Modulus Of A Path Family -- 2. The K[subscript O]-inequality -- 3. Path Lifting -- 4. Linear Dilatations -- 5. Poletskii's Lemma -- 6. Characterizations Of Quasiregularity -- 7. Proof Of Poletskii's Lemma -- 8. Poletskii's Inequality -- 9. Vaisala's Inequality -- 10. Capacity Inequalities -- Ch. Iii. Applications Of Modulus Inequalities -- 1. Global Distortion -- 2. Sets Of Capacity Zero And Singularities -- 3. The Injectivity Radius Of A Local Homeomorphism -- 4. Local Distortion -- 5. Bounds For The Local Index -- Ch. Iv. Mappings Into The N-sphere With Punctures -- 1. Coverings Averages -- 2. The Analogue Of Picard's Theorem -- 3. Mappings Of A Ball -- Ch. V. Value Distribution -- 1. Defect Relation -- 2. Coverings And Decomposition Of Balls -- 3. Estimates On Liftings -- 4. Extremal Maximal Sequences Of Liftings -- 5. Effect Of The Defect Sum On The Liftings -- 6. Completion Of The Proof Of Defect Relations -- 7. Mappings Of The Plane -- 8. Order Of Growth -- 9. Further Results -- Ch. Vi. Variational Integrals And Quasiregular Mappings -- 1. Extremals Of Variational Integrals -- 2. Extremals And Quasiregular Mappings -- 3. Growth Estimates For Extremals -- 4. Differentiability Of Quasiregular Mappings -- 5. Discreteness And Openness Of Quasiregular Mappings -- 6. Pullbacks Of General Kernels -- 7. Further Properties Of Extremals -- 8. The Limit Theorem -- Ch. Vii. Boundary Behavior -- 1. Removability -- 2. Asymptotic And Radial Limits -- 3. Continuity Results And The Reflection Principle -- 4. The Wiener Condition -- 5. F-harmonic Measure -- 6. Phragmen-lindelof Type Theorems -- 7. Asymptotic Values. Seppo Rickman. Includes Bibliographical References (p. [198]-207 And Index. 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 Front Matter....Pages I-X Introduction....Pages 1-3 Basic Properties of Quasiregular Mappings....Pages 4-23 Inequalities for Moduli of Path Families....Pages 24-58 Applications of Modulus Inequalities....Pages 59-77 Mappings into the n -Sphere with Punctures....Pages 78-93 Value Distribution....Pages 94-128 Variational Integrals and Quasiregular Mappings....Pages 129-161 Boundary Behavior....Pages 162-197 Back Matter....Pages 198-213 An introduction to the theory of quasiregular mappings in real n-dimensional space, a new field of mathematical study that has emerged during the past 20 years. A broad spectrum of results of both analytic and geometric character are presented, and the methods vary accordingly.