Softcover reprint of the original 1st ed. 1993, X, 213 pp. 5 figs.
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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.
Content Level »Research
Keywords »Extremal Length - Extremale Länge - Nichtlineare Potentialtheorie - Nonlinear Potential Theory - Quasiconformal Mappings - Quasikonforme Abbildungen - Quasiregular Mappings - Quasiregulare Abbildungen - Value Distribution - Wertverteilung
I. Basic Properties of Quasiregular Mappings.- 1. ACLp Mappings.- 2. Quasiregular Mappings.- 3. Examples.- 4. Discrete Open Mappings.- II. Inequalities for Moduli of Path Families.- 1. Modulus of a Path Family.- 2. The KO-Inequality.- 3. Path Lifting.- 4. Linear Dilatations.- 5. Poletski?’s Lemma.- 6. Characterizations of Quasiregularity.- 7. Proof of Poletski?’s Lemma.- 8. Poletski?’s Inequality.- 9. Väisälä’s Inequality.- 10. Capacity Inequalities.- 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.- IV. Mappings into the n-Sphere with Punctures.- 1. Coverings Averages.- 2. The Analogue of Picard’s Theorem.- 3. Mappings of a Ball.- 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.- 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.- 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. Phragmén-Lindelöf Type Theorems.- 7. Asymptotic Values.- List of Symbols.