Stability Theorems in Geometry and Analysis

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Springer Science & Business Media, Apr 9, 2013 - Mathematics - 394 pages
This is one of the first monographs to deal with the metric theory of spatial mappings and incorporates results in the theory of quasi-conformal, quasi-isometric and other mappings. The main subject is the study of the stability problem in Liouville's theorem on conformal mappings in space, which is representative of a number of problems on stability for transformation classes. To enable this investigation a wide range of mathematical tools has been developed which incorporate the calculus of variation, estimates for differential operators like Korn inequalities, properties of functions with bounded mean oscillation, etc. Results obtained by others researching similar topics are mentioned, and a survey is given of publications treating relevant questions or involving the technique proposed. This volume will be of great value to graduate students and researchers interested in geometric function theory.
 

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Contents

Möbius Transformations
63
Integral Representations and Estimates
106
Integral Representations by Means of Special Differential
129
Some Estimates for Operators Qi and Q2
145
Some Classes of Domains in R
156
Some Auxiliary Function Classes
175
Integral Representations of Differentiable Functions in Domains
194
Stability in Liouvilles Theorem on Conformal
204
Additional Remarks
295
Stability Estimates for Isometric Transformations in
318
Stability in Darbouxs Theorem
336
Proofs of Theorems 1 11 3
343
Differential Properties of Mappings with Bounded
356
Proof of Theorem 1 1
365
Proof of Theorem 1 2
373
References
383

A Preliminary Theorem on Stability in Liouvilles Theorem
237
Local Estimates for Stability in Liouvilles Theorem
273
Global Stability in Liouvilles Theorem
283

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