## Stability Theorems in Geometry and AnalysisThis 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 |

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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a e Q arbitrary point arclength Assign Assume ball B(a belongs bounded distortion bounded mean oscillation Chapter class J(d coefficient compact set compactly-supported completely integrable conformal mappings consequently converges Corollary cube Q defined definition denote derivatives domain of class equality estimate finite Fix an arbitrary follows function f function of class function with bounded generalised function hence homeomorphism implies included infer integral representation Lemma lemma is proved let f linear mapping Liouville's theorem Lp(U mapping f mapping of class mapping with bounded matrix-valued function Möbius transformation moreover obtain obviously open set orthogonal matrix orthogonal transformation p e Mn quasi-isometric quasiconformal mappings rectifiable curve result right-hand side satisfies segment sequence skew-symmetric matrix solution to system sphere stability starlike with respect Subsection Suppose Take an arbitrary Theorem 2.1 valid values vector-function whence zero