Stability Theorems in Geometry and Analysis

Transcription

1 Stability Theorems in Geometry and Analysis

2 Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 304

3 Stability Theorems in Geometry and Analysis by Yu. G. Reshetnyak Institute of Mathematics, Russian Academy of Sciences, Siberian Branch, Novosibirsk, Russia SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

4 Library of Congress Cataloging-in-Publication Data Reshetn rak, fur i1 Gr igor 'ev i ch. [TeoreNY usto1chivosti v geo.etrii 1 analize. Engllsh] Stabillty theorems 1n geolletry and analysis / by Yu. G. Reshetnyak. p. CN. -- (Mathematics and its applications; v. 304) Includes bibliographlcal references and index. ISBN ISBN (ebook) DOI / Geolletry, Dlfferentlal. 2. Stabil1ty. 1. T1tle. II. Serles: Mathematlcs and its applicatlons (Kluwer Acadeillc Publishers) ; v QA641.R '6--dc20 ISBN CIP This is a revised and updated translation of the original work Stability Theorems in Geometry and Analysis, translated from the Russian by N, S. Dairbekov and V. N. Dyatlov, and edited by S. S, Kutateladze. Novosibirsk, Nauka 1983 Printed on acid-free paper All Rights Reserved 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 Softcover reprint of the hardcover 1st edition 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

5 Contents Foreword to the English Translation Preface to the First Russian Edition Chapter 1. Introduction 1. Preliminaries, Notation, and Terminology Orthogonal, Mobius, and Pseudo-Orthogonal Transformations A Survey of the Content and Statement of the Main Results 18 Chapter 2. Mobius Transformations 1. Hypersphere Bundle and Linear Representations of Mobius Transformations Differential-Geometric Characterisation of Motions and Mobius Transformations Topologies on Groups of Mobius Transformations. Infinitesimal Operators for Groups of Mobius Transformations. Infinitesimal Operators for Groups of Mobius Transformations Proof of Liouville's Theorem Under Minimal Hypotheses on Smoothness Chapter 3. Integral Representations and Estimates for Differentiable Functions 1. Representations of Functions by Means of Operators Enjoying the Condition of Complete Integrability Integral Representations by Means of Special Differential Operators Some Estimates for Operators Ql and Q Some Classes of Domains in]rn Some Auxiliary Function Classes Integral Representations of Differentiable Functions in Domains with Nonsmooth Boundary vii ix

6 VI Contents Chapter 4. Stability in Liouville's Theorem on Conformal Mappings in Space 1. Mappings with Bounded Distortion A Preliminary Theorem on Stability in Liouville's Theorem Local Estimates for Stability in Liouville's Theorem Global Stability in Liouville's Theorem Additional Remarks Chapter 5. Stability of Isometric Transformations of the Space ]Rn 1. Stability with Respect to the Uniform Norm Stability Estimates for Isometric Transformations in the Class Wi Chapter 6. Stability in Darboux's Theorem 1. Principal Results Proofs of Theorems Chapter 7. Differential Properties of Mappings with Bounded Distortion and Conformal Mappings of Riemannian Spaces 1. Formulation of Results and Auxiliary Facts Proof of Theorem Proof of Theorem The Image of a Submanifold Under a Mapping Conformal at the Points of the Submanifold References 383 Subject Index 392

7 Foreword to the English Translation The proposed book presents one of the actual trends in modern mathematics. It deals with the metric theory of spatial mappings and incorporates results in the theory of quasi conformal, quasi-isometric and other mappings. Solving problems in the field requires a broad arsenal of contemporary mathematical tools and techniques. The bulk of the book is devoted to thoroughly exposing the author's results on the stability problem in Liouville's theorem on conformal mappings in space. According to this theorem, the set constituted by all conformal mappings of a domain in space is exhausted by the so-called Mobius transformations which form a finite-dimensional Lie group. The stability problem consists in proving that any mapping for which the condition of conformality is satisfied within accuracy less than c; differs from an appropriate Mobius transformation by a quantity not exceeding c5(c;) = O(c;) (with the point being temporarily obtained of how the indicated discrepancy and measure of deviation from a Mobius transformation are defined). Verifying that c5( c;) = 0 as c; = 0 is relatively immediate. The main difficulty is in establishing that in effect one may assume that c5(c;) = O(c;) as c; ---+ o. There exist other problems stated in a similar fashion; and, therefore, the problem of stability in Liouville's theorem is just a representative of a number of problems on stability for transformation classes. The solution to the problem had required the development of a theory whose separate fragments are of independent importance. In particular, solving the stability problem for the Liouville theorem employs a specific technique for constructing integral representations of functions and deriving, via them, inequalities of Korn type similar to those from elasticity theory; theorems on semi continuity and convergence with a functional in the calculus of variations; and theorems concerning properties of functions in BMO classes and their analogues.

8 Vlll Foreword to the English Translation In addition the English version of the book incorporates the results that are attributed to other authors and devoted to parallel investigations, in particular, the F. John theorem on quasi-isometric mappings. Proving these results follows the scheme expounded by the author in solving the stability problem for the Liouville theorem. In the world literature, any monographs addressing the question that is considered in the book are absent. The main results of the latter have been exposed only in periodicals, and by technical reasons (the language barrier being one of them) are relatively nonavailable to readers outside Russia. The only book tackling close topics is that by A. P. Kopylov, "Stability of Mapping Classes"; its overlapping with the present monograph is negligible. The English edition of the book contains a survey of research either treating relevant questions or involving the technique that is displayed in the book. The style of the Russian exposition is very austere, succinct and magically illuminative. In translating, my colleagues and I tried our best to preserve the rare particularity. S. S. Kutateladze

9 Preface to the First Russian Edition Uniqueness theorems are plentiful in differential geometry. Such a theorem claims that some class of geometric objects is uniquely characterised by a definite property common to all the elements of the class. An example of the situation is given by the classical Darboux theorem: each spatial surface with all the points umbilical is a part of a sphere or a plane. Another example is provided by the Liouville theorem on conformal mappings in space. Visually, a mapping from a domain in n dimensional Euclidean space is conformal whenever it transforms each infinitesimal sphere into an infinitesimal sphere. Mobius transformations, i.e. compositions of finitely many inversions with respect to a sphere, serve as the instances of conformal mappings. The Liouville theorem asserts that every conformal mapping in space is Mobius. Say that stability resides in a uniqueness theorem granted the following claim is valid: Suppose that, for a given geometric object, the hypotheses of the uniqueness theorem are satisfied approximately. Then, in the large, the object in question differs slightly from those described by the uniqueness theorem. The "approximate" satisfaction, or, as it is referred to, satisfaction within c, where c > 0 is small, of the hypothesis of the uniqueness theorem may be understood in many senses. Closeness to the object of a certain class may also be defined in various fashions. Therefore, different stability problems are possible in relation to a specific uniqueness theorem. The principal results of the book treat the stability problem for the Liouville theorem on conformal mappings in space. The problem is considered in the formulation due to M. A. Lavrent'ev and basing on the concept of a mapping with bounded distortion. Roughly speaking, such a mapping is characterised by the properties that it is orientation-preserving and transforms every infinitesimal ball into an infinitesimal ellipsoid with the ratio between the major and minor semi axes not exceeding some constant K ~ O. In the case K = 1, the mapping is Mobius.

10 x Preface to the First Russian Edition The stability problem for the Liouville theorem consists in proving that, as K tends to 1, the mapping closely approaches a Mobius one and in estimating the order of discrepancy between the mapping and the set of Mobius transformations in terms of the quantity K - 1. The articles [12,13,68,95,99,102, ) are devoted to the problem. The solution to the stability problem in the Liouville theorem, presented by the book, incorporates all the known results in this direction. Together with studying the deviation of the given mapping, we consider the deviation of respective derivatives (apropos, with the estimates of the latter deviation available, it is possible, via the embedding theorems, to obtain the estimates of deviation for the mapping). For the initial mapping, we have the next result: the deviation, of a mapping with bounded distortion, from a Mobius one is the quantity of order C(K - 1) (C constant) in the uniform metric. As far as derivatives are concerned, we can say that they differ from the derivatives of a Mobius mapping by the quantity of the same order but the distance between the derivatives is, however, measured in the metric of L p, where K < Cj(K - 1). By way of deliberation we infer some information on the behavior of derivatives of a mapping with bounded distortion for K close to 1. In a general case, the indicated derivatives are discontinuous and, leaning on the definition of a mapping with bounded distortion, we may conclude only that they are locally integrable to the nth power (n standing for the dimension of the space considered). Therein we exhibit more delicate properties of the derivatives; namely, it turns out that the derivatives of a mapping with bounded distortion are locally integrable to the power p < C(K - 1) and, besides, the order, as K -t 1, of the stated exponent of summability, p, is exact. As an application of the stability theorem in Liouville's theorem, we formulate some results on differentiability of quasi conformal mappings and conformal mappings of Riemannian spaces. In proving stability in Liouville's theorem we employ results from the theory of mappings with bounded distortion. For convenience, all necessary prerequisites for the mappings are incorporated into the contents of the book; their justifications can be found, for instance, in (110). Furthermore, we apply integral representations of functions via differential operators; the technique for their constructing presented in the book is grounded on the concept of a completely integrable system of differential operators and is of interest by itself. This technique is applicable to other situations; for instance, it provides an opportunity to derive the classical integral representation, of a function via derivatives, established by S. L. Sobolev and playing an important role in the theory of functions with weak (= generalised) derivatives. Finally, solving the stability problem, we make use of convergence

11 Preface to the First Russian Edition Xl theorems for functionals in the calculus of variations and, besides, properties of some function class which is analogous to the class of functions with bounded mean oscillation. Another problem addressed by the book is the stability problem in the Darboux theorem. The question of stability for this theorem was also raised by M. A. Lavrent'ev. In the book we expose a general result, due to S. K. Vodop'yanov, which renders a solution to the Lavrent'ev problem. Outside the frames of the book lie F. John's interesting results [56) on stability of isometric transformations, as well as those of L. G. Gurov [42-45) on stability of Lorentz mappings; the exposition of the results involves an additional analytic technique similar to that presented. The book also leaves aside one of the most noticeable stability theorems in differential geometry, the Yu. A. Volkov theorem, which contains estimates of form deviation for a convex surface whose intrinsic metric undergoes changes. The proof of the theorem requires a technique principally different from that developed in the book. Supplement to the English Translation Yu. G. Reshetnyak The author seizes the opportunity to express his sincere gratitude to all the colleagues who lent him a hand with preparation of the English translation of the book. Yu. G. Reshetnyak