The Geometry of Domains in Space
The analysis of Euclidean space is well-developed. The classical Lie groups that act naturally on Euclidean space-the rotations, dilations, and trans lations-have both shaped and guided this development. In particular, the Fourier transform and the theor
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Edited by Herbert Amman, University of Zurich Ranee Brylinski, Penn State University
Steven G. Krantz Harold R. Parks
The Geometry of Domains in Space
Springer-Science+Business Media, LLC
Steven G. Krantz Department of Mathematics Washington University SI. Louis, MO 63130
Harold R. Parks Department of Mathematics Oregon State University Corvallis, OR 97331
Library of Congress Cataloging-in-Publication Data Krantz, Steven G. (Steven George), 1951The geometry of domains in space / Steven G. Krantz, Harold R. Parks. p. cm. - (Birkhäuser Advanced Texts) Includes bibliographical references and index. ISBN 978-1-4612-7199-4 ISBN 978-1-4612-1574-5 (eBook) DOI 10.1007/978-1-4612-1574-5 1. Mathematical analysis. 2. Geometry. I. Parks, Harold R., 1949II. Title. III. Series: Birkhäuser Advanced Texts (Boston, Mass.) QA300.K644 1999 98-44619 515---dc21
CIP AMS Subject Classifications: 26-01, 26-02, 26A24, 26A45, 26A51, 26B05, 26B15, 26B20, 26B25, 26B30, 26B35, 28-01, 28A05, 25A12M, 28A15, 28A25, 28A75, 28A78, 30C62, 30C65, 33B15, 35105, 46E20, 46E30, 46E35, 49K20, 49R05, 51M25, 52A20, 53-01, 53A04, 53A05, 53AlO, 52A20, 53B20, 53C45 Printed on acid-free paper. © 1999 Springer Science+Business Media New York Origina11y published by Birkhäuser Boston in 1999 Softcover reprint of the hardcover 1st edition 1999 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, eic., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely byanyone. ISBN 978-1-4612-7199-4
Typeset by the author in U\TEX.
9 8 765 432 1
To Herbert Federer
Contents Preface 1.
ix
Elementary Topics
1
1.1 Smooth Functions ........................................... 1 1.2 The Concept of Defining Function ........................... 8 1.3 Measure Theory ............................................ 15 2.
3.
Domains with Smooth Boundaries
27
2.1
The Tangent Bundle and Normal Bundle of the Boundary .......................................... 27
2.2 2.3
The Second Fundamental Form and Curvature .............. 34 Surfaces with Constant Mean Curvature .................... 45
Measures 57 3.1 The CaratModory Construction ............................ 57
3.2 3.3 3.4
Rectifiability ............................................... 66 Minkowski Content ......................................... 74 A Space-Filling Curve ...................................... 81
3.5
Covering Lemmas .......................................... 83
3.6
Functions of Bounded Variation ..
