Classics in Mathematics Herbert Federer Geometric Measure Theory

Transcription

1 Classics in Mathematics Herbert Federer Geometric Measure Theory

2 Springer-Verlag Berlin Heidelberg GmbH

3 Herbert Federer was born on July 23, 1920, in Vienna. After emigrating to the US in 1938, he studied mathematics and physics at the University of California, Berkeley. Affiliated to Brown University, Providence since 1945, he is now Professor Emeritus there. The major part of Professor Federer's scientific effort has been directed to the development of the subject of Geometric Measure Theory, with its roots and applications in classical geometry and analysis, yet in the functorial spirit of modern topology and algebra. His work includes more than thirty research papers published between 1943 and 1986, as well as this book.

4 Herbert Federer Geometric Measure Theory Reprint of the 1969 Edition " Springer

5 Herbert Federer (Professor Emeritus) Department of Mathematics Brown University Providence, RI USA Originally published as VoI. 153 of the Grundlehren der mathematischen Wissenschaften Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Federer, Herbert: Geometric measure theory / Herbert Federer. - Reprint of the 1969 ed. - Berlin; Heidelberg ; New York; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris; Santa Clara; Singapore; Tokyo: Springer, 1996 (Grundlehren der mathematischen Wissenschaften ; VoI. 153) (Classies in mathematics) NE: 1. GT Mathematics Subject Classification (1991): 53C65, 46AXX ISBN DOI / ISBN (ebook) This work is subject to copyright. AlI rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustration, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provision of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-V erlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1996 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. SPIN / Printed on acid-free paper

6 Herbert Federer Geometric Measure Theory Springer-Verlag Berlin Heidelberg GmbH 1969

7 Herbert Federer Florence Pirce Grant University Proressor Brown University, Providence, Rhode Island Geschaftsftihrende Herausgeber: Prof. Dr. B, Eckmann Eidgenossische Technische Hochschule ZUrich Prof. Dr. B. L. van def Waerden Mathematisches lnstitut der UniversiHit ZUrich All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. by Springer-Verlag Berlin Heidelberg Library of Congress Catalog Card Number Title No. 5136

8 To my friends Charles Donald Shane Mary Lea Shane

9 Preface During the last three decades the subject of geometric measure theory has developed from a collection of isolated special results into a cohesive body of basic knowledge with an ample natural structure of its own, and with strong ties to many other parts of mathematics. These advances have given us deeper perception of the analytic and topological foundations of geometry, and have provided new direction to the calculus of variations. Recently the methods of geometric measure theory have led to very substantial progress in the study of quite general elliptic variational problems, including the multidimensional problem of least area. This book aims to fill the need for a comprehensive treatise on geometric measure theory. It contains a detailed exposition leading from the foundations of the theory to the most recent discoveries, including many results not previously published. It is intended both as a reference book for mature mathematicians and as a textbook for able students. The material of Chapter 2 can be covered in a first year graduate course on real analysis. Study of the later chapters is suitable preparation for research. Some knowledge of elementary set theory, topology, linear algebra and commutative ring theory is prerequisite for reading this book, but the treatment is selfcontained with regard to all those topics in multilinear algebra, analysis, differential geometry and algebraic topology which occur. The formal presentation of the theory in Chapters 1 to 5 is preceded by a brief sketch of the main theme in the Introduction, which contains also some broad historical comments. A systematic attempt has been made to identify, at the beginning of each chapter, the original sources of all relatively new and important material presented in the text. References to literature on certain additional topics, which this book does not treat in detail, appear in the body of the text. Some further related publications are listed only in the bibliography. All references to the bibliography are abbreviated in square brackets; for example [C 1] means the first listed work byc.caratheodory. The index is supplemented by a list of basic notations defined in the text, and a glossary of some standard notations which are used but not defined in the text.

10 VIII Preface I wish to thank Brown University and the National Science Foundation for supporting my work on this book, and to express appreciation of the efforts of my fellow mathematicians who helped with the project. Frederick J. Almgren Jr. and I had many stimulating discussions, in particular about his ideas presented in Section 5.3. Casper Goffman posed some interesting questions which inspired part of Katsumi Nomizu showed me an elegant treatment of William K. Allard read the whole manuscript with great care and contributed significantly, by many valuable queries and comments, to the accuracy of the final version. John E. Brothers, Lawrence R. Ernst, Josef Knil, Arthur Sard and William P.Ziemer read parts of the manuscript and supplied very useful lists of errata. The editors and personnel of Springer-Verlag have been unfailingly cooperative in all phases of publication of this book. I am grateful, in particular, to David Mumford for inviting me to contribute my work to the Grundlehren series, and to Klaus Peters for planning all the necessary arrangements with utmost consideration. Providence, Rhode Island January 1969 Herbert Federer

11 Contents Introduction CHAPTER ONE Grassmann algebra 1.1. Tensor prod ucts Graded algebras The exterior algebra of a vectors pace 1.4. Alternating forms and duality 1.5. Interior multiplications 1.6. Simple m-vectors. I. 7. Inner products 1.8. Mass and comass 1.9. The symmetric algebra of a vectors pace Symmetric forms and polynomial functions Measures and measurable sets Numerical summation Measurable sets Measure hulls Ulam numbers. CHAPTER TWO General measure theory Borel and S uslin sets Borel families Approximation by closed subsets Nonmeasurable sets Radon measures The space of sequences of positive integers Lipschitzian maps O S uslin sets Borel and Baire functions Separability of supports Images of Radon measures Measurable functions Basic properties Approximation theorems Spaces of measurable functions

12 x Contents 2.4. Lebesgue integration Basic properties Limit theorems O Integrals over subsets Lebesgue spaces Compositions and image measures Jensen's inequality 2.5. Linear functionals Lattices of functions Daniell integrals Linear functionals on Lebesgue spaces Riesz's representation theorem Curve length Riemann-Stieltjes integration Spaces of Daniell integrals Decomposition of Daniell integrals 2.6. Product measures Fubini's theorem Lebesgue measure Infinite cartesian products Integration by parts 2.7. Invariant measures Definitions Existence and uniqueness of invariant integrals Covariant measures are Radon measures Examples Nonmeasurable sets Ll continuity of group actions 2.8. Covering theorems Adequate families Coverings with enlargement Centered ball coverings Vitali relations Derivates Existence of derivates Indefinite integrals. Density and approximate continuity Additional results on derivation using centered balls Derivatives of curves with finite length O. Caratheodory's construction The general construction The measures J'fm, gm, g-m, <;gm, <{jm, J,m,!!l.~ Relation to Riemann-Stieltjes integration Partitions and multiplicity integrals Curve length Integralgeometric measures Densities Remarks on approximating measures

13 Contents Spaces of Lipschitzian functions and closed subsets Approximating measures of increasing sequences Direct construction of the upper integral Integrals of measures of counter images Sets of Cantor type Steiner symmetrization Inequalities between basic measures Lipschitzian extension of functions Cartesian products Subsets of finite Hausdorff measure XI CHAPTER THREE Rectifmbility 3.1. Differentials and tangents Differentiation and approximate differentiation Higher differentials Partitions of unity Differentiable extension of functions Factorization of maps near generic points Submanifolds of Euclidean space Tangent vectors Relative differentiation Locaillattening of a submanifold Analytic functions Area and coarea of Lipschitzian maps lacobians Area of maps of Euclidean spaces Co area of maps of Euclidean spaces Applications; Euler's function r Rectifiable sets Approximate tangent vectors and differentials Area and coarea of maps of rectifiable sets Cartesian products Equality of measures of rectifiable sets Areas of projections of rectifiable sets Examples Rectifiable sets and manifolds of class Further results on coarea Steiner's formula and Minkowski content Brunn-Minkowski theorem Relations between the measures f2~ Hausdorff measures in Riemannian manifolds Integralgeometry on spheres Structure theory Tangential properties of arbitrary Suslin sets Rectifiability and projections Examples of unrectifiable sets Rectifiability and density

14 XII Contents 3.4. Some properties of highly differentiable functions Measures ofj{x: dim im DJ(x)~v} Analytic varieties 318 CHAPTER FOUR Homological integration theory 4.1. Differential forms and currents Distributions Regularization Distributions representable by integration Differential forms and m-vectorfields Currents Cartesian products Homotopies Joins, oriented simplexes Flat chains Relation to integralgeometry measure Polyhedral chains and flat approximation Rectifiable currents Lipschitz neighborhood retracts Transformation formula Oriented submanifolds Projective maps and polyhedral chains Duality formulae Lie product of vectorfields Deformations and compactness Slicing normal currents by real valued functions Maps with singularities Cubical subdivisions Deformation theorem Isoperimetric inequality Flat chains and integralgeometric measure Closure theorem Compactness theorem Approximation by polyhedral chains Indecomposable integral currents Flat chains modulo v Locally rectifiable currents Analytic chains Slicing Slicing flat chains by maps into R" Homotopies, continuity of slices Slicing by maps into manifolds Oriented cones Oriented cylinders Oriented tangent cones Intersections of flat chains Homology groups Homology theory with coefficient group Z Isoperimetric inequalities 466

15 Contents XIII Compactness properties of homology classes Homology theories with coefficient groups Rand Zv Two simple examples Homotopy groups of cycle groups Cohomology groups Normal currents of dimension n in R" Sets with locally finite perimeter Exterior normals Gauss-Green theorem Functions corresponding to locally normal currents Densities and locally finite perimeter Examples and applications CHAPTER FIVE Applications to the calculus of variations 5.1. Integrands and minimizing currents Parametric integrands and integrals Ellipticity of parametric integrands Convexity, parametric Legendre condition Diffeomorphic invariance of ellipticity Lowersemicontinuity of the integral Minimizing currents Isotopic deformations, variations Nonparametric integrands Nonparametric Legendre condition Euler-Lagrange formulae Regularity of solutions of certain differential equations L2 and Holder conditions Strongly elliptic systems Sobolev's inequality Generalized harmonic functions Convolutions with essentially homogeneous functions Elementary solutions HOlder estimate for linear systems Nonparametric variational problems Maxima of real valued solutions One dimensional variational problems 5.3. Excess and smoothness Estimates involving excess A limiting process The decrease of excess Regularity of minimizing currents Minimizing currents of dimension m in R.,+I Minimizing currents of dimension 1 in R n Minimizing flat chains modulo v 5.4. Further results on area minimizing currents Terminology Weak convergence of variation measures

16 XIV Contents O Density ratios and tangent cones Regularity of area minimizing currents Cartesian products Study of cones by differential geometry Currents of dimension m in R m Lack of uniqueness and symmetry Nonparametric surfaces, Bernstein's theorem Holomorphic varieties Boundary regularity Bibliography. 655 Glossary of some standard notations List of basic notations defined in the text Index