弗兰克·摩根（Frank Morgan）是一位享誉国际的美国数学家，以其深厚的学术造诣和突破性的研究成果在数学界享有盛誉。摩根教授在哈佛大学完成了他的学士学位学习，随后在普林斯顿大学攻读硕士和博士学位，师从几何分析学家威廉·特劳布里奇（William H. Trotter）教授。目前，摩根教授担任美国威廉斯学院的数学教授，同时，他还曾在多所知名大学担任访问教授和客座教授，积累了丰富的数学教学和研究经验。他的研究兴趣广泛，聚焦于曲线和曲面的几何性质、测度论在高维空间中的创新应用，以及微分几何中的复杂变分问题。摩根教授的研究不仅深入理论层面，还积极探索其在多个领域的实际应用，展现了其跨学科的综合能力。

Preface

Singular geometry governs the physical universe: soap bubble clusters meeting along singular curves, black holes, defects in materials, chaotic turbulence, crystal growth. The governing principle is often some kind of energy minimization. Geometric measure theory provides a general framework for understanding such minimal shapes, a priori allowing any imaginable singularity and then proving that only certain kinds of structures occur.

Jean Taylor used new tools of geometric measure theory to derive the singular structure of soap bubble clusters and sea creatures, recorded by J. Plateau over a century ago (see Section 13.9). R. Schoen and S.-T. Yau used minimal surfaces in their original proof of the positive mass conjecture in cosmology, extended to a proof of the Riemannian Penrose Conjecture by H. Bray. David Hoffmanand his collaborators used modern computer technology to discover some of the ??rst new complete embedded minimal surfaces in a hundred years (Figure 6.1.3), some of which look just like certain polymers. Other mathematicians are now investigating singular dynamics, such as crystal growth. New software computes crystals growing amidst swirling ??uids and temperatures, as well as bubbles in equilibrium, as on the Chapter headings of this book. (See Section 16.8.)

In 2000, Hutchings, Morgan, Ritoré, and Ros announced a proof of the Double Bubble Conjecture, which says that the familiar double soap bubble provides the least-area way to enclose and separate two given volumes of air. The planar case was proved by my 1990 Williams College NSF “SMALL” undergraduate research Geometry Group [Foisy et al.]. The case of equal volumes in R3 was proved by Hass, Hutchings, and Schla??y with the help of computers in 1995. The general R3 proof was generalized to Rn by Reichardt. There are partial results in spheres, tori, and Gauss space, an important example of a manifold with density (see Chapters 18 and 19).

This little book provides the newcomer or graduate student with an illustrated introduction to geometric measure theory: the basic ideas, terminology, and results. It developed from my one-semester course at MIT for graduate students with a semester of graduate real analysis behind them. I have included a few fundamental arguments and a super??cial discussion of the regularity theory, but my goal is merely to introduce the subject and make the standard text, Geometric Measure Theory by H. Federer, more accessible.

Other good references include L. Simon’s Lectures on Geometric Measure Theory, E. Guisti’s Minimal Surfaces and Functions of Bounded Variation, R. Hardt and Simon’s Seminar on Geometric Measure Theory, Simon’s Survey Lectures on Minimal Submanifolds, J. C. C. Nitsche’s Lectures on Minimal Surfaces (now available in English), R. Osserman’s updated Survey of Minimal Surfaces, H. B. Lawson’s Lectures on Minimal Submanifolds, A. T. Fomenko’s books on The Plateau Problem, and S. Krantz and H. Parks’s Geometric Integration Theory. S. Hildebrandt and A. Tromba offer a beautiful popular gift book for your friends, reviewed by Morgan [14, 15]. J. Brothers and also Sullivan and Morgan assembled lists of open problems. There is an excellent Questions and Answers about Area Minimizing Surfaces and Geometric Measure Theory by F. Almgren [4], who also wrote a review [5] of the ??rst edition of this book. The easiest starting place may be the Monthly article “What is a Surface?” [Morgan 24].

It was from Fred Almgren, whose geometric perspective this book attempts to capture and share, that I ??rst learned geometric measure theory. I thank many students for their interest and help, especially Benny Cheng, Gary Lawlor, Robert McIntosh, Mohamed Messaoudene, Marty Ross, Stephen Ai, David Ariyibi, John Bihn, John Herrera, Nam Nguyen, and Gabriel Ngwe. I also thank typists Lisa Court, Louis Kevitt, and Marissa Barschdorf. Jim Bredt??rst illustrated an article of mine as a member of the staff of Link, a one-time MIT student newspaper. I feel very fortunate to have him with me again on this book. I am grateful for help from many friends, notably Tim Murdoch, Yoshi Giga and his students, who prepared the Japanese translation, and especially John M. Sullivan. I would like to thank my new editor, Graham Nisbet, production manager Poulouse Joseph, and my original editor and friend Klaus Peters. A ??nal thankyou goes to all who contributed to this book at MIT, Rice, Stanford, and Williams. Some support was provided by National Science Foundation grants, by my Cecil and Ida Green Career Development Chair at MIT, and by my Dennis Meenan and Webster Atwell chairs at Williams.

This ??fth edition includes updated material and references and a new Chapter20on the recently proved Log-Convex Density Theorem, one of many recent advances on manifolds with density and metric measure spaces.

Bibliographic references are simply by author’s name, sometimes with an identifying numeral or section reference in brackets. Following a useful practice of Nitsche [2], the bibliography includes cross-references to each citation.

Frank Morgan

Williamstown, MA

Contents

Preface vii

Part I: Basic Theory 1

1Geometric Measure Theory 3

2Measures 11

3Lipschitz Functions and Recti?able Sets 25

4Normal and Recti?able Currents 39

5The Compactness Theorem and the Existence of Area-Minimizing Surfaces 61

6Examples of Area-Minimizing Surfaces 69

7The Approximation Theorem 79

8Survey of Regularity Results 83

9Monotonicity and Oriented Tangent Cones 89

10The Regularity of Area-Minimizing Hypersurfaces 97

11Flat Chains Moduloν, Varifolds, and -Minimal Sets 105

12Miscellaneous Useful Results 111

Part II: Applications 119

13Soap Bubble Clusters 121

14Proof of Double Bubble Conjecture 143

15The Hexagonal Honeycomb and Kelvin Conjectures 159

16Immiscible Fluids and Crystals 173

17Isoperimetric Theorems in General Codimension 179

18Manifolds with Density and Perelman’s Proof of the Poincaré Conjecture 183

19Double Bubbles in Spheres, Gauss Space, and Tori 197

20The Log-Convex Density Theorem 205

Solutions to Exercises 213

Bibliography 235

Index of Symbols 255

Name Index 257

Subject Index 259