GEOMETRIC MEASURE THEORY

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.