Geometric Measure Theory
Herbert FedererDuring 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 varia-
tions. 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 geo-
metric 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 re-
search. Some knowledge of elementary set theory, topology, linear algebra
and commutative ring theory is prerequisite for reading this book, but
the treatment is selfconthined with regard to all those topics in multi-
linear 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 addi-
tional 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 by C. 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.