Many years ago H. Federer's book on geometric measure theory appeared [Geometric measure theory, Springer, Berlin, 1969; MR0257325] and immediately became a reference for those working in and interested in the field. Its very size and comprehensiveness lent it more the nature of a reference book than a textbook. In the meantime there has also been considerable progress in the field, including deeper regularity theorems and a closer connection with differential-geometric and analytic objects. The book under review, which the author says is a preliminary version of a more complete book he hopes to write, is both an introduction to geometric measure theory and related variational problems and to the regularity theory. The exposition is clear throughout (at least as clear as can be hoped for in a field as technical as this); although it is not as self-contained as Federer's book, some basic measure theory is reviewed at the beginning. Unfortunately, there is neither an index nor a list of notations, which makes it difficult to read only the portions in which one is interested. In addition to Federer's book, W. Allard's paper [Ann. of Math. (2) 95 (1972), 417–491; MR0307015] has influenced the content of this volume, but much of the treatment is new. In particular, Allard's regularity theorem is discussed in the special case of rectifiable varifolds, which makes the ideas in the proof much more transparent. In sum, the book should assist people to become better acquainted with this area of mathematics. The main topics are the area and co-area formulae, the monotonicity formulae, Allard's regularity theorem, the regularity theory for codimension one area-minimizing currents, and the theory of general varifolds.
Reviewer: Joel, J. S. [form MathSciNet]