*(English)*Zbl 0546.49019

For many years now, the standard reference in geometric measure theory has been *H. Federer*’s treatise, [”Geometric measure theory” (1969; Zbl 0176.00801)]. Federer’s book is so self-contained and complete that it has been difficult reading for newcomers to the field. They should welcome the author’s book. This is not to say that readers will find the author’s book easy going, either for geometric measure theory, and especially the regularity theory, is a very technical field relying on many delicate estimates. However, the author’s treatment will bring the reader more quickly to the heart of the matter. Very much missed will be the index and glossary of notation which was so helpful in Federer’s book.

Topics covered include the monotonicity formula, Allard’s regularity theorem, the theory of currents, regularity theory for codimension one area minimizing currents, and the theory of general varifolds. Appendices cover Federer’s dimension reducing argument for regularity theorems and the non-existence of stable minimizing cones of dimension n, $n\le 6$.