Advanced Mathematics (Beijing/Boston) 1. Beijing: Science Press; Boston, MA: International Press (ISBN 7-03-010271-1/hbk; 1-57146-125-6/hbk). x, 237 p. (2002).
The authors undertake a wide-ranging introduction to geometric measure theory and its applications. Many nice topics and ideas are discussed in this book, but it has only 227 pages of text, so the coverage is often more a sketch than a detailed account. There are some minor inconsistencies of notation, but all of them can be deciphered. The authors are not native English speakers, so the usage is a bit unusual in places, but the reviewer always found the meaning clear enough.
The book is divided into eight chapters. The first chapter develops the theory of Hausdorff measure and related topics. A highlight is the remarkable theorem of J. Marstand that tells us that if a Radon measure has finite positive -density at almost all points, then is an integer. The second chapter concerns fine properties of functions and sets. A notable feature of this chapter is a section devoted to Federer’s dimension reduction principle, which is used in proving partial regularity results. Chapter 3 is about Lipschitz functions and rectifiable sets, and includes the structure theorem for sets of finite Hausdorff measure. Chapter 4 contains the proof of the area and co-area formulas and related results. Chapter 4 concludes with calculations of the formulas for the first and second variation of the area of a submanifold of the Euclidean space. The fifth chapter introduces results concerning functions of bounded variation. Chapter 6 develops the theory of varifolds including the regularity theory. Chapter 7 concerns the theory of rectifiable currents, both rectifiable currents with integer densities and those with real densities. Based on recent work of L. Ambrosio and B. Kirchheim [Acta Math. 185, No. 1, 1–80 (2000; Zbl 0984.49025)], the authors give a proof of the compactness theorem that does not require the structure theorem for sets of finite Hausdorff measure – even though the structure theorem was proved in Chapter 3. The eighth and final chapter sketches the regularity theory for mass-minimizing currents. The bibliography is excellent, but citations could have been more specific. The index is far from exhaustive.