*(English)*Zbl 1179.49050

The geometric measure theory refers to the study of the generalized $k$-dimensional surfaces (integral currents) in ${\mathbb{R}}^{n}$ and their compactness and approximation properties. They make integral currents suitable for use in employing the so-called direct method of the calculus of variations for studying geometrical problems such as the problem of least area. This well-written book was developed from the author’s one-semester course at MIT for graduate students. The first goal was to provide an introduction to the fundamentals of the geometric measure theory and to make the standard text on the subject by *H. Federer* [Geometric measure theory. Berlin-Heidelberg-New York: Springer-Verlag (1969; Zbl 0176.00801)] more accessible. The second goal was to present applications of geometric measure theory to certain geometrical problems in the calculus of variations. The results of the author, Federer, Fleming, Almgren, and others on regularity of solutions to the least area problem and soap bubble clusters are outlined. Proofs of the double bubble, the hexagonal honeycomb, and Kelvin conjectures are discussed. A brief sketch of Almgren’s proof of the general isoperimetric inequality in Euclidean spaces is given.

This fourth edition includes updated material and references, recent results on planar soap films, and new chapters. Manifolds with density and Perelman’s proof of the Poincaré conjecture, and double bubbles in spheres, Gauss space, and tori are discussed. The author presents all main results without detailed proofs but the fundamental arguments are always given. Beautiful illustrations help readers understand basic concepts easier. Exercises with solutions which follow each chapter, and the author’s lively narrative make this book a real pleasure to read.

##### MSC:

49Q15 | Geometric measure and integration theory, integral and normal currents (optimization) |

49-01 | Textbooks (calculus of variations) |

49Q05 | Minimal surfaces (calculus of variations) |

49Q20 | Variational problems in a geometric measure-theoretic setting |

53-01 | Textbooks (differential geometry) |

53C42 | Immersions (differential geometry) |