Geometric measure theory. A beginner’s guide.

*(English)*Zbl 0671.49043
Boston, MA etc.: Academic Press, Inc. viii, 145 p. $ 19.95 (1988).

Any mathematician who knows H. Federer’s monograph “Geometric measure theory” (1969; Zbl 0176.008) will welcome this little book. It is subtitled “A beginner’s guide” and this is exactly what it is. The author’s aim is to make Federer’s book more assessible instead of writing an alternative.

He introduces the relevant terminology and describes the basic ideas of geometric measure theory. The book is divided into twelve chapters. The individual titles are: Geometric measure theory; Measures; Lipschitz functions and rectifiable sets; Normal and rectifiable currents; The compactness theorem and the existence of area-minimizing surfaces; Examples of area minimizing surfaces; The approximation theorem; Survey of regularity results; Monotonicity and oriented tangent cones; The regularity of area-minimizing hypersurfaces; Flat chains modulo \(\nu\) ; Varifolds; (\(\underset {=} M,\epsilon,\delta)\)-minimal sets; and Miscellaneous useful results.

The main results of course appears without detailed proofs but the author always succeeds in giving the fundamental arguments. A large number of beautiful illustrations helps the reader to get the right intuition and makes it easier for him not to get confused by the necessary definitions. Another advantage of this book are the exercises (with solutions!) which follow each chapter, and the author’s lively style adds to the reader’s pleasure.

Anyone interested in geometry and analysis will certainly enjoy this wonderful book.

He introduces the relevant terminology and describes the basic ideas of geometric measure theory. The book is divided into twelve chapters. The individual titles are: Geometric measure theory; Measures; Lipschitz functions and rectifiable sets; Normal and rectifiable currents; The compactness theorem and the existence of area-minimizing surfaces; Examples of area minimizing surfaces; The approximation theorem; Survey of regularity results; Monotonicity and oriented tangent cones; The regularity of area-minimizing hypersurfaces; Flat chains modulo \(\nu\) ; Varifolds; (\(\underset {=} M,\epsilon,\delta)\)-minimal sets; and Miscellaneous useful results.

The main results of course appears without detailed proofs but the author always succeeds in giving the fundamental arguments. A large number of beautiful illustrations helps the reader to get the right intuition and makes it easier for him not to get confused by the necessary definitions. Another advantage of this book are the exercises (with solutions!) which follow each chapter, and the author’s lively style adds to the reader’s pleasure.

Anyone interested in geometry and analysis will certainly enjoy this wonderful book.

Reviewer: M.Grüter

##### MSC:

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

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

49Q05 | Minimal surfaces and optimization |

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

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |