Geometric Measure Theory: A Beginner's Guide

This is the fifth edition of an introductory text for graduate students. Morgan describes geometric measure theory as “differential geometry, generalized through measure theory to deal with maps and surfaces that are not necessarily smooth, and applied to the calculus of variations”. He calls the book an illustrated introduction, and his treatment is friendly enough (though still rigorous) that it does not intimidate. (Immediate intimidation was the effect on me some years ago by Federer’s book on the same subject.)

It’s still not an easy subject. But Morgan’s treatment skillfully draws on the reader’s intuition, and uses historical connections to develop the critical ideas and place them in a broader context. Morgan is very good at explaining how all the pieces fit together, what matters most and why the difficult parts are difficult. He is adept at using pictures to explain the importance of the main ideas. His focus is on geometry. Most of the hard results (largely analytical in character) are not proved here.

The book is divided into two parts of about equal length. The first part develops the basic theory, and the second takes up applications. Using the archetypal example of finding the surface of least area in \(\mathbb{R}^n\) with a given boundary, Morgan describes how an approach using the concept of rectifiable currents gives a natural topology on the space of surfaces that is dual to an appropriate topology on differential forms. A key result of the first part is the existence of area-minimizing surfaces in \(\mathbb{R}^n\). Its proof depends on Federer’s compactness theorem, which Federer proved using his own amazing result on the structure of arbitrary subsets of \(\mathbb{R}^n\).

A highlight of the second part is a sketch of the proof in \(\mathbb{R}^3\) of the Double Bubble conjecture: the familiar standard double soap bubble consisting of three spherical caps meeting at 120 degrees along a common circle encloses two prescribed volumes in \(\mathbb{R}^3\) with the minimum surface area. (An analogous result holds in \(\mathbb{R}^n\).)

Other conjectures and failed conjectures appear here too. The Hexagonal Honeycomb Conjecture says that regular hexagons provide the most efficient (least perimeter) way to divide the plane into unit areas. Thomas Hales proved the result in 1999, and a sketch is given here; its proof depends on a fairly intricate isoperimetric inequality. A failed conjecture due to Lord Kelvin states that truncated octahedrons partition three-dimensional space into regions of unit volumes with the least surface area. Weaire and Phelan, inspired by the natural crystal structure of a clathrate compound, found a counterexample in 1999. It beats Kelvin’s idea by a bare 0.3%.

Two other topics here might be of potentially broader interest. Both concern the current hot topic of manifolds with density. The first is a brief treatment of the use of this idea by Perelman in his proof of the Poincaré conjecture. The second is Chambers log-convex density theorem: balls about the origin are isoperimetric for a smooth radial density if and only if the log of the density is convex.

Morgan’s book offers the best access I know into this difficult subject. It won’t make the reader an expert, but it does open the door to more detailed treatments. An excellent bibliography is provided.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.