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Almgren’s big regularity paper. $Q$-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2. Edited by V. Scheffer and Jean E. Taylor. (English) Zbl 0985.49001
World Scientific Monograph Series in Mathematics. 1. Singapore: World Scientific. xvi, 955 p. \$ 147.00; £51.00 (2000).

This is the first time Almgren’s fundamental regularity result for area-minimizing surfaces of codimension greater than one becomes available to the wider mathematical community. Written over a period of more than ten years it originally appeared in 1984 as a preprint (1728 pages) consisting of three volumes each resembling a large telephone directory. Shortly after Almgren’s untimely death in 1997 Vladimir Scheffer (his third doctoral student) began the monumental task of converting the typed manuscript into files which finally resulted in the publication of this impressive book.

Its main object is to establish the following regularity result in the calculus of variations: A mass-minimizing $m$-dimensional rectifiable current is regular away from its boundary except for a closed subset whose dimension does not exceed $m-2$. This dimension is optimal since the surface $\left\{\left({z}^{2},{z}^{3}\right):z\in ℂ\right\}\subset {ℂ}^{2}\cong {ℝ}^{4}$ has an isolated singularity at the origin. In fact, from an earlier result by Federer every complex subvariety is automatically area-minimizing. This example already indicates the main difficulty in Almgren’s result. In the case of an area-minimizing hypersurface a regular point is characterized by its tangent cone – it is regular if and only if it has a plane as a tangent cone. But in the example above the tangent cone at the origin is a plane with multiplicity two. Thus, the problem is multiplicity which makes it necessary to consider multiple-valued functions when approximating currents by graphs of functions over tangent cones. Chapter 1 is called: Basic properties of $Q$ and $Q$-valued functions. Another well-known feature in regularity is that the approximating functions are closely related to harmonic functions. Therefore, the topic of Chapter 2 is: Properties of Dir minimizing $Q$-valued functions and tangent cone stratification of mass minimizing rectifiable currents. Consequently, Chapter 3 is called: Approximations in mass of nearly flat rectifiable currents which are mass minimizing in manifolds by graphs of Lipschitz $Q$-valued functions which can be weakly nearly Dir minimizing. The construction of a center manifold is one of the fundamental tools in Almgren’s regularity theory and Chapter 4 shows how this can be achieved: Approximation in mass of a nearly flat rectifiable current which is mass minimizing in a manifold by the image of a Lipschitz $Q\left({ℝ}^{m+n}\right)$ valued function defined on a center manifold. Finally, to carry over the regularity of multiple-valued functions minimizing Dirichlet’s integral to the currents under investigation, a second fundamental tool is the so-called frequency function $N\left(r\right)=\frac{r{\int }_{{B}_{r}\left(a\right)}{|Df|}^{2}}{{\int }_{\partial {B}_{r}\left(a\right)}{|f|}^{2}}$, already introduced in Chapter 2. For Dir minimization functions, $N\left(r\right)$ turns out to be nondecreasing in $r$. Estimates for the frequency functions are proved by certain range and domain deformations, called “squashing” and “squeezing”. The title of the final Chapter 5 is: Bounds on the frequency functions and the main interior regularity theorem. The book closes with a number of appendices which also are of independent interest, and it starts with a beautiful Introduction (16 pages) which contains a Summary of the principal themes by chapters. Among others, one of the questions addressed in the Introduction is Why this volume is so long. Let me close by saying: This work is a monument.

##### MSC:
 49-02 Research monographs (calculus of variations) 49N60 Regularity of solutions in calculus of variations 49Q20 Variational problems in a geometric measure-theoretic setting