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Regularity of Minimal Surfaces

Ulrich Dierkes, Stefan Hildebrandt, and Anthony J. Tromba
Publisher: 
Springer
Publication Date: 
2010
Number of Pages: 
623
Format: 
Hardcover
Series: 
Grundlehren der mathematischen Wissenschaften 340
Price: 
129.00
ISBN: 
978-3-642-11699-5
Category: 
Monograph
[Reviewed by
Matthias Weber
, on
06/1/2011
]

Review of

In his note “Essai d’une nouvelle methode pour determiner les maxima et les minima des formules intégrales indéfinies” from 1760, Joseph Lagrange writes, after establishing the minimal surface equation of a graph and observing that planar graphs do indeed satisfy his equation, that “la solution générale doit ètre telle, que le périmètre de la surface puisse ètre détermine a volonté” — the general solution ought to be such that the perimeter of the surface can be prescribed arbitrarily.

The three volumes at hand are at this moment the most complete and thorough record of the ongoing efforts to justify Lagrange’s optimism. Despite their broad titles, the books are mainly concerned with Plateau’s problem and its ramifications. In its simplest form, it asks to find a disk of least area that is bounded by a given simple closed curve in space. More general forms of the problem allow more general boundary constraints, other topological types of spanning surfaces, and ask about regularity of the solutions.

Hence you will not find in these books anything about the curvature estimates of Colding and Minicozzi or about recent developments in the theory complete minimal surfaces by Meeks, Perez, Ros, Rosenberg and many others. This material would easily fill another volume of the same size.

If, on the other hand, Plateau’s problem is an area of interest to you, then yes, of course, these are the books to get — with one caveat: If this is really an area of interest to you, then you probably already know and own the two volumes by Dierkes, Hildebrandt, Küster, Wohlrab in the same series from 1991, and the question arises whether the upgrade is worth it.

The three new books all contain parts from the earlier volumes in the sense that the two earlier volumes are replicated. Most chapters are taken with only small changes. However, all volumes contain a wealth of new material in the form of newly written chapters and sections:

In the first volume, we find new chapters on stability, graphs with prescribed mean curvature, and the Douglas problem. The second volume has a chapter outlining Tromba’s new method to exclude interior branch points. The third volume has the most new material: Chapters about partially free boundary problems, the Bernstein problem, index theory and Morse theory justify the word “global” in the title. In addition to that, many chapters in all books have been augmented by sections presenting brand new research material.

The rearrangement of the material is partially motivated by an attempt to make the three books somewhat independent from each other. I view this as a success concerning books two and three, but would still consider large parts of the first book a prerequisite for the others.

The total of about 1700 pages is indeed so overwhelming that I am tempted to complain that there is too much. On the other hand, there are still topics that are missing and would have deserved inclusion. For instance, on my personal wish list would be a section about the classical theorem of Jenkins and Serrin about graphs with infinite boundary values (which is stated), and a chapter about the very recent completion by Desideri of Garnier’s program to solve the Plateau problem via polygonal approximation (which is mentioned in proof).

These three volumes are more than just a compilation of results and proofs from a vast subject. Here were true scholars in the best sense of the word at work, creating their literary lifetime achievements. They wrote with love for detail, clarity and history, which makes them a pleasure to read. These books will become instantaneous classics.

All three books have a gorgeous bibliography of which I’d love to have a bibTeX database. My only complaint is the rather brief index. In these computerized days, it should have been easy for Springer to compile a longer list of index words. Moreover, not every occurrence of a name in the text is referenced, making it sometimes hard to find theorems associated to people.


Matthias Weber is associate professor of Mathematics at Indiana University, Bloomington

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