Main References
1. Frank Morgan's Geometric Measure Theory: A beginners guide, 2000
2. Evans and Gariepy's Measure theory and Fine Properties of Functions, 1999
3. Krantz and Parks' Geometric Integration Theory, preprint 2006
4. Mattila's Geometry of Sets and Measures in Euclidean Spaces 1995
5. Leon Simon's Lectures on Geometric Measure Theory, 1983
6. Lin and Yang's Geometric Measure theory 2002
7. Federer's Geometric Measure Theory, 1969
As hinted at above, if you are only going to buy two texts to begin with, get Morgan and Evans and Gariepy. Krantz and Parks' is an AIM preprint and free. Download it from here . I will update this page when Krantz and Parks is in print, with a more discriminating recommendation. I expect to like the book quite a bit form the brief look I have had so far. [Update: I used Krantz and Parks — several years ago now — as the text for a course. I think it is fine and sometimes very good, but the figures can be really bad at points and it has its idiosyncracies and less well written chapters. I do like many pieces of their presentation of the fundamental theory quite a bit. Overall, I recommend that anyone interested in GMT should have a copy.] If you are serious about the subject, get all 7.
Notes on these books and others:
Geometric Measure Theory - A Beginner's Guide, Frank Morgan 2000. This is a introduction to GMT that is intended to be an interface with more advanced treatises such as Federer's (below). There are lots of pictures and he puts a great deal of effort into explaining the intuitions behind things.
Measure Theory and Fine Properties of Functions, L. Craig Evans and Ronald F. Gariepy 1999. This text is the best reference to anything it contains. In particular it does a beautiful job covering the theory of BV functions and Cacioppoli sets. This theory was introduced by De Giorgi in 1960 to solve co-dimension 1 minimal surface problems. Other topics covered including measures, densities, covering theorems, the area and co-area formulas, the theory of Sobolev functions, the Whitney extension theorem, and other useful and usually-not-covered topics.
Geometry of Sets and Measures in Euclidean Spaces, Pertti Mattila 1995. This is the reference for the rectifiability, fractal measure, etc branch of the subject. Nothing about currents and varifolds here. But it is a favorite with students for its excellent exposition.
Geometric Integration Theory, Steven Krantz and Harold Parks, 2006 preprint. The existence of this book was pointed out to me very recently. I have not had time to look at it very carefully, but it looks nice. And you can get it here.
Geometric Measure Theory, Herbert Federer 1969. This is the standard, comprehensive (but in some places dated) reference for the subject. It is notoriously difficult in spots. Precision and generality (and often you need it) are characteristics of the text. Remembering that what is being written about is most often geometric can help reading it. Frank Morgan's text was constructed as an interface to this text.
Lectures on Geometric Measure Theory, Leon Simon 1980. A more readable GMT text that does get to the details, but is not of course comprehensive. This is the text that many people first learn GMT from (often in combination with an initial read of Frank Morgan's text).
Seminar on Geometric Measure Theory, Bob Hardt and Leon Simon 1986. Nice, fairly brief notes from a 10 lecture short course taught by Bob and Leon in Dusseldorf in 1984. This is a nice first look at the subject.
Geometric Measure Theory - An Introduction, Fanghua Lin and Xiaoping Yang 2002. This is a rich reference that is more up to date in some ways than Federer's book. It is not as complete of course and it has frequent typos and even some typos that seem like errors. On the other hand, Fanghua told me that he had sent the publisher corrections which they refused to use to update the book. With a bit more background, it is a great reference.
Riemannian Geometry, Manfredo Do Carmo 1992. My favorite book on Riemannian Geomtry. Beautifully written. Really gives you a feel for the tools and the intuitions without holding your hand too much. And it is thin!
Geometric Inequalities, Yu.D. Burago and V.A. Zalgaller 1988 (Russian 1980). This is a very interesting book ranging all over the geometric analytic landscape with numerous facts and theorem at the intersection of geometry and analysis. I have used it for its look at mixed volumes and the Steiner-Minkowski formula as well as various other misc. Highly recommended!
Minimal Surfaces and Functions of Bounded Variation, Enrico Giusti 1984. Giusti's book is a standard reference on the BV function route into minimal surfaces. I found it very nice to study and went through a significant part of it when I was looking at mild regularity properties of the minimizers to image functionals. For the theory of Sets of Finite Perimeter, Evans and Gariey (above) is also excellent, probably better. But for the combination of BV and variational problems, this is the best.
Singular Integrals and Differentiability Properties of Functions, Elias M. Stein 1970. This is the standard reference, the bible, on singular integrals. It is beautifully written. Of course it is a bit dated, but on the advice of harmonic analysts and geometric analysts I trust, I intend to study it carefully. What I have seen of it is compelling.
Introduction to Smooth Manifolds, John M. Lee 2003. John (Jack) M. Lee's writing is so clear and well motivated that this really should be in the library of anybody learning differential geometry. He has two other books, one on Riemannian Manifolds and another on topological manifolds that are also highly recommended. Anecdote: When I was in X-division at LANL, I recruited Gary Sandine to come and work with me on various mathematically driven methods for inverse problems. He found a draft of Jack's book on smooth manifolds and studied it with great care: in fact he ended up producing a big and detailed errata list as well as a great number of the illustrations appearing in the final book. One can find Jack's acknowledgment of this work in the preface.
Nonlinear Functional Analysis and its Applications: volumes I, IIa, IIb, III, and IV, Eberhard Zeidler mid-1980's. These five volumes are an amazing corpus on nonlinear functional analysis and everything touching it. In particular, I often consult the appendix of the first volume Nonlinear Functional Analysis and its Applications I: Fixed Point Theorems because it contains a lot of details in a very concise form on functional analysis and facts from topology needed in analysis. In general, these volumes are fun to read, filled with an enormous amount of information (how in the world did he write these volumes they must be over 4000 pages together!) and such delightful extras as a large collection of quotes, the most amazing and complete set of indices I have ever seen and problem sets that guide you through the literature and to the cutting edge. I have used Volumes I (Fixed Point Theorems) and III (Variational Methods and Optimization) a fair bit and enjoy it every time I dip into these books.
Papers and Notes
The first four lectures given by Urs Lang at a Spring school in Switzerland in 2005 are useful to have: Here is a link to the pdf: Urs Langs Lectures. The last lecture concerns recent developments that are less relevant to what we will be talking about.
Another useful -- though brief -- set of notes from the same Spring School are those by Giovanni Alberti: A link to those notes is here: Alberti 2005 Les Diablerets Notes. Albteri also wrote a short article for the Encyclopedia of Mathematical Physics about GMT. You can download those notes from here.
Nice notes by Camillo De Lellis on Preiss' theorem: The Notes (pdf). This is an exposition of Priess' big paper. Along the way he also proves Marstrand's Theorem.
Notes from my Spring 2007 GMT class at UCLA and LANL:
1. The first lecture was an overview and panorama that I did not write up.
