Geometric measure theory
Here are some resources I used over the years to learn geometric measure theory. These include books, lecture notes, papers and surveys. What works best for you is a matter of personal taste, and what I wrote below reflects my own opinion.
- Lecture Notes: I was the TA for the second half of a summer school held by SLmath on June 17-26 2024. The second week was an introduction to geometric measure theory taught by Alessandro Pigati and we used the following Lecture notes (written by Alessandro):
- Sets of finite perimeter: Considering surfaces of codimension one as boundaries of sets of finite perimeter, was pioneered by Renato Caccioppoli in 1927. They are also named Caccioppoli sets.
- Sets of Finite Perimeter and Geometric Variational Problems, Francesco Maggi. link: This book builds from the basics, with lots of examples and proofs. It is my favorite one to start learning from. The pre-requisites are multidimensional calculus and a bit of functional analysis.
- Currents
- Cartesian Currents in the Calculus of Variations I: Mariano Giaquinta, Giuseppe Modica, Jiri Soucek. link: This book is intended for the (specialized) topic of Cartesian currents, however it is an excellent introduction to the theory of currents. It also contains a very nice crash course on measure theory and real analysis.
- Geometric integration theory, Steven Krantz, Harold Parks. link: This book starts from the basics of measure theory and it is a great reference for (a detailed proof of) the epsilon regularity theorems for integral currents. It is written in the modern language of GMT and it is very accessible.
- De-Giorgi's epsilon regularity theorem: There are multiple references where one can learn, one already mentioned above. Here I will put some more:
- Center manifold: A case study, Camillo De Lellis, Emanuele Nunzio Spadaro: [Appendix A], arxiv link.
- Varifolds
- On the First Variation of a Varifold, William K. Allard. link: Perhaps the most cited paper in this field. In this paper Allard introduces the concept of a varifold and proves the compactness and regularity theorems (among other results).
- Introduction to Geometric Measure Theory, Leon Simon. link: This books introduces Geometric measure theory in a condensed text and it is a great pleace to begin for slighly more advanced graduate students. It also presents a detailed version of the regularity theorm of Allard, with a more modern notation compared to the orignal paper of Allard.
- Allard's interior regularity theorem: An invitation to stationary Varifolds, Camillo De Lellis. link. This is a gentle introduction to the thoery of integral varifolds, especially the Allard regularity theorem. It contains a soft presentation of the proof of the tilt-excess decay lemma.