geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfac ...

, a field of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Almgren regularity theorem, proved by , states that the singular set of a mass-minimizing surface has

codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals ...

at least 2. Almgren's proof of this was 955 pages long. Within the proof many new ideas are introduced, such as monotonicity of a ''frequency function'' and the use of a ''center manifold'' to perform a more intricate blow-up procedure. A streamlined and more accessible proof of Almgren's regularity theorem, following the same ideas as Almgren, was given by

Camillo De Lellis Camillo De Lellis (born 11 June 1976) is an Italian mathematician who is active in the fields of calculus of variations, hyperbolic systems of conservation laws, geometric measure theory and fluid dynamics. He is a permanent faculty member in ...

and Emanuele Spadaro in a series of three papers.De Lellis, Camillo; Spadaro, Emanuele Regularity of area minimizing currents III: blow-up. Ann. of Math. (2) 183 (2016), no. 2, 577–617.

References

* * * * Theorems in measure theory Theorems in geometry {{mathanalysis-stub