Geometric measure theory

Summary

Geometric measure theory studies the geometry of sets and measures in Euclidean space and in manifolds with the tools of measure theory. Its basic objects are Hausdorff measure, rectifiable sets, currents and varifolds, which act as surfaces that may be singular, have multiplicities, or arise as limits. It provides compactness theorems for such generalised surfaces, which give existence results for variational problems such as Plateau's problem in every dimension and codimension. It also studies the regularity of the minimisers and the structure of their singular sets.