Geometric measure theory

Definition

Let $n \ge 1$ and $0 \le m \le n$. For $E \subseteq \R^n$, let $\mathcal{H}^m(E)$ be its $m$-dimensional Hausdorff measure. A set $E \subseteq \R^n$ is countably $m$-rectifiable if there are Lipschitz maps $f_j \colon \R^m \to \R^n$, $j = 1, 2, \dots$, with $\mathcal{H}^m\left( E \setminus \bigcup_j f_j(\R^m) \right) = 0$. Geometric measure theory is the branch of analysis that studies:

1. the measure-theoretic structure of subsets of $\R^n$ and of Riemannian manifolds, measured by Hausdorff measures, in particular the rectifiable sets and their approximate tangent planes;
2. generalised surfaces built from them, such as currents and varifolds;
3. variational problems for these objects, such as the minimisation of area, with the existence and regularity of their solutions.