Varifold

Summary

An $m$-dimensional varifold in an open set $U \subseteq \R^n$ is a Radon measure on a product space. The product is $U$ times the space of $m$-dimensional linear subspaces of $\R^n$. It records where a generalised surface lies and which tangent planes it has there, but not its orientation. L. C. Young introduced them in 1951 as generalised surfaces. Almgren gave them the name varifold in the 1960s, and Allard developed them as a setting for minimal surfaces with singularities. Their compactness properties make them a main tool in the calculus of variations for area.