Varifold

In mathematics, a varifold is, loosely speaking, a measure-theoretic generalization of the concept of a differentiable manifold, by replacing differentiability requirements with those provided by rectifiable sets, while maintaining the general algebraic structure usually seen in differential geometry. More closely, the varifold generalize the idea of a rectifiable current. Varifolds are one of the topics of study in geometric measure theory.

Varifolds were first introduced by L.C. Young in (Young 1951), under the name "generalized surfaces".Frederick Almgren slightly modified the definition in his mimeographed notes (Almgren 1965) and coined the name varifold: he wanted to emphasize that these objects are substitutes for ordinary manifolds in problems of the calculus of variations. The modern approach to the theory was based on Almgren's notes and laid down by William Allard, in the paper (Allard 1972).

Given an open subset ${\displaystyle \Omega }$ of Euclidean space ℝⁿ, an m-dimensional varifold on ${\displaystyle \Omega }$ is defined as a Radon measure on the set

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