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. Varifolds generalize the idea of a rectifiable current, and are studied in geometric measure theory.Varifolds were first introduced by Laurence Chisholm Young in (Young 1951), under the name 'generalized surfaces'. Frederick J. Almgren Jr. 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 K. Allard, in the paper (Allard 1972).Given an open subset Ω {displaystyle Omega }   of Euclidean space R n {displaystyle mathbb {R} ^{n}}  , an m-dimensional varifold on Ω {displaystyle Omega }   is defined as a Radon measure on the set