# 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, varifolds generalize the ideas of a rectifiable current. Varifolds are the topic of study in geometric measure theory.

## Historical note

Varifolds were first introduced by Frederick Almgren in 1964:[1] he coined the name varifold meaning that these objects are substitutes for ordinary manifolds in problems of the calculus of variations.[2] The modern approach to the theory was laid down by William Allard, in the paper Template:Harv.

## Definition

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

${\displaystyle \Omega \times G(n,m)}$

where ${\displaystyle G(n,m)}$ is the Grassmannian of all m-dimensional linear subspaces of an n-dimensional vector space. The Grassmannian is used to allow the construction of analogs to differential forms as duals to vector fields in the approximate tangent space of the set ${\displaystyle \Omega }$.

The particular case of a rectifiable varifold is the data of a m-rectifiable set M (which is measurable with respect to the m-dimensional Hausdorff measure), and a density function defined on M, which is a positive function θ measurable and locally integrable with respect to the m-dimensional Hausdorff measure. It defines a Radon measure V on the Grassmannian bundle of ℝn

${\displaystyle V(A):=\int _{\Gamma _{M,A}}\!\!\!\!\!\!\!\theta (x){\mathrm {d} }{\mathcal {H}}^{m}(x)}$

where

Rectifiable varifolds are weaker objects than locally rectifiable currents: they do not have any orientation. Replacing M with more regular sets, one easily see that differentiable submanifolds are particular cases of rectifiable manifolds.

Due to the lack of orientation, there is no boundary operator defined on the space of varifolds.

## Notes

1. The first widely circulated exposition of Almgren's ideas is the book Template:Harv: however, the first systematic exposition of the theory is contained in the book Template:Harv, which had a far lower circulation, even if it is cited in Herbert Federer's classic text on geometric measure theory. See also the brief, clear survey by Template:Harvtxt.
2. Probably the acronym is variational manifold, as Template:Harvtxt describes the coining of the name with the following exact words:-"I called the objects "varifolds" having in mind that they were a measure-theoretic substitute for manifolds created for the variational calculus."

## Historical references

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## References

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|CitationClass=citation }}. A set of mimeographed notes where Frederick J. Almgren, Jr. introduces varifolds for the first time.

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|CitationClass=citation }}. The first widely circulated book describing the concept of a varifold and its applications to the Plateau's problem, particularly in chapter 4, section 6 "A solution to the existence portion of Plateau's problem".

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|CitationClass=citation }}. The second edition of the book Template:Harv.

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|CitationClass=citation }}, ISBN 7-03-010271-1 (Science Press), ISBN 1-57146-125-6 (International Press).