Varifold Explained

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.

Historical note

Varifolds were first introduced by Laurence Chisholm Young in, under the name "generalized surfaces".^{[1] [2]} Frederick J. Almgren Jr. slightly modified the definition in his mimeographed notes and coined the name varifold: he wanted to emphasize that these objects are substitutes for ordinary manifolds in problems of the calculus of variations.^[3] The modern approach to the theory was based on Almgren's notes^[4] and laid down by William K. Allard, in the paper .

Definition

Given an open subset

\Omega of Euclidean space R ^{n} , an m-dimensional varifold on \Omega is defined as a Radon measure on the set\Omega x G(n,m)

where

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 \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

R ^{n} V(A) : = \int _{\Gamma _{M,A}} \theta(x) d l{H} ^{m(x)}

where

\Gamma _{M,A} = M \cap \{x : (x, Tan ^{m(x,M))} \in A \} l{H} ^{m(x)} is the m −dimensional Hausdorff measureRectifiable 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.

See also

• Current
• Geometric measure theory
• Grassmannian
• Plateau's problem
• Radon measure

References

• . This paper is also reproduced in .
• .
• .
• .
• .
• . A set of mimeographed notes where Frederick J. Almgren Jr. introduces varifolds for the first time: the linked scan is available from Albert - The Digital Repository of the IAS.
• . The first widely circulated book describing the concept of a varifold. In chapter 4 is a section titled "A solution to the existence portion of Plateau's problem" but the stationary varifolds used in this section can only solve a greatly simplified version of the problem. For example, the only stationary varifolds containing the unit circle have support the unit disk. In 1968 Almgren used a combination of varifolds, integral currents, flat chains and Reifenberg's methods in an attempt to extend Reifenberg's celebrated 1960 paper to elliptic integrands. However, there are serious errors in his proof. A different approach to the Reifenberg problem for elliptic integrands has been recently provided by Harrison and Pugh without using varifolds.
• .
• . The second edition of the book .
• .
  • .
• , (Science Press), (International Press).
• .
• . An extended version of with a list of Almgren's publications.
• .

Notes and References

1. In his commemorative papers describing the research of Frederick Almgren, writes that these are "essentially the same class of surfaces".
2. See also the 2015 unpublished essay of Wendell Fleming.
3. exactly writes:-"I called the objects "varifolds" having in mind that they were a measure-theoretic substitute for manifolds created for the variational calculus". As a matter of fact, the name is a portmanteau of variational manifold.
4. The first widely circulated exposition of Almgren's ideas is the book : however, the first systematic exposition of the theory is contained in the mimeographed notes, 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 .