Varifold

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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 (Allard 1972).

Definition[edit]

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

\Omega \times 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 ℝn

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.

See also[edit]

Notes[edit]

  1. ^ The first widely circulated exposition of Almgren's ideas is the book (Almgren 1966): however, the first systematic exposition of the theory is contained in the book (Almgren 1965), 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 De Giorgi (1968, p. 400).
  2. ^ Probably the acronym is variational manifold, as Almgren (1993, p. 46) 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[edit]

References[edit]