Varifold
Encyclopedia
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 set
s, 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
.
in 1964:: he coined the name varifold meaning that these objects are substitutes for ordinary manifolds in problems of the calculus of variations
. The modern approach to the theory was lay down by William Allard, in the paper .
of Euclidean space
ℝn, an m-dimensional varifold on
is defined as a Radon measure
on the set
where
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 form
s as duals to vector fields in the approximate tangent space of the set
.
Worth mentioning is the particular case of a rectifiable varifold: it 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 Grassmanian bundle of ℝn
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 submanifold
s are particular cases of rectifiable manifolds
.
Due to the lack of orientation
, there is no boundary operator defined on the space of varifolds.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a varifold is, loosely speaking, a measure-theoretic generalization of the concept of a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
, by replacing differentiability requirements with those provided by rectifiable set
Rectifiable set
In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set...
s, while maintaining the general algebraic structure usually seen in differential geometry. More closely, varifolds generalize the ideas of a rectifiable current
Current (mathematics)
In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Formally currents behave like Schwartz...
. Varifolds are the topic of study in geometric measure theory
Geometric measure theory
In mathematics, geometric measure theory is the study of the geometric properties of the measures of sets , including such things as arc lengths and areas. It uses measure theory to generalize differential geometry to surfaces with mild singularities called rectifiable sets...
.
Historical note
Varifolds were first introduced by Frederick AlmgrenFrederick J. Almgren, Jr.
Frederick Justin Almgren, Jr. was a mathematician working in geometric measure theory...
in 1964:: he coined the name varifold meaning that these objects are substitutes for ordinary manifolds in problems of the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
. The modern approach to the theory was lay down by William Allard, in the paper .
Definition
Given an open subset of Euclidean spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
ℝn, an m-dimensional varifold on
is defined as a Radon measureRadon measure
In mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...
on the set
where
is the GrassmannianGrassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...
of all m-dimensional linear subspaces of an n-dimensional vector space. The Grassmannian is used to allow the construction of analogs to differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s as duals to vector fields in the approximate tangent space of the set
.Worth mentioning is the particular case of a rectifiable varifold: it 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 Grassmanian bundle of ℝn
where
is the 
−dimensional Hausdorff measureHausdorff measureIn mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in Rn or, more generally, in any metric space. The zero dimensional Hausdorff measure is the number of points in the set or ∞ if the set is infinite...
Rectifiable varifolds are weaker objects than locally rectifiable currents: they do not have any orientation
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
. Replacing M with more regular sets, one easily see that differentiable submanifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
s are particular cases of rectifiable manifolds
Rectifiable set
In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set...
.
Due to the lack of orientation
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
, there is no boundary operator defined on the space of varifolds.
See also
- CurrentCurrent (mathematics)In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Formally currents behave like Schwartz...
- Geometric measure theoryGeometric measure theoryIn mathematics, geometric measure theory is the study of the geometric properties of the measures of sets , including such things as arc lengths and areas. It uses measure theory to generalize differential geometry to surfaces with mild singularities called rectifiable sets...
- Grassmanian
- Plateau's problemPlateau's problemIn mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who was interested in soap films. The problem is considered part of the calculus of variations...
- Radon measureRadon measureIn mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...