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
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Re ...
, and are studied in
geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
.
Historical note
Varifolds were first introduced by
Laurence Chisholm Young
Laurence Chisholm Young (14 July 1905 – 24 December 2000) was a British mathematician known for his contributions to measure theory, the calculus of variations, optimal control theory, and potential theory. He was the son of William Henry Yo ...
in , under the name "''generalized surfaces''".
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. The modern approach to the theory was based on Almgren's notes
[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
Herbert Federer (July 23, 1920 – April 21, 2010) was an American mathematician. He is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis.Parks, H. (2012''Remembering Herbert ...
's classic text on geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
. See also the brief, clear survey by . and laid down by
William K. Allard, in the paper .
Definition
Given an open subset
of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, an ''m''-dimensional varifold on
is defined as a
Radon measure
In mathematics (specifically in measure theory), 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 finite on all compact sets, outer regular on all Borel ...
on the set
:
where
is the
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...
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
.
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''
:
where
*
*
is the
−dimensional Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that as ...
Rectifiable varifolds are weaker objects than locally rectifiable currents: they do not have any
orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
. 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
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
*
Geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
*
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...
*
Plateau's problem
In 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 experimented with soap films. The problem ...
*
Radon measure
In mathematics (specifically in measure theory), 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 finite on all compact sets, outer regular on all Borel ...
Notes
References
*. This paper is also reproduced in .
*.
*.
*.
*.
*. A set of
mimeograph
A mimeograph machine (often abbreviated to mimeo, sometimes called a stencil duplicator) is a low-cost duplicating machine that works by forcing ink through a stencil onto paper. The process is called mimeography, and a copy made by the proc ...
ed notes where
Frederick J. Almgren Jr. introduces varifolds for the first time.
*. 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.
*{{Citation
, last = Young
, first = Laurence C.
, author-link = Laurence Chisholm Young
, title = Surfaces parametriques generalisees
, journal =
Bulletin de la Société Mathématique de France
'' Bulletin de la Société Mathématique de France'' is a mathematics journal published quarterly by Société Mathématique de France.
Founded in 1873, the journal publishes articles on mathematics.
It publishes articles in French and English. ...
, volume = 79
, pages = 59–84
, year=1951
, url =http://www.numdam.org/item?id=BSMF_1951__79__59_0
, mr = 46421
, zbl = 0044.10203
, doi = 10.24033/bsmf.1419
, doi-access = free
.
Measure theory
Generalized manifolds