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 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
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.
^In his commemorative papers describing the research of
Frederick Almgren,
Brian White (
1997, p.1452, footnote 1,
1998, p.682, footnote 1) writes that these are "essentially the same class of surfaces".
^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 mimeographed notes (
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
Ennio De Giorgi (
1968).
References
Almgren, Frederick J. Jr. (1993), "Questions and answers about area-minimizing surfaces and geometric measure theory.", in
Greene, Robert E.;
Yau, Shing-Tung (eds.), Differential Geometry. Part 1: Partial Differential Equations on Manifolds. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8–28, 1990, Proceedings of Symposia in Pure Mathematics, vol. 54, Providence, RI:
American Mathematical Society, pp. 29–53,
ISBN978-0-8218-1494-9,
MR1216574,
Zbl0812.49032. This paper is also reproduced in (
Almgren 1999, pp. 497–521).
Almgren, Frederick J. Jr. (1966), Plateau's Problem: An Invitation to Varifold Geometry, Mathematics Monographs Series (1st ed.), New York–Amsterdam: W. A. Benjamin, Inc., pp. XII+74,
MR0190856,
Zbl0165.13201. 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 (
HarrisonPugh 2016) without using varifolds.
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 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
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.
^In his commemorative papers describing the research of
Frederick Almgren,
Brian White (
1997, p.1452, footnote 1,
1998, p.682, footnote 1) writes that these are "essentially the same class of surfaces".
^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 mimeographed notes (
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
Ennio De Giorgi (
1968).
References
Almgren, Frederick J. Jr. (1993), "Questions and answers about area-minimizing surfaces and geometric measure theory.", in
Greene, Robert E.;
Yau, Shing-Tung (eds.), Differential Geometry. Part 1: Partial Differential Equations on Manifolds. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8–28, 1990, Proceedings of Symposia in Pure Mathematics, vol. 54, Providence, RI:
American Mathematical Society, pp. 29–53,
ISBN978-0-8218-1494-9,
MR1216574,
Zbl0812.49032. This paper is also reproduced in (
Almgren 1999, pp. 497–521).
Almgren, Frederick J. Jr. (1966), Plateau's Problem: An Invitation to Varifold Geometry, Mathematics Monographs Series (1st ed.), New York–Amsterdam: W. A. Benjamin, Inc., pp. XII+74,
MR0190856,
Zbl0165.13201. 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 (
HarrisonPugh 2016) without using varifolds.