# Varifold

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In [mathematics](/source/Mathematics), a **varifold** is, loosely speaking, a [measure-theoretic](/source/Measure_theory) generalization of the concept of a [differentiable manifold](/source/Differentiable_manifold), by replacing differentiability requirements with those provided by [rectifiable sets](/source/Rectifiable_set), while maintaining the general algebraic structure usually seen in [differential geometry](/source/Differential_geometry). Varifolds generalize the idea of a [rectifiable current](/source/Current_(mathematics)), and are studied in [geometric measure theory](/source/Geometric_measure_theory).

## Historical note

Varifolds were first introduced by [Laurence Chisholm Young](/source/Laurence_Chisholm_Young) in (Young 1951), under the name "*generalized surfaces*".[1][2] [Frederick J. Almgren Jr.](/source/Frederick_J._Almgren_Jr.) slightly modified the definition in his mimeographed notes (Almgren 1965) and coined the name *varifold*: he wanted to emphasize that these objects are substitutes for ordinary manifolds in problems of the [calculus of variations](/source/Calculus_of_variations).[3] The modern approach to the theory was based on Almgren's notes[4] and laid down by [William K. Allard](/source/William_K._Allard), in the paper (Allard 1972).

## Definition

Given an open subset \Omega of [Euclidean space](/source/Euclidean_space) \mathbb{R}^n, an *m*-dimensional varifold on \Omega is defined as a [Radon measure](/source/Radon_measure) on the set

- \Omega \times G(n,m)

where G(n,m) is the [Grassmannian](/source/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](/source/Differential_form) as duals to vector fields in the [approximate tangent space](/source/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 \mathbb{R}^n

- V(A) := \int_{\Gamma_{M,A}}\!\!\!\!\!\!\!\theta(x) \mathrm{d} \mathcal{H}^m(x)

where

- \Gamma_{M,A}=M \cap \{x : (x, \mathrm{Tan}^m(x,M)) \in A \}
- \mathcal{H}^m(x) is the [m−dimensional](/source/Dimension_(mathematics)) [Hausdorff measure](/source/Hausdorff_measure)

Rectifiable varifolds are weaker objects than locally rectifiable currents: they do not have any [orientation](/source/Orientability). Replacing *M* with more regular sets, one easily see that [differentiable submanifolds](/source/Differentiable_manifold) are particular cases of [rectifiable manifolds](/source/Rectifiable_set).

Due to the [lack of orientation](/source/Orientability), there is no [boundary operator](/source/Boundary_operator) defined on the space of varifolds.

## See also

- [Current](/source/Current_(mathematics))
- [Geometric measure theory](/source/Geometric_measure_theory)
- [Grassmannian](/source/Grassmannian)
- [Plateau's problem](/source/Plateau's_problem)
- [Radon measure](/source/Radon_measure)

## Notes

1. In his commemorative papers describing the research of [Frederick Almgren](/source/Frederick_J._Almgren_Jr.), txt writes that these are "*essentially the same class of surfaces*".

1. See also the [2015 unpublished essay](https://www.dam.brown.edu/people/documents/Geometic_000.pdf) of [Wendell Fleming](/source/Wendell_Fleming).

1. Almgren (1993, p. 46) exactly writes:-"*I called the objects "varifolds" having in mind that they were a [measure-theoretic](/source/Measure_theory) substitute for [manifolds](/source/Manifold) created for the [variational calculus](/source/Calculus_of_variations)*". As a matter of fact, the name is a [portmanteau](/source/Portmanteau) of ***vari**ational* *man**ifold***.

1. The first widely circulated exposition of [Almgren](/source/Frederick_J._Almgren_Jr.)'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](/source/Herbert_Federer)'s classic text on [geometric measure theory](/source/Geometric_measure_theory). See also the brief, clear survey by txt, p. 400.

## References

- Almgren, Frederick J. Jr. (1993), "Questions and answers about area-minimizing surfaces and geometric measure theory.", *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*, Vol. 54, Proceedings of Symposia in Pure Mathematics, Greene, Robert E. (ed.), Providence, RI: [American Mathematical Society](/source/American_Mathematical_Society), pp. 29–53, ISBN 978-0-8218-1494-9. MR 1216574. Zbl 0812.49032. This paper is also reproduced in (Almgren 1999, pp. 497–521).
- Almgren, Frederick J. Jr. (1999), [*Selected works of Frederick J. Almgren, Jr.*](https://books.google.com/books?isbn=0821810677), Vol. 13, Collected Works, Providence, R.I.: [American Mathematical Society](/source/American_Mathematical_Society), ISBN 978-0-8218-1067-5. MR 1747253. Zbl 0966.01031.
- De Giorgi, Ennio (1968), ["Hypersurfaces of minimal measure in pluridimensional euclidean spaces"](https://web.archive.org/web/20150706164436/http://www.mathunion.org/ICM/ICM1966.1/), "Trudy Mezhdunarodnogo kongressa matematikov. Proceedings of International Congress of Mathematicians (Moscow−1966)", [ICM Proceedings](/source/International_Congress_of_Mathematicians), Petrovsky, Ivan G. (ed.), [Moscow](/source/Moscow): [Mir Publishers](/source/Mir_Publishers), pp. 395−401, MR 0234329. Zbl 0188.17503, archived from [the original](http://www.mathunion.org/ICM/ICM1966.1/) on 2015-07-06, retrieved 2011-07-18.
- Allard, William K. (May 1972), "On the first variation of a varifold", *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **95** (3): 417–491, Second Series, [doi:10.2307/1970868](https://doi.org/10.2307/1970868). [JSTOR 1970868](https://www.jstor.org/stable/1970868). MR 0307015. Zbl 0252.49028.
- Allard, William K. (May 1975), "On the first variation of a varifold: Boundary Behavior", *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **101** (3): 418–446, Second Series, [doi:10.2307/1970934](https://doi.org/10.2307/1970934). [JSTOR 1970934](https://www.jstor.org/stable/1970934). MR 0397520. Zbl 0319.49026.
- Almgren, Frederick J. Jr. (1965), ["The theory of varifolds: A variational calculus in the large for the k-dimensional area integrand"](https://hdl.handle.net/20.500.12111/8025), [Princeton](/source/Princeton,_New_Jersey): [Princeton University Library](/source/Princeton_University_Library), p. 178. A set of [mimeographed](/source/Mimeograph) notes where [Frederick J. Almgren Jr.](/source/Frederick_J._Almgren_Jr.) introduces varifolds for the first time: the linked scan is available from [Albert - The Digital Repository of the IAS](https://albert.ias.edu/home).
- 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, MR 0190856. Zbl 0165.13201. 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.
- Harrison, Jenny & Pugh, Harrison (2016), "General Methods of Elliptic Minimization", p. 22, [arXiv:1603.04492](https://arxiv.org/abs/1603.04492). [Bibcode:2016arXiv160304492H](https://ui.adsabs.harvard.edu/abs/2016arXiv160304492H).
- Almgren, Frederick J. Jr. (2001 [1966]), [*Plateau's Problem: An Invitation to Varifold Geometry*](https://books.google.com/books?id=Tk8zHPGVmBsC), Vol. 13, Student Mathematical Library, 2nd ed., [Providence, RI](/source/Providence,_RI): [American Mathematical Society](/source/American_Mathematical_Society), pp. xvi+78, ISBN 978-0-8218-2747-5. MR 1853442. Zbl 0995.49001. The second edition of the book (Almgren 1966).
- Đào, Trọng Thi & Fomenko, A. T. (1991), [*Minimal Surfaces, Stratified Multivarifolds, and the Plateau Problem*](https://books.google.com/books?id=mncIV2c5Z4sC), Vol. 84, Translations of Mathematical Monographs, Providence, RI: [American Mathematical Society](/source/American_Mathematical_Society), pp. ix+404, ISBN 978-0-8218-4536-3. MR 1093903. Zbl 0716.53003.
- Simon, Leon (1984), [*Lectures on Geometric Measure Theory*](http://maths.anu.edu.au/research/symposia-proceedings/lectures-geometric-measure-theory), Vol. 3, Proceedings of the Centre for Mathematical Analysis, [Canberra](/source/Canberra): [Centre for Mathematics and its Applications (CMA)](/source/Centre_for_Mathematics_and_its_Applications), [Australian National University](/source/Australian_National_University), pp. VII+272 (loose errata), ISBN 978-0-86784-429-0. MR 0756417. Zbl 0546.49019.
- Lin, Fanghua & Yang, Xiaoping (2002), "Geometric Measure Theory – An Introduction", Vol. 1, Advanced Mathematics (Beijing/Boston), [Beijing](/source/Beijing)–New York / Boston, MA: [Science Press](/source/Science_Press) / [International Press](/source/International_Press), pp. x+237, MR 2030862. Zbl 0546.49019, ISBN 7-03-010271-1 (Science Press), ISBN 1-57146-125-6 (International Press).
- White, Brian (1997), ["The Mathematics of F. J. Almgren Jr."](http://www.ams.org/notices/199711/index.html), *[Notices of the American Mathematical Society](/source/Notices_of_the_American_Mathematical_Society)*. **44** (11): 1451–1456, [ISSN 0002-9920](https://www.worldcat.org/issn/0002-9920). MR 1488574. Zbl 0908.01017.
- White, Brian (1998), "The mathematics of F. J. Almgren, Jr.", *[The Journal of Geometric Analysis](/source/The_Journal_of_Geometric_Analysis)*. **8** (5): 681–702, [CiteSeerX 10.1.1.120.4639](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.120.4639). [doi:10.1007/BF02922665](https://doi.org/10.1007/BF02922665). [ISSN 1050-6926](https://www.worldcat.org/issn/1050-6926). MR 1731057. [S2CID 122083638](https://api.semanticscholar.org/CorpusID:122083638). Zbl 0955.01020. An extended version of (White 1997) with a list of Almgren's publications.
- Young, Laurence C. (1951), ["Surfaces parametriques generalisees"](http://www.numdam.org/item?id=BSMF_1951__79__59_0), *[Bulletin de la Société Mathématique de France](/source/Bulletin_de_la_Soci%C3%A9t%C3%A9_Math%C3%A9matique_de_France)*. **79**: 59–84, [doi:10.24033/bsmf.1419](https://doi.org/10.24033/bsmf.1419). MR 46421. Zbl 0044.10203.

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