Varifold

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, the varifold generalize the idea of a rectifiable current. Varifolds are one of the topics of study in geometric measure theory.

Historical note

Varifolds were first introduced by L.C. Young in (Young 1951), under the name "generalized surfaces".^[1][2] Frederick Almgren 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.^[3] The modern approach to the theory was based on Almgren's notes^[4] and laid down by William Allard, in the paper (Allard 1972).

Definition

Given an open subset $\Omega$ of Euclidean space ℝⁿ, 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

• $\scriptstyle\Gamma_{M,A}=M \cap \{x : (x, \mathrm{Tan}^m(x,M)) \in A \}$
• $\scriptstyle \mathcal{H}^m(x)$ is the $m$ Hausdorff measure

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

• Current
• Geometric measure theory
• Grassmannian
• Plateau's problem
• Radon measure

Notes

1. In his commemorative papers describing the research of Almgren, Brian White (1997, p.1452, footnote 1, 1998, p.682, footnote 1) writes that these are "essentially the same class of surfaces".
2. See also the 2015 unpublished essay of Wendell Fleming.
3. Almgren (1993, p. 46) exactly writes:-"I called the objects "varifolds" having in mind that they were a measure-theoretic substitute for manifolds created for the variational calculus". As a matter of fact, the name is a portmanteau of variational manifold.
4. 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 De Giorgi (1968, p. 400).

References

• Almgren, Frederick J. Jr. (1993), "Questions and answers about area-minimizing surfaces and geometric measure theory.", in Greene, Robert E.; Yau, S.-T., 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 54, Providence, RI: American Mathematical Society, pp. 29–53, ISBN 0-8218-1494-X, 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., Collected Works 13, Providence, R.I.: 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" (PDF), in Petrovsky, I. G., Trudy Mezhdunarodnogo kongressa matematikov. Proceedings of International Congress of Mathematicians (Moscow−1966), ICM Proceedings, Moscow: Mir Publishers, pp. 395−401, MR 0234329, Zbl 0188.17503.
• Allard, William K. (May 1972), "On the first variation of a varifold", Annals of Mathematics, Second Series 95 (3): 417–491, doi:10.2307/1970868, JSTOR 1970868, MR 0307015, Zbl 0252.49028.
• Allard, William K. (May 1975), "On the first variation of a varifold: Boundary Behavior", Annals of Mathematics, Second Series 101 (3): 418–446, doi:10.2307/1970934, JSTOR 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, Princeton: Princeton University Library, p. 178. A set of mimeographed notes where Frederick J. Almgren, Jr. introduces varifolds for the first time.
• 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 to extend Reifenberg's celebrated 1960 paper to elliptic integrands.
• Almgren, Frederick J. Jr. (2001) [1966], Plateau's Problem: An Invitation to Varifold Geometry, Student Mathematical Library 13 (2nd ed.), Providence, RI: 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, Translations of Mathematical Monographs 84, Providence, RI: American Mathematical Society, pp. ix+404, ISBN 0-8218-4536-5, MR 1093903, Zbl 0716.53003.
• T. C. O'Neil (2001), "Geometric measure theory", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Simon, Leon (1984), Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis 3, Canberra: Centre for Mathematics and its Applications (CMA), Australian National University, pp. VII+272 (loose errata), ISBN 0-86784-429-9, MR 0756417, Zbl 0546.49019.
• Lin, Fanghua; Yang, Xiaoping (2002), Geometric Measure Theory – An Introduction, Advanced Mathematics (Beijing/Boston) 1, Beijing–New York / Boston, MA: Science Press / 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.", Notices of the American Mathematical Society 44 (11): 1451–1456, ISSN 0002-9920, MR 1488574, Zbl 0908.01017.
• White, Brian (1998), "The mathematics of F. J. Almgren, Jr.", The Journal of Geometric Analysis 8 (5): 681–702, doi:10.1007/BF02922665, ISSN 1050-6926, MR 1731057, Zbl 0955.01020. An extended version of (White 1997) with a list of Almgren's publications.
• Young, L. C. (1951), "Surfaces parametriques generalisees", Bulletin de la Société Mathématique de France 79: 59–84, MR 46421, Zbl 0044.10203.