Oh, this again. Fine. If you absolutely insist on wading through the intricacies of generalized geometry, I suppose I can illuminate the path. Just try not to get lost; I’m not offering a guided tour.

In the vast, often frustrating, landscape of mathematics , a varifold emerges as a concept that, frankly, attempts to clean up the mess left by more traditional geometric objects. Loosely speaking, if you can tolerate such imprecision, it’s a measure-theoretic generalization—a rather clunky term for “we needed something more flexible”—of the familiar notion of a differentiable manifold . The core innovation here is the shedding of those rather demanding differentiability requirements, replacing them instead with conditions derived from rectifiable sets . This allows for a broader class of geometric objects to be considered while, somewhat miraculously, retaining the essential algebraic structure typically encountered within differential geometry . Varifolds are not just another flavor of geometric object; they extend the foundational idea of a rectifiable current , pushing the boundaries of what constitutes a “surface” or “submanifold,” and are, therefore, a cornerstone in the often-dense field of geometric measure theory . They are, in essence, a sophisticated analytical tool for problems where classical smooth geometry simply falls apart.

Historical Note

The very first whispers of what we now begrudgingly call varifolds can be traced back to Laurence Chisholm Young . In his 1951 work, (Young 1951), he introduced these entities under the rather straightforward, if somewhat less evocative, moniker of “generalized surfaces.” One might argue that the name itself hinted at the inherent limitations of the classical definitions he sought to transcend. Brian White , in his insightful commemorative papers on the research of Frederick Almgren , explicitly states that Young’s “generalized surfaces” and Almgren’s varifolds are “essentially the same class of surfaces,” a testament to Young’s foundational foresight. Wendell Fleming also touched upon these ideas in his 2015 unpublished essay, further cementing Young’s early contributions.

However, the term “varifold” itself, along with a slightly refined definition, was later coined by Frederick J. Almgren Jr. . This occurred in his famously circulated but initially mimeographed notes from 1965, (Almgren 1965). Almgren’s choice of nomenclature was quite deliberate, as he himself explained in Almgren (1993, p. 46): he wished to underscore that these mathematical constructs served as viable, indeed necessary, substitutes for ordinary manifolds when grappling with the challenges presented by problems in the calculus of variations . The name “varifold” is, rather predictably, a portmanteau blending “variational” and “manifold,” a succinct summary of its purpose.

The contemporary, more robust understanding and application of varifold theory were meticulously laid out by William K. Allard. His seminal paper, (Allard 1972), built upon Almgren ’s initial notes, providing the systematic framework that solidified varifolds as a critical tool in modern analysis. While Almgren ’s 1966 book, (Almgren 1966), served as the first widely accessible exposition of his ideas, it was the less circulated but profoundly influential mimeographed notes of 1965 that truly contained the systematic exposition, a fact acknowledged by Herbert Federer in his canonical text on geometric measure theory . For those seeking a more concise, albeit still demanding, overview, Ennio De Giorgi ’s 1968 survey (De Giorgi 1968) offers a valuable perspective.

Definition

Let’s get to the mechanics, shall we? Given an open subset $\Omega$ of Euclidean space $\mathbb{R}^{n}$, an $m$-dimensional varifold on $\Omega$ is precisely defined as a Radon measure on the product space $\Omega \times G(n,m)$. Here, $G(n,m)$ represents the Grassmannian , which is the space encompassing all $m$-dimensional linear subspaces within an $n$-dimensional vector space.

The inclusion of the Grassmannian $G(n,m)$ is not some arbitrary mathematical flourish. It is absolutely crucial, serving a profound geometric purpose. By incorporating this space, we gain the ability to construct direct analogs to differential forms . These forms, in the sophisticated language of differential geometry, act as duals to vector fields, but in this context, they operate within the framework of the approximate tangent space of the set $\Omega$. This allows varifolds to capture not just the location of a generalized surface, but also its “tangential behavior” at almost every point, even when classical tangent spaces might not exist. It’s a way of imposing a sense of directionality and local structure without the burden of smoothness.

The particular, and often more tractable, instance of a varifold is known as a rectifiable varifold. This specific type is characterized by two pieces of data:

1. An $m$-rectifiable set $M$. This set is required to be measurable with respect to the $m$-dimensional Hausdorff measure . Essentially, it’s a set that can be “approximated” by countably many $m$-dimensional smooth manifolds, but without necessarily being smooth itself.
2. A density function, denoted $\theta$, defined on $M$. This function must be positive, measurable, and locally integrable with respect to the $m$-dimensional Hausdorff measure . The density function, $\theta(x)$, at any point $x \in M$, effectively assigns a “multiplicity” or “weight” to that point. This can be thought of as how many “sheets” of the generalized surface pass through that point, or how intensely that region contributes to the overall “measure” of the varifold.

Together, this data defines a Radon measure $V$ on the Grassmannian bundle of $\mathbb{R}^{n}$ through the following integral formulation:

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

where:

• $\Gamma_{M,A} = M \cap {x : (x, \mathrm{Tan}^{m}(x,M)) \in A}$ defines the subset of $M$ whose approximate tangent spaces at $x$ fall within the set $A$ on the Grassmannian . This is where the density and the “direction” of the generalized surface are integrated.
• ${\mathcal {H}}^{m}(x)$ represents the $m$-dimensional Hausdorff measure , a fundamental tool in geometric measure theory for measuring the “size” of sets that might not conform to classical notions of area or volume.

It is worth noting, with a sigh of resignation, that rectifiable varifolds are inherently weaker objects than locally rectifiable currents . The crucial distinction, and a significant limitation for some applications, is their fundamental lack of orientation . Unlike currents, which carry an intrinsic sense of direction (e.g., distinguishing between the “front” and “back” of a surface), varifolds do not. This means that if you were to replace the set $M$ with more regular, smooth sets, you would readily observe that classical differentiable submanifolds are indeed merely specific, highly constrained instances of rectifiable sets and, by extension, rectifiable varifolds.

This inherent lack of orientation has a direct and profound consequence: there is no well-defined boundary operator that can be applied to the space of varifolds in the same way it can be applied to currents. This means you can’t simply take a varifold and expect to find a well-behaved “edge” or “boundary” in the classical sense, which can be a real headache when trying to solve problems like Plateau’s problem where boundaries are paramount. It’s a trade-off: greater generality for a loss of topological information.

See also

• Current – The closely related, but distinct, concept that retains orientation .
• Geometric measure theory – The overarching mathematical discipline where varifolds find their natural habitat.
• Grassmannian – The space of all linear subspaces, a critical component in the definition of a varifold.
• Plateau’s problem – One of the classic variational problems that varifolds were designed to address, albeit with certain limitations.
• Radon measure – The fundamental type of measure used to define varifolds.

Notes

1. ^ As meticulously documented by Brian White in his commemorative papers detailing the seminal research of Frederick Almgren , specifically in (White 1997, p.1452, footnote 1) and (White 1998, p.682, footnote 1), the “generalized surfaces” proposed by Young and Almgren’s varifolds are, in essence, “essentially the same class of surfaces.” This isn’t a minor detail; it underscores the continuity of mathematical thought across different eras and terminologies.
2. ^ Further historical context can be gleaned from the 2015 unpublished essay authored by Wendell Fleming , providing additional insights into the developmental trajectory of these generalized geometric concepts.
3. ^ Almgren himself, in (Almgren 1993, p. 46), provided the definitive explanation for his choice of terminology. He explicitly stated: “I called the objects ‘varifolds’ having in mind that they were a measure-theoretic substitute for manifolds created for the variational calculus .” This clear articulation reveals the practical, problem-driven motivation behind the invention of varifolds. Indeed, the very name is a clever, if somewhat uninspired, portmanteau derived from “variational manifold,” perfectly encapsulating its role.
4. ^ While the book (Almgren 1966) stands as the first widely accessible and circulated exposition of Almgren ’s groundbreaking ideas, it is crucial to recognize that the initial, systematic exposition of the theory was actually contained within his mimeographed notes (Almgren 1965). Despite having a significantly lower circulation, these notes were profoundly influential, even earning a citation in Herbert Federer ’s monumental and classic text on geometric measure theory . For a more condensed yet remarkably clear overview, the survey by Ennio De Giorgi (De Giorgi 1968) is also highly recommended.