1 Varifolds in Geometric Measure Theory

1.1 Motivation from weak limits of surfaces

Many geometric variational problems, such as those involving surface area, produce sequences of surfaces that do not converge smoothly. Singularities, oscillations, or concentration effects can prevent pointwise or smooth convergence. Varifolds provide a framework in which a sequence of hypersurfaces can still converge in a weak sense: rather than tracking every geometric detail, one records how much “mass” the surfaces carry and what their approximate tangent directions look like in the limit. This enables the study of limiting objects even when the underlying set fails to be a smooth manifold.

1.2 The basic definition of a varifold

At a high level, a (d-dimensional) varifold in an open set of Euclidean space is a measure-theoretic object that represents both:

  1. a distribution of mass on space, and
  2. an associated distribution of approximate tangent planes at those points.

Formally, one encodes this information in a Radon measure on the product of the ambient space with the Grassmannian of d-dimensional unoriented planes. Weak convergence of varifolds corresponds to convergence of integrals of suitable test functions defined on position and tangent-plane data.

1.3 Grassmannian and orientationless tangent data

The Grassmannian is the parameter space of all d-dimensional linear subspaces of the ambient space. In the orientationless (usual) varifold setting, one does not distinguish a plane from its opposite orientation; tangent information is recorded as an element of the unoriented Grassmannian. This choice fits naturally with problems where reversing orientation does not change the energy or the geometric quantity being studied.

1.4 Mass measures and first variations (high-level)

A varifold’s mass is obtained by projecting its measure from the Grassmannian factor back to the ambient space. Under additional regularity or boundedness assumptions, one can define the first variation of the varifold: a weak notion of how the varifold responds to smooth deformations. In geometric measure theory, first variation plays the role analogous to the Euler–Lagrange equation in the smooth setting, allowing definitions of generalized minimality or stationarity for nonsmooth limits.

2 Rectifiability

2.1 Countably rectifiable sets

A set is countably rectifiable if, up to a set of Hausdorff measure zero, it can be covered by countably many images of Lipschitz maps from d-dimensional Euclidean space. Intuitively, this means the set looks like a union of countably many “pieces of d-dimensional surfaces” at almost every point, even though globally it may have complicated structure.

2.2 Approximate tangent planes

Rectifiability implies the existence of approximate tangent planes for almost every point with respect to the appropriate Hausdorff measure. Approximate tangent planes are defined through measure-theoretic density: in small neighborhoods, the set becomes close to a linear subspace in an averaged sense. This tangent-plane information is crucial for associating a canonical Grassmannian datum to a rectifiable set.

2.3 Hausdorff measure and density

The Hausdorff measure provides the canonical d-dimensional measure used in rectifiability theory. Density—roughly, the limit of the ratio of Hausdorff measure of the set inside balls to the measure of the ball in d dimensions—captures how much of the set is present near a point. Density values are fundamental for distinguishing smooth-like points from more singular behavior.

2.4 Rectifiable charts and parametrizations

Rectifiability is often described using rectifiable charts: Lipschitz parametrizations whose images cover the set up to negligible error. These charts allow one to transfer many analytic questions to the parameter domain, where measure-theoretic tools and differentiation along Lipschitz maps are available. In rectifiable varifolds, these parametrizations underpin the construction of both the mass distribution and the tangent-plane structure.

3 Rectifiable Varifolds: Definition and Structure

3.1 Decomposition into rectifiable components

A rectifiable varifold is built so that its underlying support is countably rectifiable with respect to its associated mass measure. More concretely, the varifold’s mass concentrates on a set that can be decomposed into countably many Lipschitz pieces (again up to a negligible set for the induced measure). This decomposition is not unique but enables a consistent geometric interpretation.

3.2 Associated varifold measures from rectifiable sets

Given a countably rectifiable set, one can define a varifold by pushing forward the relevant Hausdorff measure using the map that assigns to almost every point its approximate tangent plane. The resulting measure on the space–Grassmannian product reflects both where the set lies and which tangent directions are present at each point.

3.3 Weight (multiplicity) functions

Rectifiable varifolds typically include a nonnegative weight (multiplicity) function defined on the rectifiable set. This weight records how many times the varifold covers a tangent plane at a point, or, more generally, the local density of mass relative to Hausdorff measure. In applications, multiplicity distinguishes, for example, a single surface from a surface that is “stacked” with higher intensity.

3.4 Tangent-plane encoding on the Grassmannian

The varifold’s measure on the Grassmannian factor is concentrated on the approximate tangent planes of the rectifiable support. In the simplest rectifiable case, the tangent-plane dependence is deterministic at almost every point: for almost every location, the varifold assigns all its tangent-plane mass to the unique approximate tangent plane there (possibly scaled by multiplicity). This tangent encoding is the key structural property that characterizes rectifiable varifolds among general varifolds.

4 Examples and Canonical Constructions

4.1 Smooth submanifolds as rectifiable varifolds

If a d-dimensional submanifold of Euclidean space is embedded with sufficient smoothness, it is rectifiable. One can define a rectifiable varifold by taking the Hausdorff measure on the submanifold and pairing it with the tangent d-plane at each point. With unit multiplicity, the varifold corresponds closely to integration over the submanifold and reproduces standard formulas from differential geometry in a weak measure-theoretic language.

4.2 Lipschitz images and graph-type examples

Lipschitz parametrized sets provide canonical examples of countably rectifiable objects. For instance, the graph of a Lipschitz function from a d-plane into an orthogonal complement defines a rectifiable set. The approximate tangent plane exists almost everywhere on such graphs, and the induced varifold uses these planes together with Hausdorff measure. Graph-type examples illustrate how nonsmoothness in the underlying set can still yield well-defined tangent data almost everywhere.

4.3 Finite unions with multiplicity

Another standard construction uses finite unions of rectifiable pieces. If a set is the union of finitely many Lipschitz images, one can define a rectifiable varifold by adding the contributions of each piece. Multiplicity can be constant on each component, or more generally described by measurable weights. At intersection regions, the varifold naturally superposes the masses and tangent-plane contributions from the different components.

4.4 Convergence scenarios producing rectifiable limits

Rectifiable varifolds arise naturally as limits of sequences of smooth or Lipschitz surfaces under suitable compactness assumptions, such as uniform bounds on mass and appropriate control of generalized curvature through first variation. In many geometric settings, minimizing sequences or critical sequences have subsequences converging (in the varifold sense) to rectifiable varifolds. The limit may still contain singularities, but rectifiability ensures that its tangent-plane structure remains meaningful almost everywhere.

5 Properties and Basic Estimates

5.1 Mass and support relationships

The mass measure of a varifold is obtained by projecting the varifold measure from the Grassmannian back to the ambient space. For a rectifiable varifold, this mass measure is absolutely continuous with respect to the d-dimensional Hausdorff measure restricted to the rectifiable support. Consequently, the support of the mass measure aligns with the set where the rectifiable structure holds, up to negligible sets.

5.2 Density and rectifiability implications

Density properties provide insight into how close the varifold is to being smooth. For rectifiable varifolds, densities are finite almost everywhere on the support under typical assumptions, and density bounds are connected to quantitative rectifiability. In applications, lower and upper density estimates often serve as tools to rule out pathological behavior and to locate potential singular sets.

5.3 Typical regularity versus singular sets

Even when the varifold is rectifiable, it may fail to be regular everywhere. However, rectifiability implies that the set behaves like a manifold at almost every point in a measure-theoretic sense. The remaining exceptional set—where approximate tangent planes or smooth structure may not match well—forms the singular set, whose size can often be estimated using monotonicity and dimension-reduction principles in specific variational contexts.

5.4 Behavior under pushforward by Lipschitz maps

Rectifiable varifolds behave well under Lipschitz mappings. If one pushes a rectifiable set forward by a Lipschitz map, the image remains countably rectifiable under mild conditions. The corresponding varifold pushforward transforms both the mass distribution and tangent-plane information via the differential (where it exists) and an appropriate geometric Jacobian factor. This allows geometric operations to be performed without leaving the rectifiable category.

6 Operations on Rectifiable Varifolds

6.1 Restriction to subsets

One can restrict a varifold to a measurable subset of the ambient space. For rectifiable varifolds, the restriction corresponds to limiting the integration to points of the rectifiable support lying in the chosen subset, while maintaining the tangent-plane encoding inherited from those points. This operation is useful for local analysis, such as deriving estimates in balls or on regions where test functions are supported.

6.2 Pushforward and pullback viewpoints (where applicable)

Pushforward maps transport a varifold along a map of the ambient space, producing a new varifold whose mass and tangent data reflect the geometry of the transformation. Pullback is more delicate and depends on the structure of the map; nonetheless, for suitable smoothness and transversality, one can define meaningful preimage-type constructions. In practice, pushforward is the primary robust operation for rectifiable varifolds, especially in geometric approximation schemes.

6.3 Adding varifolds and multiplicity scaling

Varifolds form a linear structure at the level of measures: one can add varifolds and scale multiplicities by nonnegative constants. For rectifiable varifolds, addition corresponds to superposing weighted rectifiable sets, including cases where tangent planes differ at the same spatial location. This superposition principle is foundational for describing limits where multiple sheets appear, vanish, or coalesce.

6.4 Disintegration with respect to densities (conceptual)

A conceptual way to analyze rectifiable varifolds is to disintegrate the varifold measure relative to its mass density with respect to Hausdorff measure. This separates the “where” information (the rectifiable set and its distribution of points) from the “how much weight” information (multiplicity) and the “what tangent” information (Grassmannian concentration). Such disintegration clarifies which components of the varifold are determined by the underlying set and which reflect multiplicity or other weighting effects.

7 Connections to Variational Problems

7.1 Weak formulations for surface energies (overview)

Many surface energies admit weak formulations suitable for nonsmooth limits. Instead of requiring differentiability of the surface, one tests the energy against variations expressed in terms of flows or deformations. Varifolds provide the natural object upon which these weak formulations are defined: integrals of geometric quantities are interpreted as integrals against the varifold measure, and tangent-plane information guides how curvature-type terms are modeled.

7.2 Role of rectifiable varifolds in compactness

Compactness results often show that from a sequence of surfaces with uniform energy control, one can extract a convergent subsequence whose limit is a rectifiable varifold. Rectifiability is significant because it ensures the limit is geometrically interpretable: tangent planes exist almost everywhere and the limiting object does not become purely unstructured measure. This property is frequently a key step toward proving existence of generalized minimizers.

7.3 Approximation of minimizers by rectifiable objects

When a generalized minimizer exists in the varifold category, rectifiable approximations can often be constructed using sequences of smooth or Lipschitz competitors. Conversely, minimizers in geometric measure theory can sometimes be approximated by rectifiable objects with controlled multiplicity and tangent behavior. This approximation theme links the abstract measure-based framework to more concrete geometric constructions.

7.4 First variation as a weak geometric derivative (overview)

First variation provides a way to characterize stationarity: a varifold is stationary if its first variation vanishes for all admissible deformations. In the rectifiable setting, first variation can be interpreted using the weak geometric derivative of the associated mass distribution, with tangent-plane structure supplying the correct notion of directionality. While smooth surfaces lead to classical mean curvature, the varifold approach extends the concept to nonsmooth and potentially singular configurations.

8 Measure-Theoretic Tools Used in the Theory

8.1 Integration with respect to varifold measures

The central computational device is integration of test functions against the varifold measure on space–Grassmannian. If a test function depends on position and a d-plane, its integral against the varifold gives the weak quantity corresponding to evaluating how the surface “looks” at each point and orientation. For rectifiable varifolds, integration can often be rewritten as integrals over the underlying rectifiable set with the tangent-plane assignment inserted.

When associating tangent planes to almost every point, one must ensure measurability of the tangent-plane map into the Grassmannian. Rectifiability theory provides frameworks guaranteeing that approximate tangent planes can be chosen in a measurable manner, enabling the construction of the varifold measure and the validity of disintegration and pushforward operations.

8.3 Typical uses of densities and tangent measures

Densities and tangent measures are used to zoom in on the varifold at small scales and to detect limiting structures near a point. In rectifiable contexts, tangent measures can help classify blow-up limits and identify whether the varifold resembles a plane with multiplicity or a more complex structure. These tools are especially relevant for understanding singularities and for proving partial regularity statements.

8.4 Compactness and semicontinuity frameworks (overview)

The theory relies on compactness principles for Radon measures and on semicontinuity of energy-like functionals under varifold convergence. Typically, one shows that mass is lower semicontinuous and that certain curvature-controlled quantities behave well under limits. Together with rectifiability preserved (under appropriate bounds), these frameworks ensure that limiting variational objects exist and inherit key analytic properties from the approximating sequence.

9 Notation and Conventions

9.1 Common symbols for Grassmannians and mass

Standard conventions denote the Grassmannian of d-dimensional (unoriented) planes in an n-dimensional ambient space by a symbol such as \(G(n,d)\). The mass of a varifold is commonly denoted by a measure or scalar derived from the projection onto the ambient space. Consistent notation is important because integrands depend on both spatial variables and Grassmannian elements.

9.2 Naming conventions for rectifiable varifolds

Rectifiable varifolds are often named by their dimension and ambient setting, and sometimes by whether they are integer-multiplicity (a stronger condition) or merely real-valued weights. In this entry, the emphasis is on the general rectifiable structure: countable rectifiability of the underlying support combined with a tangent-plane measure supported on approximate tangents.

9.3 Dimensional parameters and scaling

Dimensions determine which Hausdorff measure is used, which Grassmannian factor is relevant, and how scaling behaves. Under dilation of the ambient space, the d-dimensional Hausdorff measure scales predictably, and varifold measures transform accordingly. Scaling laws are frequently used in monotonicity-type arguments and blow-up analyses.

9.4 Orientationless versus oriented variants (comparative overview)

Orientationless varifolds treat planes as elements of an unoriented Grassmannian, so reversing a normal orientation does not change the tangent data. Oriented variants enrich the tangent information by including a choice of orientation, which can be useful for sign-sensitive quantities. Many foundational compactness and rectifiability results carry over with modifications, but the orientationless framework is the most common baseline for general geometric measure theory.