a varifold

Imagine you want a notion of 'surface' so flabby that the limit of any reasonable sequence of surfaces is still one, that does not need an orientation, and that remembers not just where the surface is but which way it is tilted at each point. A varifold is exactly that: a generalized surface that records, simultaneously, a distribution of mass in space and the tangent-plane direction attached to each bit of that mass — but with no orientation, so opposite-facing sheets do not cancel. It is the right object for studying surfaces that are merely critical for area (not necessarily minimizing) and for surfaces evolving by mean-curvature flow.

Precisely, a k-varifold in R^n is a Radon measure on the Grassmannian bundle R^n x G(n, k), where G(n, k) is the set of k-dimensional planes through the origin. Unpack that: at each point of space the varifold carries a measure on possible tangent k-planes, so it records position-with-direction. An integral varifold is one coming from a k-rectifiable set with integer-valued multiplicity — the generalized-surface case. The crucial operation is the first variation: push the varifold around by a vector field and measure the rate of change of its total mass; the result is a linear functional, and when it vanishes for all compactly supported fields the varifold is stationary — the weak form of 'zero mean curvature'. More generally the first variation, if it is a measure, gives a generalized mean curvature, and Allard's regularity theorem says a stationary (or bounded-mean-curvature) integral varifold is, near most points and under a density bound, a smooth surface. Because varifolds forget orientation, mass cannot cancel: two oppositely oriented sheets lying on top of each other have mass 2, not 0, which is exactly the behaviour you want for min-max constructions where currents would wrongly cancel.

Why varifolds, and not just currents: currents carry orientation and a boundary, which is perfect for minimizing area with fixed boundary; but for unstable critical surfaces (saddle-type minimal surfaces produced by min-max / Almgren-Pitts theory) and for mean-curvature flow with surgery (Brakke flow), orientation is a liability and cancellation is fatal, so the unoriented varifold is the correct setting. Honest cautions. First, a stationary varifold need not be smooth everywhere — Allard regularity gives smoothness only on an open dense set with a density-one bound, and genuine singularities (triple junctions, as in soap-film networks) are allowed and expected; do not assume 'stationary' means 'smooth minimal surface'. Second, the variational theory of varifolds is weaker than that of currents in one respect — there is no good boundary operator and no homology, so varifolds answer 'is there a critical surface?' but currents are still the tool for 'is there a least-area surface with this boundary?'