Sharp singular-set bounds for stationary integral varifolds

Claims that the singular points of a stationary integral varifold have dimension at most one less than the varifold.

2 papers September 23, 2026 Explained by AI

A stationary integral varifold is a very general minimal surface, allowed to cross itself and to be stacked with whole-number thickness. Its singular points are where it is not a smooth sheet. A companion paper claims these points take up zero area, settling a conjecture of Brena, Decio and De Lellis. The main paper claims more: the singular points have dimension at most one less than the surface. Two flat sheets crossing along a line show this is the best possible.

First step

1972

William Allard, regularity theory for varifolds

Significance

Major

It claims a sharp answer to a central regularity question for minimal surfaces, with no stability or codimension condition.

Trust

Claimed, not checked

Field

Differential geometry

2 papers

The explanation

Take three straight rays in the plane, starting at one point and spaced 120 degrees apart. Their unit directions are (1, 0), (−1/2, √3/2) and (−1/2, −√3/2). These add up to (0, 0). So the three rays pull on their meeting point in perfect balance, like a soap-film junction. Small pushes do not lower their total length to first order. A shape with this balance is called stationary. The rays are one-dimensional, and the only bad point is the junction, of dimension zero. In space, two flat planes crossing along a line behave the same way. Their bad points form the line, one dimension less than the planes.

A varifold is a very general surface. It may cross itself and carry whole-number thickness, called multiplicity. Stationary means its area does not change to first order under any small pushSource 1. A point is regular if, near it, the varifold is a whole-number multiple of one smooth minimal sheetSource 2. All other points form the singular set.

Allard showed that the regular points are dense, and that points of thickness one are regularSource 3. The hard points sit where several layers stack on one flat planeSource 4. There Allard's theorem does not apply. Brena, Decio and De Lellis conjectured that the singular set has zero area in the varifold's own dimension. They also conjectured the sharper dimension boundSource 5.

The second paper claims the zero-area result in every dimension and codimensionSource 6. Near almost every point, the varifold is a whole-number multiple of one smooth minimal sheet. It also claims the same on round spheresSource 7. Its main estimate compares projected mass and tilt while keeping their leading terms signedSource 8.

The first paper claims the sharp bound. The singular set has Hausdorff dimension at most m − 1 for an m-dimensional stationary integral varifoldSource 9. Hausdorff dimension measures how big a rough set is; a line has dimension 1 and a point 0. The crossing planes show the bound cannot be improvedSource 10. The key obstacle is that zero area alone does not bound dimensionSource 11. The proof argues by contradiction. A too-large singular set would give measures on stacked points that survive a blow-up limit, which the paper rules out.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. Both papers rely on long, delicate estimates in geometric measure theory. No specialist review of either paper has been published.

What it doesn't show. This does not cover curved ambient metrics in general, where other examples behave differently. It has no Lean check.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if the proofs hold: they would settle two conjectures of Brena, Decio and De Lellis on minimal surfaces in Euclidean space. The proofs are long and have no Lean check or published specialist review.

Words used

Varifold: A very general surface, allowed to cross itself and to be counted with whole-number thickness.
Stationary: Having area that does not change to first order under any small push.
Integral: Carrying whole-number thickness, or multiplicity, at almost every point.
Regular point: A point near which the varifold is a whole-number multiple of one smooth minimal sheet.
Singular set: All points of the surface that are not regular.
Hausdorff dimension: A measure of how big a rough set is; a line has dimension 1 and a point 0.
Codimension: How many dimensions the surrounding space has beyond those of the surface.

The original papers (2)

A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold (PDF, opens in a new tab)
October 5, 2026
Almost-everywhere regularity of stationary integral varifolds (PDF, opens in a new tab)
September 23, 2026

All quoted lines (11)

Stationarity is the weakest variational condition naturally imposed on a minimal submanifold: the first variation of area vanishes under every compactly supported ambient deformation.
A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold Back to text
A point $p\in\spt\mu\cap U$ is regular if there are a neighborhood $W$ of $p$ , a smooth embedded minimal $m$ -submanifold $\Sigma\subset W$ , and an integer $q\geq1$ such that the restriction of the varifold to $W$ is $q|\Sigma|$ .
Almost-everywhere regularity of stationary integral varifolds Back to text
Allard's regularity theory gives a relatively open dense regular set and, in particular, regularity at density-one points; see also and.
Almost-everywhere regularity of stationary integral varifolds Back to text
The remaining difficulty is therefore the singular set at higher-multiplicity planar tangents, where Allard's multiplicity-one regularity theorem does not apply.
A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold Back to text
Brena, Decio, and De Lellis separate the dimension bound from the weaker assertion $\Hh^m(\Sing V)=0$ .
A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold Back to text
Every stationary integral $m$ -varifold in an open subset of $\R^{m+n}$ satisfies $\HH^m(\operatorname{Sing}V)=0.$
Almost-everywhere regularity of stationary integral varifolds Back to text
The corresponding statement also holds on round spheres.
Almost-everywhere regularity of stationary integral varifolds Back to text
The main estimate compares projected mass and tilt with their leading terms kept signed.
Almost-everywhere regularity of stationary integral varifolds Back to text
The singular set of every stationary integral $m$ -varifold in an open Euclidean set has Hausdorff dimension at most $m-1$ , in every positive dimension and codimension.
A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold Back to text
The estimate is sharp: the sum of the varifolds of two distinct $m$ -planes meeting along an $(m-1)$ -plane is stationary, and its singular set is exactly that intersection.
A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold Back to text
The almost-everywhere regularity conclusion itself is not an input: a set can have zero $\Hh^m$ measure and still have Hausdorff dimension greater than $m-1$ .
A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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