1 · The essentials
The manuscripts study generalized surfaces that may have irregular points and may carry several overlapping layers. They assume that these surfaces are balanced: small deformations do not change their area to first order. The main manuscript claims that, in ordinary Euclidean space, the irregular part is at least one dimension smaller than the surface itself. The claim covers every positive surface dimension and every positive number of surrounding dimensions beyond it. The bound is described as sharp, meaning that it cannot be lowered in general. A companion manuscript claims that almost every point lies on a smooth piece carrying a fixed positive whole number of layers, both in Euclidean space and on round spheres. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, think of sheets of fabric meeting along seams: the seams can be irregular even when the fabric away from them is smooth. The claimed bound resembles the observation that a seam is smaller in dimension than a sheet, but this picture does not capture the precise balance and whole-number-layer assumptions.
Why it matters
The main manuscript claims a sharp answer to the Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis. It gives a dimension bound, rather than only saying that the irregular part has zero measure in the surface's own dimension. The companion manuscript also addresses almost-everywhere regularity on round spheres.
2 · Going further
The precise Euclidean claim concerns a stationary integral -varifold in an open subset of Euclidean space, with positive dimension and positive codimension. The main manuscript states that . Here the singular set consists of the points where the stated smooth local description fails, and Hausdorff dimension measures the size of that set. The abstract describes the bound as sharp in every positive dimension and codimension.
The source identifies this as the Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis. The companion manuscript addresses their almost-everywhere regularity conjecture: . It states more concretely that near almost every support point, the varifold is a constant positive integer multiple of a smooth embedded minimal submanifold.
The two manuscripts therefore concern complementary levels of control: a sharp dimension bound for the Euclidean singular set, and a zero-measure regularity statement that is also asserted on round spheres. Neither supplied abstract describes the mechanism of the argument, so no argument outline can be supplied from this source. In particular, the spherical statement provided here is almost-everywhere regularity, not the sharper dimension bound.
Useful notions
- Smooth surfaces and submanifolds
- Stationarity under small deformations
- Whole-number multiplicity
- Hausdorff dimension and measure
- Regular and singular points
What was known before / what is new
Before
The source identifies a Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis. It also attributes an almost-everywhere regularity conjecture to them. No earlier bounds or details of prior arguments are supplied.
New
The main manuscript claims the sharp bound for stationary integral varifolds in Euclidean open sets, in every positive dimension and codimension. The companion manuscript claims zero -dimensional Hausdorff measure for the singular set and gives a smooth local description near almost every support point. It asserts the corresponding almost-everywhere statement on round spheres as well.
Limits and assumptions
The claims require stationary integral varifolds of positive dimension and codimension. The dimension bound is stated for Euclidean open sets; only the zero-measure, almost-everywhere regularity statement is supplied for round spheres. These statements allow singular points and do not assert smoothness everywhere. The abstracts provide neither argument details nor sharpness examples, and they identify no further open questions. The main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Stationary integral varifold —
- A generalized surface with positive integer multiplicities whose area has zero first-order change under small deformations.
- Singular set —
- The set of points where the varifold does not have the smooth local description stated in the manuscript.
- Codimension —
- The dimension of the surrounding space minus the dimension of the varifold.
- Hausdorff dimension —
- A notion of dimension used to quantify the size of sets, including irregular ones.
- Almost-everywhere regularity —
- Regularity outside a set of zero Hausdorff measure in the dimension of the varifold.
- Multiplicity —
- The positive whole-number weight describing how many layers the varifold carries locally.
Keywords : Stationary integral varifolds · Singular-set bounds · Almost-everywhere regularity
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves that every stationary integral m-varifold in a Euclidean open set has singular set of Hausdorff dimension at most , a sharp bound in every positive dimension and codimension. On round spheres, the family also proves almost-everywhere regularity: the singular set has zero m-dimensional Hausdorff measure.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold
05/10/2026The singular set of every stationary integral m-varifold in an open Euclidean set has Hausdorff dimension at most , in every positive dimension and codimension. This sharp bound proves the Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis.
Almost-everywhere regularity of stationary integral varifolds
23/09/2026We resolve the almost-everywhere regularity conjecture of Brena, Decio, and De Lellis for stationary integral varifolds of arbitrary positive dimension and codimension in Euclidean open sets. The singular set has zero measure in the dimension of the varifold: near almost every support point, the varifold is a constant positive integer multiple of a smooth embedded minimal submanifold. The corresponding statement also holds on round spheres.