∑math20

Differential geometry · Result 346

Sharp singular-set bounds for stationary integral varifolds

2 papers

Proves that every stationary integral m-varifold in a Euclidean open set has singular set of Hausdorff dimension at most \(m-1\), 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.

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Papers 2

First page of A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold

A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold

OpenAI · October 5, 2026

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. This sharp bound proves the Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis.

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@misc{OAI:A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026,
  author = {{OpenAI}},
  title = {{A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026/varifold-singular-dimension.pdf}{OAI:A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026}},
  year = {2026}
}
First page of Almost-everywhere regularity of stationary integral varifolds

Almost-everywhere regularity of stationary integral varifolds

OpenAI · September 23, 2026

We 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.

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@misc{OAI:Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026,
  author = {{OpenAI}},
  title = {{Almost-everywhere regularity of stationary integral varifolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026/paper.pdf}{OAI:Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026}},
  year = {2026}
}

Formal verification

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