Result 346, Differential geometry

Sharp singular-set bounds for stationary integral varifolds

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

Proof

The bigger picture

Why it matters

Surfaces in geometric equilibrium can have crossings and other singularities, even when no small deformation changes their area to first order. These manuscripts claim precise limits on how large those irregular regions can be.

What changes?

An integral m-varifold is a generalized m-dimensional surface that allows integer multiplicities, representing multiple sheets. It is stationary when every localized deformation has zero first-order effect on area. The manuscript reports that, in any Euclidean open set, its singular set has Hausdorff dimension at most m minus one. Hausdorff dimension measures size even for irregular sets. The bound applies in every positive dimension and codimension, meaning every positive difference between ambient and surface dimensions, and is sharp throughout that range.

What does that help mathematicians do?

This rules out singularities occupying a positive amount of m-dimensional surface measure in Euclidean space. The companion manuscript reports the same zero-measure conclusion on round spheres, without claiming the stronger dimensional bound there. Near almost every support point, the varifold is a constant positive integer multiple of a smooth embedded minimal submanifold. Researchers can therefore distinguish typical smooth behavior from exceptional singular behavior, while sharpness prevents any universally smaller Euclidean dimension bound.

Are there practical applications?

The immediate value is foundational for studying generalized minimal surfaces. The result gives researchers a regularity baseline that does not depend on restricting attention to surfaces with just one extra ambient dimension. It identifies how much singular behavior stationarity alone permits, helping clarify what stronger conclusions would require additional geometric assumptions.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

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

October 5, 2026 35 pages

The singular set of every stationary integral m-varifold in an open Euclidean set has Hausdorff dimension at most m−1m-1, in every positive dimension and codimension. This sharp bound proves the Euclidean singular-set conjecture recorded by Brena, Decio, and De Lellis.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Almost-everywhere regularity of stationary integral varifolds

September 23, 2026 77 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.