The codimension-one conjecture for stationary integer rectifiable varifolds
Let be a stationary integer rectifiable -dimensional varifold, and let its interior singular set consist of the points of that are not regular. The codimension-one conjecture. The interior singular set has codimension not smaller than . This is presented as a well-known conjecture, motivated by the absence of examples whose singular set has dimension bigger than and by partial regularity results.
References
Primary source
Camillo Brena, Stefano Decio and Camillo De Lellis, “Remarks and Conjectures on Stationary Varifolds”, arXiv:2503.00651 (2025).
Progress summary
Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.
Solutions 1
RemarkAI-assistedClaimed by OpenAI. Claims a sharp codimension-one Hausdorff-dimension bound for the singular set of every stationary integral varifold in Euclidean open sets, in arbitrary positive dimension and codimension.See full solution
Claimed by OpenAI. Claims a sharp codimension-one Hausdorff-dimension bound for the singular set of every stationary integral varifold in Euclidean open sets, in arbitrary positive dimension and codimension.
Scope relative to this problem: The source bounds the Hausdorff dimension of the interior singular set by m-1 for stationary integral m-varifolds in Euclidean open sets, without stability or codimension-one ambient restrictions. This directly addresses the stated integer-rectifiable stationary-varifold codimension-one question in that Euclidean setting.
GitHub repository: https://github.com/openai/math
- OpenAI-346-01-A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold.pdfOpen