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A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold
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Abstract
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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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Related prior literature mentioned by others
- Camillo Brena, Stefano Decio and Camillo De Lellis, Remarks and conjectures on stationary varifolds dedicated to Enrico Bombieri (Rendiconti Lincei. Matematica e Applicazioni 36(3), 409–443, 2025; published online 2026-02-03). States the codimension-one singular-set conjecture (Conjecture 1.1) and the measure-zero conjecture (Conjecture 1.2) addressed by the two manuscripts of this family. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Camillo De Lellis, Jonas Hirsch, Andrea Marchese, Salvatore Stuvard, Regularity of area minimizing currents mod p (Geom. Funct. Anal. 30(5), 2020). Proves the same sharp bound $\dim_{\mathcal H}\mathrm{Sing}\le m-1$ in every dimension and codimension for area-minimizing currents mod $p$; a comparison point for the sharpness discussion in Theorem 1.1.
- Spencer Becker-Kahn, Paul Minter, Neshan Wickramasekera, On the Nature of Stationary Integral Varifolds near Multiplicity 2 Planes (arXiv:2507.13148, July 2025). Proves an epsilon-regularity theorem near multiplicity-2 planes under a topological condition on the density-below-2 part; this is the higher-multiplicity planar-tangent regime that the paper calls the remaining difficulty.
- Neshan Wickramasekera, A general regularity theory for stable codimension 1 integral varifolds (Annals of Mathematics 179(3), 2014, 843–1007). Regularity theory for stable codimension-one integral varifolds under the α-Structural Hypothesis; context for the paper's statement that no stability or codimension hypothesis is required.
Digests and expositions
- Latent Seminar, Singular sets of stationary integral varifolds (2026-10-07). Explains both manuscripts’ signed mass–tilt comparison, fitting and frequency arguments, and the additional dimension-reduction steps, explicitly limiting its scope to exposition rather than independent verification of the complete proofs. [family-level]
- Gradient Damage, Yau’s Uniformization Conjecture, Blaschke Conjecture and 8 More (video compilation, 2026-10-15). The chapter beginning at 2:48 illustrates the family’s singular-set claims, with the accompanying description distinguishing the animation from a proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 346
Independent sites; their content is not reviewed here.
Cite
@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}
}Family 346: 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 $m-1$, 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.
Other papers in this family:
- Almost-everywhere regularity of stationary integral varifolds · September 23, 2026