Browse › Differential geometry › 346 Sharp singular-set bounds for stationary integral varifolds
Almost-everywhere regularity of stationary integral varifolds
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Abstract
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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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)
- 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's fitting and signed-excess argument addresses.
- Camillo De Lellis, Jonas Hirsch, Luca Spolaor, Stationary and stable varifolds with singularities (arXiv:2509.21508, v2 April 2026). Constructs minimal immersions in $C^{1,\alpha}$ metrics with catenoidal necks or floating disks accumulating at an isolated multiplicity-2 flat singular point; relevant to the paper's restriction to Euclidean and round-sphere ambients (cited in the companion codimension-one paper, not in this one).
- 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-one 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: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}
}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: