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Almost-everywhere regularity of stationary integral varifolds

Let VV be a stationary integral mm-varifold in an open set U⊂Rm+nU\subset\mathbb R^{m+n}: an integer-multiplicity rectifiable mm-dimensional measure whose first variation of area vanishes. A support point is regular if near it VV is a positive integer multiple of a single smooth embedded minimal submanifold. Allard (1972) proved that the regular set is relatively open and dense and that density-one points are regular, but at points with higher-multiplicity planar tangents his theorem does not apply. Simon's notes record the question, and Brena, Decio and De Lellis (2025, Conjecture 1.2) conjecture a positive answer. Is Hm(Sing V)=0\mathcal H^m(\mathrm{Sing}\,V)=0 for every stationary integral mm-varifold, in every dimension and codimension?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Geometric measure theory: regularity of stationary varifolds
Posed by
Leon Simon, Introduction to Geometric Measure Theory (notes, 2018), Ch. 5 Remark 6.4; stated as Conjecture 1.2 by Brena, Decio and De Lellis, Rend. Lincei Mat. Appl. 36 (2025)
Year posed
2018
Years open
8y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
58 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: every stationary integral mm-varifold in an open subset of Rm+n\mathbb R^{m+n}, m,n≥1m,n\ge1, has Hm(Sing V)=0\mathcal H^m(\mathrm{Sing}\,V)=0, with no stability, minimizing, orientability, density-bound or codimension hypothesis. Corollary 1.2 gives the same on round spheres. The main new estimate is a signed excess bound in flat cylinders. The companion A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold (5 Oct 2026) claims the stronger bound dim⁡HSing V≤m−1\dim_{\mathcal H}\mathrm{Sing}\,V\le m-1, which implies this one and is a separate entry; it uses this paper's signed-excess theorem as an input. Neither paper treats non-Euclidean ambient metrics, where De Lellis-Hirsch-Lihn-Spolaor construct exotic examples.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1 and Corollary 1.2 of the TeX source, read against the question as Simon and Brena-Decio-De Lellis state it according to the manuscript's citations; the proof was not refereed. The result is for Euclidean open sets and, by a separate cone argument, round spheres; it says nothing about general Riemannian metrics. The release has no Lean formalization for this family, and the README cautions that unformalized results could have issues.

Sources

Changelog1 change

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