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.
Manuscripts (2)
A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold
October 5, 2026 · source · cite as \cite{OAI:A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold-October-5-2026}
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.
Almost-everywhere regularity of stationary integral varifolds
September 23, 2026 · source · cite as \cite{OAI:Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026}
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.
Result 346 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md