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.
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In Almost-everywhere regularity of stationary integral varifolds, 23 September 2026
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Every stationary integral m-varifold in an open subset of \R^{m+n} satisfies
Here m,n\geq1 are arbitrary. No stability, minimizing, orientability, density upper bound, or codimension-one hypothesis is required.
In Almost-everywhere regularity of stationary integral varifolds, 23 September 2026
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Let m,n\geq1 and let V be an integral m-varifold on the full unit round sphere S^{m+n}, stationary for variations tangent to the sphere. A support point is regular if on a spherical neighborhood V=q|\Sigma| for an integer q\geq1 and a smooth embedded minimal m-submanifold \Sigma of the sphere. If \operatorname{Sing}_S V denotes the remaining support points, then
No fixed upper bound on mass, density, or multiplicity is required, nor are stability, orientability, or mod-two cyclicity. The same conclusion holds on every round sphere of positive radius.
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Why this passage is shown
Theorem 1.1
In Almost-everywhere regularity of stationary integral varifolds, 23 September 2026
Why this passage is shown
Corollary 1.2 (Round spheres)
In Almost-everywhere regularity of stationary integral varifolds, 23 September 2026
Why this passage is shown
Theorem 1.1
In A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold, 5 October 2026