Family 346Differential geometryNo LeanNot yet digested
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 , 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 in this family · 2
A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold
Oct 5, 2026·varifold-singular-dimension.pdf
The singular set of every stationary integral m-varifold in an open Euclidean set has Hausdorff dimension at most , 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
Sep 23, 2026·paper.pdf
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.