A general regularity theory for stable codimension 1 integral varifolds

TL;DR

This paper establishes a comprehensive geometric regularity criterion for stable codimension 1 integral varifolds, clarifying the nature and size of their singular sets across various dimensions without initial smallness assumptions.

Contribution

It provides a necessary and sufficient condition for regularity of stable codimension 1 varifolds, resolving a longstanding open problem in geometric measure theory.

Findings

Singular set is empty in dimension ≤6.

Singular set is discrete in dimension 7.

Singular set has Hausdorff codimension ≥7 in dimensions ≥8.

Abstract

We give a necessary and sufficient geometric structural condition for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities; when this condition is satisfied, the singular set is empty if the dimension of the varifold is 6 or smaller, discrete if the dimension is 7 and has Hausdorff codimension at least 7 if the dimension is 8 or larger. No initial smallness assumption on the singular set is necessary for these conclusions. The work in particular settles the long standing question, left open by the Schoen-Simon Regularity Theory, as to which weakest size hypothesis on the singular set guarantees the validity of the above conclusions. An optimal strong maximum principle for stationary codimension 1 integral varifolds follows.