A general regularity theory for stable codimension 1 integral varifolds

Abstract

We give a necessary and sufficient geometric structural condition, which we call the $\alpha$-Structural Hypothesis, 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. The $\alpha$-Structural Hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded $C^{1, \alpha}$ hypersurfaces-with-boundary meeting (only) along their common boundary. We establish that whenever a stable integral $n$-varifold on a smooth $(n+1)$-dimensional Riemannian manifold satisfies the $\alpha$-Structural Hypothesis for some $\alpha \in (0, 1/2)$, its singular set is empty if $n \leq 6$, discrete if $n =7$ and has Hausdorff dimension $\leq n-7$ if $n \geq 8$; in view of well-known examples, this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses. We also establish compactness of mass-bounded subsets of the class of stable codimension 1 integral varifolds satisfying the $\alpha$-Structural Hypothesis for some $\alpha \in (0, 1/2)$. The $\alpha$-Structural Hypothesis on an $n$-varifold for any $\alpha \in (0, 1/2)$ is readily implied by either of the following two hypotheses: (i) the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary, (ii) the singular set of the varifold has vanishing $(n-1)$-dimensional Hausdorff measure. Thus, our theory subsumes the well-known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions.
An optimal strong maximum principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems.