A regularity and compactness theory for immersed stable minimal hypersurfaces of multiplicity at most 2

Abstract

We prove that a stable minimal hypersurface of an open ball which is immersed away from a closed (singular) set of finite co- dimension 2 Hausdorff measure and weakly close to a multiplicity 2 hyperplane must in the interior be the graph over the hyper- plane of a 2-valued function satisfying a local C1,α estimate. This regularity is optimal under our hypotheses. As a consequence, we also establish compactness of the class of stable minimal hypersur- faces of an open ball which have volume density ratios uniformly bounded by 3 − δ for any fixed δ ∈ (0, 1) and interior singular sets of vanishing co-dimension 2 Hausdorff measure. © 2008 Applied Probability Trust.