A general regularity theory for stable codimension 1 integral varifolds

We give a necessary and sufficient geometric structural condition, which we call the α-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 α-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,α 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 α-Structural Hypothesis for some α ∈ (0, 1/2), its singular set is empty if n ≤ 6, discrete if n = 7 and has Hausdorff dimension ≤ n -7 if n ≥ 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 α-Structural Hypothesis for some α ∈ (0, 1/2).The α-Structural Hypothesis on an n-varifold for any α ∈ (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.Contents NESHAN WICKRAMASEKERA 4. Proper blow-up classes 861 5. Lipschitz approximation and coarse blow-ups 869 6.An outline of the proof of the main theorems 873 7. Nonconcentration of tilt-excess 875 8. Properties of coarse blow-ups: Part I 879 9. Properties of coarse blow-ups: Part II 888 10.Parametric L 2 -estimates in terms of fine excess 894 11.Blowing up by fine excess 919 12. Continuity estimates for the fine blow-ups and their derivatives 923 13.Improvement of fine excess 936 14.Properties of coarse blow-ups: Part III 947 15.The Sheeting Theorem 957 16.The Minimum Distance Theorem 968 17.The Regularity and Compactness Theorem 994 18. Generalization to Riemannian manifolds 996 19.A sharp varifold maximum principle 1004