A general regularity theory for stable codimension 1 integral varifolds. (English) Zbl 1307.58005

From the abstract: ”We give a necessary and sufficient geometric structural condition, which we call \(\alpha\)-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 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.”


58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49N60 Regularity of solutions in optimal control
49Q05 Minimal surfaces and optimization
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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