科研之友
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摘要
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 sing ularities.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,C-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 is an element of (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 alpha-Structural Hypothesis for some alpha is an element of ( 0, 1/2). The alpha-Structural Hypothesis on an n-varifold for any alpha is an element of (0,1/2) is readily implied by either of the following two hypotheses: (i) the vari old 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.
- 出版日期2014-5
