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 3 or more embedded C 1,α hypersurfaceswith-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.10. Parametric L 2 -estimates in terms of fine excess 40

We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals-Cahn-Hilliard energy functionals with perturbation parameter ! 0 þ is supported by an embedded stable minimal hypersurface which in low dimensions has no singularities and in general dimensions has possibly a closed singular set of co-dimension f 7.This result was previously known in case the critical points are local minimizers of energy, in which case the limit-interface is locally area minimizing and its (normalized) multiplicity is 1 a.e.Our theorem uses earlier work of the first author establishing stability of the limitinterface as an integral varifold, and relies on a recent general theorem of the second author for its regularity conclusions in the presence of higher multiplicity.Y.T. is partially supported by JSPS Grant-in-aid for scientific research 21340033. Both authors wish to thank Mathematisches Forschungsinstitut Oberwolfach for providing the opportunity to initiate this joint research.Brought to you by | Purdue University Libraries Authenticated Download Date | 5/25/15 7:28 PM W ju e i j þ E e i ðu e i Þ e c for some c f 1 independent of i, then either (a) u e i ! 1 or u e i ! À1 locally uniformly in W, or (b) there exists an embedded smooth stable minimal hypersurface M of W such that after passing to a subsequence of fe i g without changing notation, for each fixed s A ð0; 1Þ, the interface regions fx A W : ju e i ðxÞj < sg converge locally in Hausdor¤ distance to M; furthermore, in case (b), the interior singular set sing M 1 ðMnMÞ X W is empty for 2 e n e 7, sing M is (at most) a discrete set for n ¼ 8 and sing M has Hausdor¤ dimension at most n À 8 for n f 9. This regularity result was known for the limit-interfaces corresponding to sequences fu e i g of energy minimizers since in that case the limit-interfaces are area-minimizing and the well known regularity theory for locally area minimizing currents is applicable. The new result in this paper is that the 192 Tonegawa and Wickramasekera, Stable phase interfaces in the van der Waals-Cahn-Hilliard theory Brought to you by | Purdue University Libraries Authenticated Download Date | 5/25/15 7:28 PM Tonegawa and Wickramasekera, Stable phase interfaces in the van der Waals-Cahn-Hilliard theory Brought to you by | Purdue University Libraries Authenticated Download Date | 5/25/15 7:28 PM

We establish a frequency function monotonicity formula for two‐valued C1,α solutions to the minimal surface system on n‐dimensional domains. We also establish the sharp regularity result that such solutions are of class C1, 1/2, and that their branch sets, if nonempty, have Hausdorff dimension equal to n‐2.© 2016 Wiley Periodicals, Inc.

We describe a method for producing smooth 2-valued minimal graphs over the cylindrical region (D \{0})×R n−2 , where D is the disk in R 2 , subject to given continuous 2-valued boundary data on ∂D × R n−2 . Subject to appropriate symmetry assumptions, the construction produces branched minimal immersions in D × R n−2 × R with prescribed boundary and branching at every point of {0} × R n−2 , and we also discuss the nature of the possible singularities along {0} × R n−2 in case of general boundary data.

Weakly stable constant mean curvature (CMC) hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations. We establish a pointwise curvature estimate (in the non-singular dimensions) and a sheeting theorem (in all dimensions) for weakly stable CMC hypersurfaces, giving an effective version of the compactness theorem for weakly stable CMC hypersurfaces established in the recent work of the first-and third-named authors [BW18]. Our results generalize the curvature estimate and the sheeting theorem proven respectively by Schoen-Simon-Yau and Schoen-Simon for strongly stable hypersurfaces.