A general regularity theory for stable codimension 1 integral varifolds

By Neshan Wickramasekera. 2009

127 citations recorded by the source corpus.

We give a necessary and sufficient geometric structural condition, which we call the a -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 a -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,a 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 a -Structural Hypothesis for some a?(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 a -Structural Hypothesis for some a?(0,1/2) . The a -Structural Hypothesis on an n -varifold for any a?(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

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3aa95ad659384d88270d8ecc8dd1c89bf94b751a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Ancient asymptotically cylindrical flows and applications (2019, background): “By [Wic14a], the hypotheses imply that sing(M∩H) has Hausdorff dimension at most n−7 and therefore that”
Compactness of minimal hypersurfaces with bounded index (2015, background): “Moreover we mention that a highly non-trivial extension of this work, due to Neshan Wickramasekera [20] would in particular allow one to assume only that Hn−1(sing(M)) = 0 for all dimensions.”
Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates (2018, methodology): “We emphasize here that by using to our curvature estimates, we give an proof of the regularity of Σ that does not rely on the deep works of Wickramasekera [Wic14] and Tonegawa– Wickramasekera [TW12] (cf.”
On the entropy of closed hypersurfaces and singular self-shrinkers (2016, methodology): “The α-structural hypothesis here allows us to use the regularity theory of Wickramasekera [22] to control the singular set; it is automatically satisfied for instance if the singular set has vanishing codimension 1 Hausdorff measure.”
Min-max for phase transitions and the existence of embedded minimal hypersurfaces (2015, background): “In Section 18 of [42], the stability of a varifold V is defined by means of an inequality satisfied by the pushforward of V to a tangent space of M via the inverse of the exponential map.”