A general regularity theory for stable codimension 1 integral varifolds
Abstract
We give a necessary and sufficient geometric structural condition 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; when this condition is satisfied, the singular set is empty if the dimension of the varifold is 6 or smaller, discrete if the dimension is 7 and has Hausdorff codimension at least 7 if the dimension is 8 or larger. No initial smallness assumption on the singular set is necessary for these conclusions. The work in particular settles the long standing question, left open by the Schoen-Simon Regularity Theory, as to which weakest size hypothesis on the singular set guarantees the validity of the above conclusions. An optimal strong maximum principle for stationary codimension 1 integral varifolds follows.
Cite
@article{arxiv.0911.4883,
title = {A general regularity theory for stable codimension 1 integral varifolds},
author = {Neshan Wickramasekera},
journal= {arXiv preprint arXiv:0911.4883},
year = {2013}
}
Comments
Sec 16 expanded per referee's suggestion, elaborating on first (and only) non-inductive use of key structural hypothesis (in Theorem 16.1); Sec 10 expanded (more detailed proof of Theorem 10.1, better organized proof of Cor. 10.2); stability hypothesis slightly weakened; Cor. 3.2 added; minor errors corrected, cosmetic changes made; more references added. 129 pages. To appear in Annals of Math