Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stable CMC integral varifolds of codimension $1$: regularity and
  compactness   We give two structural conditions on a codimension $1$ integral $n$-varifold
with first variation locally summable to an exponent $p>n$ that imply the
following: whenever each orientable portion of the $C^{1}$-embedded part of the
varifold (which is non-empty by the Allard regularity theory) is stationarity
and the $C^{2}$-immersed part of it is stable with respect to the area
functional for volume preserving deformations, its support, except possibly on
a closed set of codimension $7$, is an immersed constant-mean-curvature (cmc)
hypersurface of class $C^{2}$ that can fail to be embedded only at points where
locally the support is the union of two $C^{2}$ embedded cmc disks with only
tangential intersection. Both structural conditions are necessary for the
conclusions and involve only those parts of the varifold that are made up of
embedded $C^{1, \alpha}$-regular pieces coming together in a regular fashion,
making them easy to check in principle. We show also that any family of
codimension 1 integral varifolds satisfying these structural and variational
hypotheses as well as locally uniform mass and mean curvature bounds is compact
in the varifold topology. Our results generalize both the regularity theory of
the second author (for stable minimal hypersurfaces) and the regularity theory
of Schoen--Simon, for hypersurfaces satisfying a priori a smallness hypothesis
on the singular set in addition to the variational hypotheses). Corollaries of
the main varifold regularity theorem are obtained for sets of locally finite
perimeter, which generalize the regularity theory of
Gonzalez--Massari--Tamanini for boundaries that locally minimize perimeter
subject to the fixed enclosed volume constraint.

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The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
0
The categories of this paper are: hep-ph
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: math-ph
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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: supr-con
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The categories of this paper are: cs