Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Partial regularity at the first singular time for hypersurfaces evolving
  by mean curvature   In this paper, we consider smooth, properly immersed hypersurfaces evolving
by mean curvature in some open subset of $\mathbb{R}^{n+1}$ on a time interval
$(0, t_0)$. We prove that $p$ - integrability with $p\ge 2$ for the second
fundamental form of these hypersurfaces in some space-time region $B_R(y)\times
(0, t_0)$ implies that the $\mathcal{H}^{n+2-p}$ - measure of the first
singular set vanishes inside $B_R(y)$. For $p=2$, this was established
independently by Han and Sun. Our result furthermore generalizes previous work
of Xu, Ye and Zhao and of Le and Sesum for $p\ge n+2$, in which case the
singular set was shown to be empty. By a theorem of Ilmanen, our integrability
condition is satisfied for $p=2$ and $n=2\,$ if the initial surface has finite
genus. Thus, the first singular set has zero $\mathcal{H}^2$- measure in this
case. This is the conclusion of Brakke's main regularity theorem for the
special case of surfaces, but derived without having to impose the area
continuity and unit density hypothesis. It follows from recent work of Head and
of Huisken and Sinestrari that for the flow of closed, $k$ - convex
hypersurfaces, that is hypersurfaces whose sum of the smallest $k$ principal
curvatures is positive, our integrability criterion holds with exponent
$p=n+3-k-\alpha$ for all small $\alpha >0$ as long as $1\le k\le n-1$.
Therefore, the first singular set of such solutions is at most $(k-1)$ -
dimensional, which is an optimal estimate in view of some explicit examples.

19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
21
The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
28
The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: solv-int
0
The categories of this paper are: hep-ph