Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp Entropy Bounds for Self-Shrinkers in Mean Curvature Flow   Let $M\subset {\mathbf R}^{m+1}$ be a smooth, closed, codimension-one
self-shrinker (for mean curvature flow) with nontrivial $k^{\rm th}$ homology.
We show that the entropy of $M$ is greater than or equal to the entropy of a
round $k$-sphere, and that if equality holds, then $M$ is a round $k$-sphere in
${\mathbf R}^{k+1}$.

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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: econ
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: physics
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int