Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Volume estimate about shrinkers   We derive a precise estimate on the volume growth of the level set of a
potential function on a complete noncompact Riemannian manifold. As
applications, we obtain the volume growth rate of a complete noncompact
self-shrinker and a gradient shrinking Ricci soliton. We also prove the
equivalence of weighted volume finiteness, polynomial volume growth and
properness of an immersed self-shrinker in Euclidean space.

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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ph
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th