Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Comparison geometry for integral Bakry-\'Emery Ricci tensor bounds   We prove mean curvature and volume comparison estimates on smooth metric
measure spaces when their integral Bakry-\'{E}mery Ricci tensor bounds,
extending Wei-Wylie's comparison results to the integral case. We also apply
comparison results to get diameter estimates, eigenvalue estimates and volume
growth estimates on smooth metric measure spaces with their normalized integral
smallness for Bakry-\'{E}mery Ricci tensor. These give generalizations of some
work of Petersen-Wei, Aubry, Petersen-Sprouse, Yau and more.

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The categories of this paper are: q-alg
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The categories of this paper are: ao-sci
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The categories of this paper are: stat
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The categories of this paper are: eess
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: physics
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ph
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The categories of this paper are: cs
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The categories of this paper are: hep-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys