Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Heat Kernel on Smooth Metric Measure Spaces with Nonnegative Curvature   We derive a local Gaussian upper bound for the $f$-heat kernel on complete
smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-\'{E}mery
Ricci curvature, which generalizes the classic Li-Yau estimate. As
applications, we obtain a sharp $L_f^1$-Liouville theorem for $f$-subharmonic
functions and an $L_f^1$-uniqueness property for nonnegative solutions of the
$f$-heat equation, assuming $f$ is of at most quadratic growth. In particular,
any $L_f^1$-integrable $f$-subharmonic function on gradient shrinking or steady
Ricci solitons must be constant. We also provide explicit $f$-heat kernel for
Gaussian solitons.

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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: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: chao-dyn
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The categories of this paper are: comp-gas
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: mtrl-th
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The categories of this paper are: eess
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The categories of this paper are: hep-ph
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The categories of this paper are: nlin
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The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: quant-ph