Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Biharmonic submanifolds with parallel mean curvature in
  $\mathbb{S}^n\times\mathbb{R}$   We find a Simons type formula for submanifolds with parallel mean curvature
vector (pmc submanifolds) in product spaces $M^n(c)\times\mathbb{R}$, where
$M^n(c)$ is a space form with constant sectional curvature $c$, and then we use
it to prove a gap theorem for the mean curvature of certain complete
proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces
in $\mathbb{S}^n(c)\times\mathbb{R}$.

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The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
0
The categories of this paper are: hep-ph
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: cond-mat
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The categories of this paper are: eess