Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Remarks on the nonexistence of biharmonic maps   In this short note we study nonexistence result of biharmonic maps from a
complete Riemannian manifold into a Riemannian manifold with nonpositive
sectional curvature. Assume that $\phi:(M,g)\to (N, h)$ is a biharmonic map,
where $(M, g)$ is a complete Riemannian manifold and $(N,h)$ a Riemannian
manifold with nonpositive sectional curvature, we will prove that $\phi$ is a
harmonic map if one of the following conditions holds: (i) $|d\phi|$ is bounded
in $L^q(M)$ and $ \int_M|\tau(\phi)|^pdv_g<\infty, $ for some $1\leq
q\leq\infty$, $1< p<\infty$; or (ii) $Vol(M)=\infty$ and $
\int_M|\tau(\phi)|^pdv_g<\infty, $ for some $1< p<\infty$.
  In addition if $N$ has negative sectional curvature, we assume that
$rank\phi(q)\geq2$ for some $q\in M$ and $\int_M|\tau(\phi)|^pdv_g<\infty, $
for some $1< p<\infty$. These results improve the related theorems due to Baird
et al.(cf. \cite{BFO}), Nakauchi et al.(cf. \cite{NUG}), Maeta(cf. \cite{Ma})
and Luo(cf. \cite{Luo}).

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The categories of this paper are: mtrl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: q-fin
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The categories of this paper are: q-alg
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The categories of this paper are: quant-ph
3
The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ex
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The categories of this paper are: stat
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph