Remarks on the nonexistence of biharmonic maps

Abstract

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 $\varphi :(M,g)\to (N,h)$ 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 $\varphi$ is a harmonic map if one of the following conditions holds: (i) $|d\varphi |$ is bounded in $L^q(M)$ and ${\int }_M|\tau (\varphi ){|}^{p}d{v}_g<\infty ,$ for some $1\le q\le \infty$, $1<p<\infty$; or (ii) $Vol(M)=\infty$ and ${\int }_M|\tau (\varphi ){|}^{p}d{v}_g<\infty ,$ for some $1<p<\infty$. In addition if $N$ has negative sectional curvature, we assume that $rank\varphi (q)\ge 2$ for some $q\in M$ and ${\int }_M|\tau (\varphi ){|}^{p}d{v}_g<\infty ,$ 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}).

arXiv commentWe add an assumption on the rank of ϕin Theorem 1.3