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 $\phi:(M,g)\to (N, h)$ 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}).