Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

Abstract

We consider biharmonic maps $\phi :(M,g)\rightarrow (N,h)$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $p$ satisfies $2\le p <\infty$. If for such a $p$, $\int _M|\tau (\phi )|^{ p }\,\mathrm{d}v_g<\infty$ and $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty ,$ where $\tau (\phi )$ is the tension field of $\phi$, then we show that $\phi$ is harmonic. For a biharmonic submanifold, we obtain that the above assumption $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty$ is not necessary. These results give affirmative partial answers to the global version of generalized Chen’s conjecture.