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

We consider biharmonic maps $ϕ:(M,g)\rightarrow (N,h)$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $α$ satisfies $1<α<\infty$. If for such an $α$, $\int_M|τ(ϕ)|^αdv_g<\infty$ and $\int_M|dϕ|^2dv_g<\infty,$ where $τ(ϕ)$ is the tension field of $ϕ$, then we show that $ϕ$ is harmonic. For a biharmonic submanifold, we obtain that the above assumption $\int_M|dϕ|^2dv_g<\infty$ is not necessary. These results give affirmative partial answers to the global version of generalized Chen's conjecture.

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