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

Abstract

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

13 pages

Cited by in corpus (2)

• Some open problems and conjectures on submanifolds of finite type: recent development
• Open problems and conjectures on submanifolds of finite type revisited