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.