Polyharmonic maps of order k with finite L^p k-energy into Euclidean spaces

Abstract

We consider polyharmonic maps $\phi:(M,g)\rightarrow $\mathbb{E}^n$ of order k from a complete Riemannian manifold into the Euclidean space and let $p$ be a real constant satisfying $1<p<\infty$. (i) If, $\int_M|W^{k-1}|^p dv_g<\infty,$ and $\int_M|\bar \nabla W^{k-2}|^2dv_g<\infty.$ Then $\phi$ is a polyharmonic map of order k-1. (ii) If, $\int_M|W^{k-1}|^p dv_g<\infty,$ and $Vol (M,g)=\infty.$ Then $\phi$ is a polyharmonic map of order k-1. Here, $W^s=\bar\Delta^{s-1}\tau(\phi) (s=1,2,...)$ and $W^0=\phi$. As a corollary, we give an affirmative partial answer to generalized Chen's conjecture.