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

Abstract

We consider polyharmonic maps $\varphi :(M,g)\to$\mathbb{E}^n$oforderkfromacompleteRiemannianmanifoldintotheEuclideanspaceandlet$p$bearealconstantsatisfying$1<p<\infty$.(i)If,$\int_M|W^{k-1}|^p dv_g<\infty,$and$\int_M|\bar \nabla W^{k-2}|^2dv_g<\infty.$Then$ϕ$isapolyharmonicmapoforderk-1.(ii)If,$\int_M|W^{k-1}|^p dv_g<\infty,$and$Vol (M,g)=\infty.$Then$ϕ$isapolyharmonicmapoforderk-1.Here,$W^s=\barΔ^{s-1}τ(ϕ) (s=1,2,...)$and$W^0=ϕ$. As a corollary, we give an affirmative partial answer to generalized Chen's conjecture.

9 pages. arXiv admin note: text overlap with arXiv:1305.7065

References in corpus (3)

• Recent developments of biharmonic conjecture and modified biharmonic conjectures
• Polyharmonic maps into the Euclidean space
• On biharmonic submanifolds in non-positively curved manifolds