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

TL;DR

This paper investigates polyharmonic maps of order k with finite L^p k-energy into Euclidean spaces, establishing conditions under which such maps reduce to lower-order polyharmonic maps and addressing a version of Chen's conjecture.

Contribution

It provides new conditions for polyharmonic maps to be of lower order, extending understanding of their structure and contributing to the generalized Chen's conjecture.

Findings

Under certain integrability conditions, polyharmonic maps of order k are of order k-1.

Finite L^p k-energy implies reduction in harmonic order under specific volume conditions.

Partial affirmative answer to a generalized Chen's conjecture.

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.