POLYHARMONIC MAPS OF ORDER k WITH FINITE Lp K-ENERGY INTO EUCLIDEAN SPACES

We consider polyharmonic maps phi : (M, g) -> E-n of order k from a complete Riemannian manifold into the Euclidean space and let p be a real constant satisfying 2 <= p < infinity. (i) If integral(M) vertical bar Wk-1 vertical bar(p)dv(g) < infinity and integral(M) vertical bar(del) over barW(k-2)vertical bar(2)dv(g) < infinity, then phi is a polyharmonic map of order k - 1. (ii) If integral(M) vertical bar Wk-1 vertical bar(p)dv(g) < infinity and Vol(M, g) = infinity, then phi is a polyharmonic map of order k - 1. Here. W-s = <(Delta)overbar>(s-1) tau(phi) (s = 1,2 ...) and W-0 = phi. As a corollary, we give an affirmative partial answer to the generalized Chen conjecture.