A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds

Abstract

In this note we prove a nonexistence result for proper biharmonic maps from complete non-compact Riemannian manifolds of dimension m=dimM≥3 with infinite volume that admit a Euclidean type Sobolev inequality into general Riemannian manifolds by assuming finiteness of ‖τ(ϕ)‖Lp(M),p>1 and smallness of ‖dϕ‖Lm(M). This is an improvement of a recent result of the first named author, where he assumed 2¡p¡m. As applications we also get several nonexistence results for proper biharmonic submersions from complete non-compact manifolds into general Riemannian manifolds.