A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds

In this note we prove a nonexistence result for proper biharmonic maps from complete non-compact Riemannian manifolds of dimension \(m=\dim M\geq 3\) with infinite volume that admit an Euclidean type Sobolev inequality into general Riemannian manifolds by assuming finiteness of $\|\tau(\phi)\|_{L^p(M)}, p>1$ and smallness of $\|d\phi\|_{L^m(M)}$. 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.