A Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with small energies

Abstract

We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension \(n\) that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the $L^p$-norm of the tension field is bounded and the $n$-energy of the map is sufficiently small then every biharmonic map must be harmonic, where \(2<p<n\).