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.