Chen's conjecture and e-superbiharmonic submanifolds of Riemannian manifolds

B.-Y. Chen famously conjectured that every submanifold of Eu- clidean space with harmonic mean curvature vector is minimal. In this note we prove a much more general statement for a large class of submanifolds sat- isfying a growth condition at innity. We discuss in particular two popular competing natural interpretations of the conjecture when the Euclidean back- ground space is replaced by an arbitrary Riemannian manifold. Introducing the notion of "-superbiharmonic submanifolds, which contains each of the pre- vious notions as special cases, we prove that "-superbiharmonic submanifolds of a complete Riemannian manifold which satisfy a growth condition at innity are minimal.