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

Abstract

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