Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature

Abstract

In this paper, we show that, for every biharmonic submanifold (M, g) of a Riemannian manifold (N, h) with non-positive sectional curvature, if $${\int_M\vert \eta \vert^2 v_g < \infty}$$ , then (M, g) is minimal in (N, h), i.e., $${\eta\equiv0}$$ , where η is the mean curvature tensor field of (M, g) in (N, h). This result gives an affirmative answer under the condition $${\int_M\vert \eta \vert^2 v_g < \infty}$$ to the following generalized Chen’s conjecture: every biharmonic submanifold of a Riemannian manifold with non-positive sectional curvature must be minimal. The conjecture turned out false in case of an incomplete Riemannian manifold (M, g) by a counter example of Ou and Tang (in The generalized Chen’s conjecture on biharmonic sub-manifolds is false, a preprint, 2010).