Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature

Abstract

In this paper, we show that, for a biharmonic hypersurface $(M,g)$ of a Riemannian manifold $(N,h)$ of non-positive Ricci curvature, if ${\int }_M|H{|}^2{v}_g<\infty$, where $H$ is the mean curvature of $(M,g)$ in $(N,h)$, then $(M,g)$ is minimal in $(N,h)$. Thus, for a counter example $(M,g)$ in the case of hypersurfaces to the generalized Chen's conjecture (cf. Sect.1), it holds that ${\int }_M|H{|}^2{v}_g=\infty$.

8 pages