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

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$H$isthemeancurvatureof$(M,g)$in$(N,h)$,then$(M,g)$isminimalin$(N,h)$\mathrm{.}Thus,foracounterexample$(M,g)$ in the case of hypersurfaces to the generalized Chen’s conjecture (cf. Sect.1), it holds that ${\int }_M\mathrm{\mid }H{\mathrm{\mid }}^2{v}_g=\mathrm{\infty }$.