  A submanifold $M^m$ of a Euclidean space $R^{m+p}$ is said to have harmonic
mean curvature vector field if $\Delta \vec{H}=0$, where $\vec{H}$ is the mean
curvature vector field of $M\hookrightarrow R^{m+p}$ and $\Delta$ is the rough
Laplacian on $M$. There is a conjecture named after Bangyen Chen which states
that submanifolds of Euclidean spaces with harmonic mean curvature vector
fields are minimal. In this paper we prove that weakly convex hypersurfaces
(i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic
mean curvature vector fields in Euclidean spaces are minimal. Furthermore we
prove that weakly convex biharmonic hypersurfaces in nonpositive curved space
forms are minimal.
