Given a domain $\Omega$ of ${ℝ}^{m+1}$ and a $k$-dimensional non-degenerate minimal submanifold $K$ of $\partial \Omega$ with $1\le k\le m-1$, we prove the existence of a family of embedded constant mean curvature hypersurfaces in $\Omega$ which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\partial \Omega$ perpendicularly along their boundaries.

In [OV] we introduced an affine curvature tensor R*. Using it we characterized some types of hypersurfaces in the affine space ${ℝ}^{n+1}$. In this paper we study hypersurfaces for which R* is parallel relative to the induced connection.