  In the paper, we construct, for $\lambda>0$, complete embedded and non-convex
$\lambda$-hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can
not expect that $\lambda$-hypersurfaces share a common conclusion on the planar
domain conjecture even if the planar domain conjecture of T. Ilmanen for
self-shrinkers of mean curvature flow are solved by Brendle \cite{B}
affirmatively. Furthermore, for a fixed $\lambda<0$ which may have small
$|\lambda|$, we can construct two compact embedded $\lambda$-hypersurfaces
which are diffeomorphic to $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$, but they
are not isometric to each other.
