Embedded cylindrical and doughnut-shaped $λ$-hypersurfaces

Abstract: 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.