Embedded cylindrical and doughnut-shaped λ-hypersurfaces

In the paper, we construct, for λ>0, complete embedded and non-convex λ-hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can not expect that λ-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 {B} affirmatively. Furthermore, for a fixed λ<0 which may have small |λ|, we can construct two compact embedded λ-hypersurfaces which are diffeomorphic to S¹× Sⁿ⁻¹, but they are not isometric to each other.