Embedded cylindrical and doughnut-shaped $\lambda$-hypersurfaces

TL;DR

This paper constructs specific non-convex, embedded $ ext{lambda}$-hypersurfaces with cylindrical and doughnut shapes, demonstrating diverse geometries and challenging existing conjectures in mean curvature flow.

Contribution

It introduces new examples of complete embedded $ ext{lambda}$-hypersurfaces with cylindrical and doughnut shapes, expanding understanding of their geometric diversity.

Findings

Constructed non-convex cylindrical $ ext{lambda}$-hypersurfaces for $ ext{lambda}>0$.

Produced compact $ ext{lambda}$-hypersurfaces diffeomorphic to $ ext{S}^1 imes ext{S}^{n-1}$ for $ ext{lambda}<0$.

Showed these hypersurfaces are not necessarily isometric, indicating geometric variability.

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.