Embedded cylindrical and doughnut-shaped $\hat{I}»$-hypersurfaces

Abstract

In the paper, we construct, for $\hat{I}»>0$, complete embedded and non-convex $\hat{I}»$-hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can not expect that $\hat{I}»$-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 $\hat{I}»<0$ which may have small $|\hat{I}»|$, we can construct two compact embedded $\hat{I}»$-hypersurfaces which are diffeomorphic to ${\mathbb{S}}^1\times {\mathbb{S}}^{n-1}$, but they are not isometric to each other.

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