Halfspace type Theorems for Self-Shrinkers

TL;DR

This paper extends classical halfspace theorems to self-shrinkers and λ-hypersurfaces, proving uniqueness results for properly immersed self-shrinkers within certain geometric constraints using a geometric approach.

Contribution

It introduces new halfspace theorems for self-shrinkers and λ-hypersurfaces, employing a geometric proof with a catenoid-type hypersurface, and characterizes complete self-shrinkers in specific cylinders.

Findings

Properly immersed self-shrinkers in a half-space are hyperplanes.

Complete self-shrinkers in certain cylinders are specific spheres times Euclidean space.

Results extend to λ-hypersurfaces with similar geometric properties.

Abstract

In this short paper we extend the classical Hoffman-Meeks Halfspace Theorem to self-shrinkers, that is: "Let $P $ be a hyperplane passing through the origin. The only properly immersed self-shrinker $\Sigma$ contained in one of the closed half-space determined by $P$ is $\Sigma = P$." Our proof is geometric and uses a catenoid type hypersurface discovered by Kleene-Moller. Also, using a similar geometric idea, we obtain that the only complete self-shrinker properly immersed in an closed cylinder $ \overline{B ^{k+1} (R)} \times \mathbb{R}^{n-k}\subset \mathbb R^{n+1}$, for some $k\in \{1, \ldots ,n\}$ and radius $R$, $R \leq \sqrt{2k}$, is the cylinder $\mathbb S ^k (\sqrt{2k}) \times \mathbb{R}^{n-k}$. We also extend the above results for $\lambda -$hypersurfaces.