Halfspace type Theorems for Self-Shrinkers

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 $k\in \{1,\dots ,n\}$ and radius $R$, $R\le \sqrt{2k}$, is the cylinder ${\mathbb{S}}^k(\sqrt{2k})\times {\mathbb{R}}^{n-k}$. We also extend the above results for $\lambda -$hypersurfaces.

References in corpus (1)

• Complete Constant Mean Curvature surfaces and Bernstein type Theorems in ${\mathbb{M}}^2\times \mathbb{R}$

Cited by in corpus (7)

• A Topological Property of Asymptotically Conical Self-Shrinkers of Small Entropy
• Bi-Halfspace and Convex Hull Theorems for Translating Solitons
• Ancient mean curvature flows and their spacetime tracks
• Smooth compactness of $f$-minimal hypersurfaces with bounded $f$-index
• The halfspace theorem for minimal hypersurfaces in regions bounded by minimal cones
• A strong Frankel Theorem for shrinkers
• Geometric properties of self-shrinkers in cylinder shrinking Ricci solitons