1. Halfspace type Theorems for Self-Shrinkers
- Author
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Marcos P. Cavalcante and José M. Espinar
- Subjects
Mathematics - Differential Geometry ,0209 industrial biotechnology ,Minimal surface ,General Mathematics ,010102 general mathematics ,Short paper ,02 engineering and technology ,Radius ,Type (model theory) ,Lambda ,01 natural sciences ,Combinatorics ,020901 industrial engineering & automation ,Hypersurface ,Differential Geometry (math.DG) ,Hyperplane ,Catenoid ,FOS: Mathematics ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics - Abstract
In this short paper, we extend the classical Hoffman-Meeks Halfspace Theorem [Hoffman and Meeks, 'The strong halfspace theorem for minimal surfaces', Invent. Math. 101 (1990) 373-377] 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 [Kleene and Moller, 'Self-shrinkers with a rotational symmetry', Trans. Amer. Math. Soc. 366 (2014) 3943-3963]. Also, using a similar geometric idea, we obtain that the only 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 $R$, $R \leqslant \sqrt {2k}$, is the cylinder ${\mathbb S}^k (\sqrt {2k}) \times {\mathbb R}^{n-k}$. We also extend the above results for $\lambda $-hypersurfaces.
- Published
- 2014