Halfspace type theorems for self-shrinkers

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 Σ contained in one of the closed half-space determined by P is Σ = 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 B k + 1 ( R ) ¯ × R n − k ⊂ R n + 1 , for some k ∈ { 1 , … , n } and radius R, R ⩽ 2 k , is the cylinder S k ( 2 k ) × R n − k . We also extend the above results for λ-hypersurfaces.