Rigidity theorems for complete $\lambda$-hypersurfaces

Abstract

In this article, we study hypersurfaces $\Sigma \subset {\mathbb {R}}^{n+1}$ with constant weighted mean curvature, also known as $\lambda$-hypersurfaces. Recently, Wei-Peng proved a rigidity theorem for $\lambda$-hypersurfaces that generalizes Le–Sesum’s classification theorem for self-shrinkers. More specifically, they showed that a complete $\lambda$-hypersurface with polynomial volume growth, bounded norm of the second fundamental form, and that satisfies $|A|^2H(H-\lambda )\le H^2/2$ must either be a hyperplane or a generalized cylinder. We generalize this result by removing the bound condition on the norm of the second fundamental form. Moreover, we prove that under some conditions, if the reverse inequality holds, then the hypersurface must either be a hyperplane or a generalized cylinder. As an application of one of the results proved in this paper, we will obtain another version of the classification theorem obtained by the authors of this article, that is, we show that under some conditions, a complete $\lambda$-hypersurface with $H\ge 0$ must either be a hyperplane or a generalized cylinder.