Rigidity theorems of $\lambda$-hypersurfaces

Abstract

Since $n$-dimensional $\lambda$-hypersurfaces in the Euclidean space ${\mathbb{R}}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of complete $\lambda$-hypersurfaces. We give a gap theorem of complete $\lambda$-hypersurfaces with polynomial area growth. By making use of the generalized maximum principle for $\mathcal{L}$ of $\lambda$-hypersurfaces, we prove a rigidity theorem of complete $\lambda$-hypersurfaces.

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