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# Communications in Analysis and Geometry

## Volume 24 (2016)

### Number 1

### Rigidity theorems of $\lambda$-hypersurfaces

Pages: 45 – 58

DOI: http://dx.doi.org/10.4310/CAG.2016.v24.n1.a2

#### Authors

#### 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 some gap theorems 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.

#### Keywords

second fundamental form, weighted area functional, $\lambda$-hypersurfaces, weighted volume-preserving mean curvature flow

#### 2010 Mathematics Subject Classification

53C42, 53C44