Communications in Analysis and Geometry

Volume 24 (2016)

Number 1

Rigidity theorems of $\lambda$-hypersurfaces

Pages: 45 – 58



Qing-Ming Cheng (Department of Applied Mathematics, Faculty of Sciences, Fukuoka University, Fukuoka, Japan)

Shiho Ogata (Department of Applied Mathematics, Graduate School of Sciences, Fukuoka University, Fukuoka, Japan)

Guoxin Wei (School of Mathematical Sciences, South China Normal University, Guangzhou, China)


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.


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

2010 Mathematics Subject Classification

53C42, 53C44

Published 6 June 2016