#### Vol. 288, No. 2, 2017

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Gap theorems for complete $\lambda$-hypersurfaces

### Huijuan Wang, Hongwei Xu and Entao Zhao

Vol. 288 (2017), No. 2, 453–474
##### Abstract

An $n$-dimensional $\lambda$-hypersurface $X:M\to {ℝ}^{n+1}$ is the critical point of the weighted area functional ${\int }_{M}{e}^{\left[t\right]-\frac{1}{4}|X{|}^{2}}\phantom{\rule{0.3em}{0ex}}d\mu$ for weighted volume-preserving variations, which is also a generalization of the self-shrinking solution of the mean curvature flow. We first prove that if the ${L}^{n}$-norm of the second fundamental form of the $\lambda$-hypersurface $X:M\to {ℝ}^{n+1}$ with $n\ge 3$ is less than an explicit positive constant $K\left(n,\lambda \right)$, then $M$ is a hyperplane. Secondly, we show that if the ${L}^{n}$-norm of the trace-free second fundamental form of $M$ with $n\ge 3$ is less than an explicit positive constant $D\left(n,\lambda \right)$ and the mean curvature is suitably bounded, then $M$ is a hyperplane. We also obtain similar results for $\lambda$-surfaces in ${ℝ}^{3}$ under ${L}^{4}$-curvature pinching conditions.

##### Keywords
gap theorem, lambda-hypersurfaces, integral curvature pinching
##### Mathematical Subject Classification 2010
Primary: 53C42, 53C44