Gap theorems for complete $\lambda$-hypersurfaces

Huijuan Wang, Hongwei Xu and Entao Zhao

Vol. 288 (2017), No. 2, 453–474

DOI: 10.2140/pjm.2017.288.453

Abstract

An $n$-dimensional $\lambda$-hypersurface $X:M\to {ℝ}^{n+1}$ is the critical point of the weighted area functional ${\int \; <!--nolimits-->}_M{e}^{\left[t\right]-\frac{1}{4}|X{|}^2}d\mu$ 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

Milestones

Received: 25 February 2016

Revised: 22 September 2016

Accepted: 26 October 2016

Published: 28 April 2017