Vol. 288, No. 2, 2017

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ISSN: 0030-8730
Gap theorems for complete $\lambda$-hypersurfaces

Huijuan Wang, Hongwei Xu and Entao Zhao

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

An n-dimensional λ-hypersurface X : M n+1 is the critical point of the weighted area functional Me[t]1 4 |X|2 dμ 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 Ln-norm of the second fundamental form of the λ-hypersurface X : M n+1 with n 3 is less than an explicit positive constant K(n,λ), then M is a hyperplane. Secondly, we show that if the Ln-norm of the trace-free second fundamental form of M with n 3 is less than an explicit positive constant D(n,λ) and the mean curvature is suitably bounded, then M is a hyperplane. We also obtain similar results for λ-surfaces in 3 under L4-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
Authors
Huijuan Wang
Center of Mathematical Sciences
Zhejiang University
Hangzhou
Zhejiang 310027
China
Hongwei Xu
Center of Mathematical Sciences
Zhejiang University
Hangzhou
Zhejiang 310027
China
Entao Zhao
Center of Mathematical Sciences
Zhejiang University
Hangzhou
Zhejiang 310027
China