Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T05:05:34.046Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on rigidity theorem of λ-hypersurfaces

Part of: Global differential geometry

Published online by Cambridge University Press:  18 January 2019

Guoxin Wei  and
Yejuan Peng
Guoxin Wei
Affiliation:
School of Mathematical Sciences, South China Normal University, 510631 Guangzhou, China (weiguoxin@tsinghua.org.cn)
Yejuan Peng
Affiliation:
School of Mathematics and Information Sciences, Henan Normal University, 453007 Xinxiang, China (yejuan666@126.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

Self-shrinkers are an important class of solutions to the mean curvature flow and their generalization is λ-hypersurfaces. In this paper, we study λ-hypersurfaces and give a rigidity result about complete λ-hypersurfaces.

Keywords

mean curvaturethe weighted area functionalλ-hypersurfacesthe weighted volume-preserving mean curvature flow

MSC classification

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1595 - 1601
Copyright
Copyright © Royal Society of Edinburgh 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Cao, H.-D. and Li, H.. A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. Partial Differ. Equ. 46 (2013), 879–889.Google Scholar
2Cheng, Q.-M. and Wei, G.. A gap theorem for self-shrinkers. Trans. Amer. Math. Soc. 367 (2015), 4895–4915.Google Scholar
3Cheng, Q.-M. and Wei, G.. Complete λ-hypersurfaces of the weighted volume-preserving mean curvature flow, arXiv:1403.3177, Calc. Var. 57 (2018), no. 2, Art. 32, 21 pp.Google Scholar
4Cheng, Q.-M. and Wei, G.. Compact embedded λ-torus in Euclidean spaces, arXiv:1512.04752.Google Scholar
5Cheng, Q.-M., Ogata, S. and Wei, G.. Rigidity theorems of λ-hypersurfaces. Comm. Anal. Geom. 24 (2016), 45–58.Google Scholar
6Colding, T. H. and Minicozzi II, W. P.. Generic mean curvature flow I; Generic singularities. Ann. of Math. 175 (2012), 755–833.Google Scholar
7Guang, Q.. Gap and rigidity theorems of λ-hypersurfaces, arXiv:1405.4871v2.Google Scholar
8Le, N. Q. and Sesum, N.. Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers. Comm. Anal. Geom. 19 (2011), 1–27.Google Scholar
9Lei, L., Xu, H. W. and Xu, Z. Y.. A new pinching theorem for complete self-shrinkers and its generalization, arXiv:1712.01899v1.Google Scholar
10Ross, J.. On the existence of a closed embedded rotational λ-hypersurface, arXiv:1709. 05020v1.Google Scholar
11Wang, H., Xu, H. W. and Zhao, E. T.. Gap theorems for complete λ-hypersurfaces. Pacific J. Math. 288 (2017), 453–474.Google Scholar
12Xu, H. W., Lei, L. and Xu, Z. Y.. The second pinching theorem for complete λ–hypersurfaces (in Chinese). Sci. Sin. Math. 48 (2018), 1–10.Google Scholar