A Rigidity Theorem of <i>λ</i>-Hypersurfaces

LI Yunchuan; LIU Yan; WEI Guoxin

doi:10.6054/j.jscnun.2020067

Journal of South China Normal University (Natural Science Edition) > 2020 > 52(4): 104-106. > DOI: 10.6054/j.jscnun.2020067

LI Yunchuan, LIU Yan, WEI Guoxin. A Rigidity Theorem of λ-Hypersurfaces[J]. Journal of South China Normal University (Natural Science Edition), 2020, 52(4): 104-106. DOI: 10.6054/j.jscnun.2020067

Citation:

LI Yunchuan, LIU Yan, WEI Guoxin. A Rigidity Theorem of λ-Hypersurfaces[J]. Journal of South China Normal University (Natural Science Edition), 2020, 52(4): 104-106. DOI: 10.6054/j.jscnun.2020067

Citation:

LI Yunchuan, LIU Yan, WEI Guoxin. A Rigidity Theorem of λ-Hypersurfaces[J]. Journal of South China Normal University (Natural Science Edition), 2020, 52(4): 104-106. DOI: 10.6054/j.jscnun.2020067

PDF (979 KB)

A Rigidity Theorem of λ-Hypersurfaces

1.
Department of Business Tourism, Henan College of Transportation, Zhengzhou 451460, China
2.
School of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou 450046, China
3.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

More Information

Received Date: March 24, 2020
Available Online: March 21, 2021

Graphical Abstract

Abstract

Abstract

λ-hypersurfaces are studied and a rigidity result about complete λ-hypersurfaces is given. If X :M→ $\mathbb{R}$ ⁿ⁺¹ is an n-dimensional complete λ-hypersurface with polynomial area growth and satisfies S bounded, then ∫_M(|▽H|²+(H-λ)(H+S(λ-H))) ${{\rm{e}}^{ - \frac{{|\mathit{\boldsymbol{X}}{|^2}}}{2}}}$ dμ=0, where H is the mean curvature of M, S is the squared norm of the second fundamental form of M. As an application of the integral equation, a rigidity result about complete λ-hypersurfaces is obtained.
- mean curvature,
- the weighted area functional,
- λ-hypersurfaces

FullText(HTML)

References (15)

References

[1]	CHENG Q M, WEI G X. Complete λ-hypersurfaces of the weighted volume-preserving mean curvature flow[J]. Calculus of Variations and Partial Differential Equations, 2018, 57(2):32/1-21. doi: 10.1007/s00526-018-1303-4
[2]	CHENG Q M, OGATA S, WEI G X. Rigidity theorems of λ-hypersurfaces[J]. Communications in Analysis and Geometry, 2016, 24:45-58. doi: 10.4310/CAG.2016.v24.n1.a2
[3]	成庆明, 魏国新.完备λ-超曲面的几何[J].中国科学:数学, 2018, 48(6):699-710. http://www.cnki.com.cn/Article/CJFDTotal-JAXK201806003.htm CHENG Q M, WEI G X. Geometry of complete λ-hypersurfaces[J]. Scientia Sinica:Mathematica, 2018, 48(6):699-710. http://www.cnki.com.cn/Article/CJFDTotal-JAXK201806003.htm
[4]	CHENG Q M, WEI G X. Examples of compact λ-hypersurfaces in Euclidean spaces[J/OL]. Science China Mathematics, (2019-08-29)[2020-02-20]. https://doi.org/10.1007/s11425-018-9464-7.
[5]	CHANG J E. 1-dimensional solutions of the λ-self shrinkers[J]. Geometriae Dedicata, 2017, 189:97-112. doi: 10.1007/s10711-017-0219-z
[6]	GUANG Q. Gap and rigidity theorems of λ-hypersurfaces[J]. Proceedings of the American Mathematical Society, 2018, 146:4459-4471. doi: 10.1090/proc/14111
[7]	LEI L, XU H W, XU Z Y. A new pinching theorem for complete self-shrinkers and its generalization[J/OL]. arXiv, (2017-12-05)[2020-02-20]. https://arxiv.org/abs/1712.01899.
[8]	LI X X, CHANG X F. A rigidity theorem of ξ-submanifolds in $\mathbb{C}^2$ [J]. Geometriae Dedicata, 2016, 185:155-169. doi: 10.1007/s10711-016-0173-1
[9]	ROSS J. On the existence of a closed embedded rotational λ-hypersurface[J]. Journal of Geometry, 2019, 110(2):26/1-12. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=59632e14f4670d2deb0d81a4acdc7ce1
[10]	WANG H, XU H W, ZHAO E T. Gap theorems for complete λ-hypersurfaces[J]. Pacific Journal of Mathema-tics, 2017, 288:453-474. doi: 10.2140/pjm.2017.288.453
[11]	许洪伟, 雷力, 许智源.关于完备λ-超曲面的第二拼挤定理[J].中国科学:数学, 2018, 48(1):245-254. http://www.cnki.com.cn/Article/CJFDTotal-JAXK201801016.htm XU H W, LEI L, XU Z Y. The second pinching theorem for complete λ-hypersurfaces[J]. Scientia Sinica:Mathe-matica, 2018, 48(1):245-254. http://www.cnki.com.cn/Article/CJFDTotal-JAXK201801016.htm
[12]	ZHU Y C, FANG Y, CHEN Q. Complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow[J]. Science China Mathematics, 2018, 61(5):929-942. doi: 10.1007/s11425-016-9020-9
[13]	LE N Q, SESUM N. Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers[J]. Communications in Analysis and Geometry, 2011, 19:1-27. http://cn.bing.com/academic/profile?id=dc585521797143da4785f5ac20cc0652&encoded=0&v=paper_preview&mkt=zh-cn
[14]	CAO H D, LI H Z. A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension[J]. Calculus of Variations and Partial Differential Equations, 2013, 46:879-889. doi: 10.1007/s00526-012-0508-1
[15]	COLDING T H, MINICOZZI II W P. Generic mean curvature flow I; generic singularities[J]. Annals of Mathema-tics, 2012, 175:755-833. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=Arxiv000000124418

Cited By

Get Citation

PDF

XML

Article views (568) PDF downloads (57)

Turn off MathJax

Article Contents

Abstract

References

A Rigidity Theorem of λ-Hypersurfaces

Abstract

References

Catalog

Export File

Citation

Format

Content