Stable Hypersurfaces in a 4-Dimensional Sphere
Received:April 21, 2015  Revised:October 12, 2016
Key Word: constant scalar curvature   1-minimal stable hypersurfaces in space forms  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11471145; 11401514) and Qing Lan Projects.
Author NameAffiliation
Peng ZHU School of Mathematics and physics, Jiangsu University of Technology, Jiangsu 213001, P. R. China 
Shouwen FANG School of Mathematical Sciences, Yangzhou University, Jiangsu 225002, P. R. China 
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      We study complete noncompact $1$-minimal stable hypersurfaces in a $4$-dimensional sphere $\mathbb{S}^{4}$. We show that there is no complete noncompact $1$-minimal stable hypersurfaces in $\mathbb{S}^{4}$ with polynomial volume growth and the restriction of the mean curvature and Gauss-Kronecker curvature. These results are partial answers to the conjecture of Alencar, do Carmo and Elbert when the ambient space is a $4$-dimensional sphere.
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