Hypersurfaces with constant th mean curvature in the spheres

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• 作者：Guoxin Wei^a ; ^{weiguoxin@tsinghua.org.cn" class="auth_mail" title="E-mail the corresponding author} ; Guohua Wen^b ; ^{wengh911@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：53C40 ; 53C20
• 刊名：Journal of Geometry and Physics
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：104
• 期：Complete
• 页码：121-127
• 全文大小：386 K

文摘

In this paper, we prove that, for 1≤m≤n−1$1\le m\le n-1$, n≥3$n\ge 3$, k≥2$k\ge 2$, given a constant b55c7c395a29b9607e19d731" title="Click to view the MathML source">c$c$ between ${(cot\frac{\pi }{k})}^m$ and $\frac{k^2-2}{n}{\left(\frac{k^2+m-2}{n-m}\right)}^{\frac{m-2}{2}}$, there exists at least one compact non-isoparametric embedded hypersurface with m$m$th mean curvature H_m=c$H_m=c$ in a unit sphere 93b28b4" title="Click to view the MathML source">Sⁿ⁺¹$S^{n+1}$. As corollaries, we also prove that for m=n−1,n−2$m=n-1,n-2$ and m≥2$m\ge 2$, given any positive constant b55c7c395a29b9607e19d731" title="Click to view the MathML source">c$c$, there exists at least one compact non-isoparametric embedded hypersurface with H_m=c$H_m=c$ in 93b28b4" title="Click to view the MathML source">Sⁿ⁺¹$S^{n+1}$. Moreover, our results are the generalization of the results of Perdomo [9] (when m=1$m=1$), Wei et al. [7] (when m=2,4$m=2,4$).