Volume 35 • Number 2 • 2012   • Hypersurfaces with Constant $k$-th Mean Curvature in a Unit Sphere and Euclidean Space Shichang Shu and Annie Yi Han Abstract. Let $M^n$ be an $n (n\geq3)$-dimensional complete connected and oriented hypersurface in a unit sphere $S^{n+1}(1)$ or Euclidean space $\mathbf{R}^{n+1}$ with constant $k$-th mean curvature $H_k>0(k < n)$ and with two distinct principal curvatures $\lambda$ and $\mu$ such that the multiplicity of $\lambda$ is $n-1$. We show that (1) in the case of $S^{n+1}(1)$, if $k\geq3$ and $|h|^2\leq (n-1)t_2^{2/k}+t_2^{-2/k}$, then $M^n$ is isometric to the Riemannian product $S^1(\sqrt{1-a^2})\times S^{n-1}(a)$, where $t_2$ is the positive real root of the function $ P_{H_k}(t)=kt^\frac{k-2}{k}-(n-k)t+nH_k$; (2) in the case of $\mathbf{R}^{n+1}$, if $|h|^2\leq(n-1)(nH_k/(n-k))^\frac{2}{k}$, then $M^n$ is isometric to the Riemannian product $S^{n-1}(a)\times \mathbf{R}$. We extend some recent results to the case $k\geq3$. 2010 Mathematics Subject Classification: 53C42, 53A10. Full text: PDF