Embedded constant $m^{\text{th}}$ mean curvature hypersurfaces on spheres

In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature $H_m$ in a unit sphere $S^{n+1}(1)$ and prove that if the $m^{\text{th}}$ mean curvature $H_m$ takes value between ${\displaystyle \frac{1}{(\tan \frac{\pi }{k}{)}^m}}$ and $\frac{k^2-2}{n}(\frac{k^2+m-2}{n-m}{)}^{\frac{m-2}{2}}$ for $1\le m\le n-1$ and any integer $k\ge 2$, then there exists at least one $n$-dimensional compact nontrivial embedded hypersurface with constant $H_m>0$ in $S^{n+1}(1)$. When $m=1$, our results reduce to the results of Perdomo \cite{[P]}; when $m=2$ and $m=4$, our results reduce to the results of Cheng-Li-Wei \cite{[WCL]}.