New examples of constant mean curvature hypersurfaces in the sphere

In this paper, firstly, we show the existence of a compact embedded constant mean curvature (CMC) hypersurface ${\mathrm{\Sigma }}_1$ in ${\mathbb{S}}^{2n}$ of the type $S^{n-1}\times S^{n-1}\times S^1$. Moreover, the hypersurface ${\mathrm{\Sigma }}_1$ exhibits $O(n)\times O(n)$ symmetry. Secondly, we show that there exists a compact embedded CMC-hypersurface ${\mathrm{\Sigma }}_2\subset {\mathbb{S}}^{3n-1}$ of the type $S^{n-1}\times S^{n-1}\times S^{n-1}\times S^1$. These results generalize the results of Carlotto and Schulz.