New examples of constant mean curvature hypersurfaces in the sphere

TL;DR

This paper constructs new examples of compact, embedded constant mean curvature hypersurfaces in spheres with specific topologies and symmetries, expanding known classes of such geometric objects.

Contribution

The paper demonstrates the existence of novel compact embedded CMC hypersurfaces in spheres with complex topologies and symmetry properties, generalizing previous results.

Findings

Existence of a CMC hypersurface of type S^{n-1}×S^{n-1}×S^{1} in S^{2n}

Existence of a CMC hypersurface of type S^{n-1}×S^{n-1}×S^{n-1}×S^{1} in S^{3n-1

Generalization of Carlotto and Schulz's results

Abstract

In this paper, firstly, we show the existence of a compact embedded constant mean curvature (CMC) hypersurface $\Sigma_1$ in $\mathbb{S}^{2n}$ of the type $S^{n-1} \times S^{n-1} \times S^{1}$. Moreover, the hypersurface $\Sigma_1$ exhibits $O(n)\times O(n)$ symmetry. Secondly, we show that there exists a compact embedded CMC-hypersurface $\Sigma_2 \subset \mathbb{S}^{3n-1}$ 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.