Embedded constant mean curvature hypertori in the $2n$-sphere

Abstract: Brendle proved Lawson conjecture about minimal embedded torus in the round three-dimensional sphere. Carlotto and Schulz constructed a minimal embedded three-dimensional hypertorus in the round four-dimensional sphere and conjectured that their hypertorus is a unique minimal embedded three-dimensional hypertorus in the round four-dimensional sphere. In this paper, we construct two different constant mean curvature embedded $(2n-1)$-dimensional hypertori (that is, topological type (\mathbb{S}^{n-1} \times \mathbb{S}^{n-1} \times \mathbb{S}^1)) which have the same negative mean curvature (H) in the round $2n$-dimensional sphere (\mathbb{S}^{2n}(1)).