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 \(\mathbb{S}^{2n}(1)\) .