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

TL;DR

The paper addresses non-uniqueness of embedded constant mean curvature hypertori in the round sphere $\mathbb{S}^{2n}(1)$ by constructing two distinct embeddings of type $\mathbb{S}^{n-1}\times\mathbb{S}^{n-1}\times\mathbb{S}^1$ with the same negative mean curvature $H$ for small $|H|$. It employs a profile-curve method: an immersion $F(p_1,p_2,s)=(x(s)p_1,y(s)p_2,z(s))$ with $x=\,\sin r\cos\theta$, $y=\,\sin r\sin\theta$, $z=\,\cos r$ reduces the CMC condition to an autonomous system in $(r,\theta,\alpha)$. By establishing a symmetry analysis and studying the dynamics for $H<0$, the authors obtain two distinct closed generating curves that yield two embeddings with identical $H$. This extends prior minimal hypertori results and provides a concrete dynamical-systems framework for generating multiple CMC hypertori in higher-dimensional spheres.

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

Figures (1)

• Figure 2.1: Profile curves with $n=2$ and $H=-0.2$, where the initial value $r_0$ of the red curve is 1.29691 and that of green curve is 1.44086.

Theorems & Definitions (23)

• Theorem 1.1: Carlotto and Schulz
• Theorem 1.2
• Remark 1.1
• Remark 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4