Embedded constant mean curvature tori in the three-sphere. (English) Zbl 1318.53059

Andrews, Ben; Li, Haizhong

J. Differ. Geom. 99, No. 2, 169-189 (2015).

The authors consider constant mean curvature (CMC) tori \(\Sigma\) embedded in \(S^3\) and prove that:

(1)

every such \(\Sigma\) is a surface of revolution;

(2)

if \(\Sigma\) is not congruent to a Clifford torus, then there exists \(m\in\{2, 3, \ldots\}\) such that \(\Sigma\) is invariant under a cyclic group \(\mathbb{Z}_m\) of isometries;

(3)

for each \(m\) there exists at most one such \(\Sigma\);

(4)

given \(m\) there exists a CMC torus \(\Sigma\) with maximal symmetry \(S^1\times\mathbb{Z}_m\) and mean curvature \(H\) if only \(|H|\) lies between \(\cot (\pi /m)\) and \((m^2 - 2)/ (2\sqrt{m^2 -1 })\);

(5)

if \(H\in \{ 0, \pm 1/\sqrt{3}\}\), then every CMC torus \(\Sigma\) of mean curvature \(H\) is congruent to the Clifford one.

The authors mention that their essential contribution are items (1) and (3); given these, items (2), (4) and (5) follow from the results of O. M. Perdomo [Asian J. Math. 14, No. 1, 73–108 (2010; Zbl 1213.53080)].

Reviewer: Paweł Walczak (Łódź)

MSC:

53C42

Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

constant mean curvature; embedded torus; 3-sphere

Citations:

Zbl 1213.53080