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

The authors consider constant mean curvature (CMC) tori \(\Sigma\) embedded in \(S^3\) and prove that:
every such \(\Sigma\) is a surface of revolution;
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;
for each \(m\) there exists at most one such \(\Sigma\);
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 \((m^2 - 2)/ (2\sqrt{m^2 -1 })\);
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)].


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)


Zbl 1213.53080
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