## 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:
(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)].

### MSC:

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

### Keywords:

constant mean curvature; embedded torus; 3-sphere

Zbl 1213.53080
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