##
**Hopf tori in \(S^ 3\).**
*(English)*
Zbl 0585.53051

Let \(\pi\) : \(S^ 3\to S^ 2\) be the Hopf fibration. The author proves that the inverse image of any closed curve on \(S^ 2\) is an immersed torus in \(S^ 3\). This torus is called a Hopf torus. Using Hopf tori, he obtains: Every compact Riemann surface of genus one can be conformally embedded in the unit sphere \(S^ 3\subset R^ 4\) as a flat torus. The embedding can be chosen as the intersection of \(S^ 3\) with a quartic hypersurface in \(R^ 4\). As a corollary, he gets: Every compact Riemann surface of genus one can be conformally embedded in \(R^ 3\) as an algebraic surface of degree eight.

A surface in \(R^ 3\) is called a Willmore surface if it is an extremal surface for the variational functional \(\int H^ 2dA\) (H the mean curvature). The only known examples of compact Willmore surfaces are the stereographic projections of compact minimal surfaces in \(S^ 3\). By showing that Hopf tori of some kind are Willmore surfaces, the author obtains an infinite series of compact Willmore surfaces that do not stem from minimal surfaces in \(S^ 3\).

A surface in \(R^ 3\) is called a Willmore surface if it is an extremal surface for the variational functional \(\int H^ 2dA\) (H the mean curvature). The only known examples of compact Willmore surfaces are the stereographic projections of compact minimal surfaces in \(S^ 3\). By showing that Hopf tori of some kind are Willmore surfaces, the author obtains an infinite series of compact Willmore surfaces that do not stem from minimal surfaces in \(S^ 3\).

Reviewer: T.Ishihara

### MSC:

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

53A05 | Surfaces in Euclidean and related spaces |

57R40 | Embeddings in differential topology |

### Keywords:

conformal embedding; Hopf fibration; Hopf torus; flat torus; algebraic surface; Willmore surface; mean curvature### References:

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