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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 \(S^ 3\).
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
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References:
[1] Bryant, R.L.: A duality theorem for Willmore surfaces. (Preprint 1984) · Zbl 0555.53002
[2] Garsia, A.M.: On the conformal types of algebraic surfaces of euclidean space. Comment. Math. Helv.37, 49-60 (1962) · Zbl 0105.34702 · doi:10.1007/BF02566961
[3] Langer, J., Singer, D.A.: Curve straightening in Riemannian manifolds (In prep.) · Zbl 0653.53032
[4] Lawson, H.B.: Complete minimal surfaces inS 3. Ann. Math.92, 335-374 (1970) · Zbl 0205.52001 · doi:10.2307/1970625
[5] Palais, R.S.: The principle of symmetric criticality. Commun. Math. Phys.69, 19-30 (1979) · Zbl 0417.58007 · doi:10.1007/BF01941322
[6] R?edy, R.: Embeddings of open Riemann Surfaces. Comment. Math. Helv.46, 214-225 (1971) · Zbl 0224.30039 · doi:10.1007/BF02566840
[7] Santalo, L.A.: Integral geometry and geometric probability. London: Addison-Wesley 1976
[8] Singer, I.M., Thorpe, J.A.: Lecture notes on elementary topology and geometry. Glenview 1967 · Zbl 0163.44302
[9] Weiner, J.L.: On a problem of Chen, Willmore, et al. Indiana Univ. Math. J.27, 19-35 (1978) · Zbl 0368.53043 · doi:10.1512/iumj.1978.27.27003
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