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A relation between the right triangle and circular tori with constant mean curvature in the unit 3-sphere. (English) Zbl 1073.53004

The standard circular torus \(T (r, a)\) in \(\mathbb R^3\) obtained by rotating the circle of radius \(a\) centered at \((r, 0, 0)\) in the \(xz\)-plane about the \(z\)-axis has the equation \((\sqrt{x^2 + y^2} - r)^2 + z^2 = a^2.\) The stereographic projection of the unit 3-sphere \(S^3 \subset R^4\) onto \(\mathbb R^3\) is denoted by \(\rho\). The author shows that the inverse image \(T^2\) in the unit 3-sphere \(S^3\) of the torus of revolution \(T(r, a)\) under the stereographic projection has constant mean curvature if and only if \(r\) and \(a\) are respectively the hypothenuse and one leg of a right triangle whose other leg is 1. This means that there is a parametrization via a central angle \(\alpha\) of a certain unit circle so that \(a = \tan \alpha \,\) and \(r = \sec \alpha\). More precisely, the inverse image is given as \(T^2 = \rho^{-1} (T (\sec \alpha , \tan \alpha )) = S^1 (\cos\alpha) \times S^1 (\sin\alpha),\) with constant mean curvature \(\overline H\) computed to be \(\overline H = \frac{\tan^2\alpha - 1}{2 \tan\alpha}.\)

MSC:

53A05 Surfaces in Euclidean and related spaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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References:

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