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Helicoidal minimal surfaces in \(\mathbb H^{2} \times \mathbb R\). (English) Zbl 1252.53016

Screw motions in \(\mathbb H^{2} \times \mathbb R\) are rigid motions generated by rotations in \(\mathbb H^2\) and vertical translations along \(\mathbb R\). A surface in \(\mathbb H^{2} \times \mathbb R\) is called helicoidal it it is invariant under a \(1\)-parameter group of screw motions. The authors obtain a complete description of the minimal surfaces that are helicoidal. An important result is that conjugate surfaces of the parabolic and hyperbolic helicoids in \(\mathbb H^{2} \times \mathbb R\) are certain types of catenoids.

MSC:

53A35 Non-Euclidean differential geometry
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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References:

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