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arXiv · 2303.08043

Helicoidal minimal surfaces in the 3-sphere: An approach via spherical curves

Ildefonso Castro·Ildefonso Castro-Infantes·Jesús Castro-Infantes
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Abstract

We prove an existence and uniqueness theorem about spherical helicoidal (in particular, rotational) surfaces with prescribed mean or Gaussian curvature in terms of a continuous function depending on the distance to its axis. As an application in the case of vanishing mean curvature, it is shown that the well-known conjugation between the belicoid and the catenoid in Euclidean three-space extends naturally to the 3-sphere to their spherical versions and determine in a quite explicit way their associated surfaces in the sense of Lawson. As a key tool, we use the notion of spherical angular momentum of the spherical curves that play the role of profile curves of the minimal helicoidal surfaces in the 3-sphere.

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Ildefonso Castro, Ildefonso Castro-Infantes, Jesús Castro-Infantes. 2023-03-14. Helicoidal minimal surfaces in the 3-sphere: An approach via spherical curves. https://doi.org/10.1007/s13398-024-01574-3

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