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

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.

arXiv commentAccepted in Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM (2024)

References in corpus (1)

• Extremal spectral properties of Otsuki tori

Cited by in corpus (1)

• Screw motion surfaces of constant mean curvature in homogeneous 3-manifolds