Abstract
We study helix surfaces with parallel mean curvature vector in Euclidean space from a local point of view. Our main result says that they are either part of a cylinder of revolution or a plane. One way to prove this is with the generalization we found about the Laplacian of a support function of a hypersurface. This allows us to study the constant mean curvature surfaces in space forms which have constant angle with respect to a closed and conformal vector field. The result we find says that these surfaces are totally umbilic.
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Acknowledgments
G. Ruiz-Hernández thanks the hospitality of DISMA at Politecnico di Torino.
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A. J. Di Scala and G. Ruiz-Hernández were partially supported by Ministero degli Affari Esteri from Italy and CONACYT from Mexico. C. Barrera Cadena was supported by DGAPA-UNAM, under project PAPIIT IA100412.
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Cadena, C.B., Di Scala, A.J. & Ruiz-Hernández, G. Helix surfaces in Euclidean spaces. Beitr Algebra Geom 56, 551–573 (2015). https://doi.org/10.1007/s13366-014-0226-2
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DOI: https://doi.org/10.1007/s13366-014-0226-2