Abstract
We prove that the only surface in 3-dimensional Euclidean space \({\mathbb {R}}^3\) with constant and non-zero mean curvature H, constructed by the sum of a planar curve and a space curve, is the circular cylinder of radius \(\frac{1}{2|H|}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Darboux, G.: Lecons sur la Théorie Générale des Surfaces, 4 tomes. Gauthier Villars, Paris (1914)
do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice Hall, Upper Saddle River (1976)
Hasanis, T., López, R.: Translation surfaces in Euclidean space with constant Gaussian curvature. Commun. Anal. Geom. arXiv:1809.02758
Hasanis, T., López, R.: Classification and construction of minimal translation surfaces in Euclidean space. (2018). arXiv:1809.02759
Liu, H.: Translation surfaces with constant mean curvature in 3-dimensional spaces. J. Geom. 64, 141–149 (1999)
López, R., Perdomo, O.: Minimal translation surfaces in Euclidean space. J. Geom. Anal. 27, 2926–2937 (2017)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hasanis, T. Translation surfaces with non-zero constant mean curvature in Euclidean space. J. Geom. 110, 20 (2019). https://doi.org/10.1007/s00022-019-0476-0
Received:
Published:
DOI: https://doi.org/10.1007/s00022-019-0476-0