Type
ArticleKAUST Department
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) DivisionApplied Mathematics and Computational Science Program
Visual Computing Center (VCC)
Date
2011-10-30Online Publication Date
2011-10-30Print Publication Date
2012-10Permanent link to this record
http://hdl.handle.net/10754/561903
Metadata
Show full item recordAbstract
A Laguerre minimal surface is an immersed surface in ℝ 3 being an extremal of the functional ∫ (H 2/K-1)dA. In the present paper, we prove that the only ruled Laguerre minimal surfaces are up to isometry the surfaces ℝ (φλ) = (Aφ, Bφ, Cφ + D cos 2φ) + λ(sin φ, cos φ, 0), where A,B,C,D ε ℝ are fixed. To achieve invariance under Laguerre transformations, we also derive all Laguerre minimal surfaces that are enveloped by a family of cones. The methodology is based on the isotropic model of Laguerre geometry. In this model a Laguerre minimal surface enveloped by a family of cones corresponds to a graph of a biharmonic function carrying a family of isotropic circles. We classify such functions by showing that the top view of the family of circles is a pencil. © 2011 Springer-Verlag.Citation
Skopenkov, M., Pottmann, H., & Grohs, P. (2011). Ruled Laguerre minimal surfaces. Mathematische Zeitschrift, 272(1-2), 645–674. doi:10.1007/s00209-011-0953-0Sponsors
The authors are grateful to S. Ivanov for useful discussions. M. Skopenkov was supported in part by Mobius Contest Foundation for Young Scientists and the Euler Foundation. H. Pottmann and P. Grohs are partly supported by the Austrian Science Fund (FWF) under grant S92.Publisher
Springer Science and Business Media LLCJournal
Mathematische ZeitschriftarXiv
1011.0272ae974a485f413a2113503eed53cd6c53
10.1007/s00209-011-0953-0