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    Ruled Laguerre minimal surfaces

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    Type
    Article
    Authors
    Skopenkov, Mikhail
    Pottmann, Helmut
    Grohs, Philipp
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Applied Mathematics and Computational Science Program
    Visual Computing Center (VCC)
    Date
    2011-10-30
    Online Publication Date
    2011-10-30
    Print Publication Date
    2012-10
    Permanent link to this record
    http://hdl.handle.net/10754/561903
    
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    Abstract
    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/s00209-011-0953-0
    Sponsors
    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 LLC
    Journal
    Mathematische Zeitschrift
    DOI
    10.1007/s00209-011-0953-0
    arXiv
    1011.0272
    Additional Links
    http://link.springer.com/10.1007/s00209-011-0953-0
    http://arxiv.org/pdf/1011.0272
    ae974a485f413a2113503eed53cd6c53
    10.1007/s00209-011-0953-0
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Visual Computing Center (VCC); Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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