Ruled Laguerre minimal surfaces

Mikhail Skopenkov, Helmut Pottmann, Philipp Grohs

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)645-674
Number of pages30
JournalMathematische Zeitschrift
Volume272
Issue number1-2
DOIs
StatePublished - Oct 30 2011

ASJC Scopus subject areas

  • Mathematics(all)

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