Ruled Laguerre minimal surfaces

Handle URI:
http://hdl.handle.net/10754/561903
Title:
Ruled Laguerre minimal surfaces
Authors:
Skopenkov, Mikhail; Pottmann, Helmut ( 0000-0002-3195-9316 ) ; Grohs, Philipp
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.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Applied Mathematics and Computational Science Program; Visual Computing Center (VCC)
Publisher:
Springer Verlag
Journal:
Mathematische Zeitschrift
Issue Date:
30-Oct-2011
DOI:
10.1007/s00209-011-0953-0
Type:
Article
ISSN:
00255874
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.
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program; Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorSkopenkov, Mikhailen
dc.contributor.authorPottmann, Helmuten
dc.contributor.authorGrohs, Philippen
dc.date.accessioned2015-08-03T09:33:43Zen
dc.date.available2015-08-03T09:33:43Zen
dc.date.issued2011-10-30en
dc.identifier.issn00255874en
dc.identifier.doi10.1007/s00209-011-0953-0en
dc.identifier.urihttp://hdl.handle.net/10754/561903en
dc.description.abstractA 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.en
dc.description.sponsorshipThe 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.en
dc.publisherSpringer Verlagen
dc.subjectBiharmonic functionen
dc.subjectLaguerre geometryen
dc.subjectLaguerre minimal surfaceen
dc.subjectRuled surfaceen
dc.titleRuled Laguerre minimal surfacesen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentVisual Computing Center (VCC)en
dc.identifier.journalMathematische Zeitschriften
dc.contributor.institutionInstitute for Information Transmission Problems of the Russian Academy of Sciences, Bolshoy Karetny per. 19, bld. 1, Moscow 127994, Russian Federationen
dc.contributor.institutionSeminar for Applied Mathematics, ETH Zentrum, Rämistrasse 101, 8092 Zurich, Switzerlanden
kaust.authorSkopenkov, Mikhailen
kaust.authorPottmann, Helmuten
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