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Zbl 1125.53008
Vassilev, Vassil M.; Djondjorov, Peter A.; Mladenov, Iva\"ilo M.
On the translationally-invariant solutions of the membrane shape equation. (English)
[A] Mladenov, Iva\"ilo (ed.) et al., Proceedings of the 8th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 9--14, 2006. Sofia: Bulgarian Academy of Sciences. 312-321 (2007). ISBN 978-954-8495-37-0/pbk

Summary: The membrane shape equation derived by Helfrich and Ou-Yang describes the equilibrium shapes of biomembranes, built by bilayers of amphiphilic molecules, in terms of the mean and Gaussian curvatures of their middle-surfaces. Here, we present a new class of translationally-invariant solutions to this equation in terms of the elliptic functions which completes the solutions found earlier. In this way, all translationally-invariant solutions to the membrane shape equation are determinded. Special attention is paid to those translationally-invariant solutions of the membrane shape equation which determine closed cylindrical (tube-like) surfaces (membrane shapes). Several examples of such surfaces are presented.
[Kaarin Riives (Tartu)]

MSC 2000:: *53A10 Minimal surfaces, surfaces with prescribed mean curvature
53B50 Appl. of local differential geometry to physics
76Z05 Physiological flows

Keywords: membrane shape equations; closed cylindrical (tube-like) surfaces; mean and Gaussian curvatures

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