On the translationally-invariant solutions of the membrane shape equation. (English) Zbl 1125.53008

Vassilev, Vassil M.; Djondjorov, Peter A.; Mladenov, Ivaïlo M.

Mladenov, Ivaïlo (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 (ISBN 978-954-8495-37-0/pbk). 312-321 (2007).

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.
For the entire collection see [Zbl 1108.53003].

Reviewer: Kaarin Riives (Tartu)

MSC:

53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53B50	Applications of local differential geometry to the sciences
76Z05	Physiological flows

Keywords:

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