The equilibrium shape equation of biomembrane vesicles, a bilayer system, and
its solution are studied with Helfrich curvature elastic model, viewed as a surface
variation problem in differential geomethry. The biconcave disc shape of red blood
cell is obtained analyticly. The torus solutions of membranes are predicted and
confirmed by several Labs. The formation of carbon nanotubes is studied as the
result with the similar elastic model as biomembranes by which the helical carbon
nanotubes are well predicted as those observed in experiment. Equilibrium shapes of domains
in Smectic-A liquid crystals, a typical multibilayer system (MBS), have been investigated
theoretically. The surface-integral equation and stability condition are
obtained by minimizing the free energy of the MBS with their thickness. A surface
differential equation is derived from the variation of the energy. The latter concerns
the most general differential equation of surface which is the Euler-Lagrange equation
for a variational problem.
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