Axially Symmetric Willmore Surfaces Determined by Quadratures

Abstract

The work is concerned with a special family of axially symmetric surfaces providing local extrema to the so-called Willmore functional, which assigns to each surface its total squared mean curvature. The components of the position vector of the profile curves of the regarded Willmore surfaces satisfy a system of first-order ordinary differential equations. The solutions of this system are expressed by quadratures in terms of the tangent angle and, in this way, the corresponding Willmore surfaces are determined.

First integrals of the axisymmetric shape equation of lipid membranes

“…This form of Willmore equation was studied in some other works [41][42][43] through the geometric (point) Lie symmetry group analysis. Although we reproduce the same solution, we show that it can be found by constructing the first integral (34).…”

“…where S = S(ρ, ψ) is the principal function. When c 0 , λ , p, and Ω 0 vanish, equation (43) becomes…”

Combined effects of headgroup charge and tail unsaturation of lipids on lateral organization and diffusion of lipids in model biomembranes

“…This form of Willmore equation was studied in some other works [41][42][43] through the geometric (point) Lie symmetry group analysis. Although we reproduce the same solution, we show that it can be found by constructing the first integral (34).…”