Compatibility between shape equation and boundary conditions of lipid membranes with free edges

Only some special open surfaces satisfying the shape equation of lipid membranes can be compatible with the boundary conditions. As a result of this compatibility, the first integral of the shape equation should vanish for axisymmetric lipid membranes, from which two theorems of non-existence are verified: (i) There is no axisymmetric open membrane being a part of torus satisfying the shape equation; (ii) There is no axisymmetric open membrane being a part of a biconcave discodal surface satisfying the shape equation. Additionally, the shape equation is reduced to a second-order differential equation while the boundary conditions are reduced to two equations due to this compatibility. Numerical solutions to the reduced shape equation and boundary conditions agree well with the experimental data [A. Saitoh \emph{et al.}, Proc. Natl. Acad. Sci. USA \textbf{95}, 1026 (1998)].