Challenges in theoretical investigations on configurations of lipid membranes

This review reports some key results in theoretical investigations on configurations of lipid membranes and presents several challenges in this field which involve (i) exact solutions to the shape equation of lipid vesicles; (ii) exact solutions to the governing equations of open lipid membranes; (iii) neck condition of two-phase vesicles in the budding state; (iv) nonlocal theory of membrane elasticity; (v) relationship between symmetry and the magnitude of free energy.