Challenges in theoretical investigations of configurations of lipid membranes

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