The follow-up paper by Helfrich you are searching for is
Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders, Ou-Yang Zhong-can and Wolfgang Helfrich, Physical Review A 39, 5280-5288 (1989).
A general equation of mechanical equilibrium of fluid membranes subject to bending elasticity [reported in Phys. Rev. Lett. 59, 2486 (1987)] is derived in detail.