The geometric Cauchy problem for the membrane shape equation

We address the geometric Cauchy problem for surfaces associated to the membrane shape equation describing equilibrium configurations of vesicles formed by lipid bilayers. This is the Euler--Lagrange equation of the Canham-Helfrich-Evans elastic curvature energy subject to constraints on the enclosed volume and the surface area. Our approach uses the method of moving frames and techniques from the theory of exterior differential systems