Bending of a surface with spontaneous curvature

We interest to curvature deformations that can be described by Helfrich's energy: a quadratic mean curvature term, and a gaussian curvature term. When the surface is not strictly incompressible and presents a nonzero spontaneous mean curvature, we focus on simple cases to show that a priori determination of key features (spontaneous curvature, equilibrium area) may be biased according the expression taken for the energy.