  The Helfrich functional, denoted by H^{c_0}, is a mathematical expression
proposed by Helfrich (1973) for the natural free energy carried by an elastic
phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid
bilayers minimise H^{c_0} among all possible configurations. The functional
integrates a spontaneous curvature parameter c_0 together with the mean
curvature of the bilayer and constraints on area and volume, either through an
inclusion of osmotic pressure difference and tensile stress or otherwise. Using
the mathematical concept of embedded orientable surface to represent the
configuration of the bilayer, one might expect to be able to adapt methods from
differential geometry and the calculus of variations to perform a fine analysis
of bilayer configurations in terms of the parameters that it depends upon. In
this article we focus upon the case of spherical red blood cells with a view to
better understanding spherocytes and spherocytosis. We provide a complete
classification of spherical solutions in terms of the parameters in the
Helfrich model. We additionally present some further analysis on the rigidity
and stability of spherocytes.
