Global minimizers for the doubly-constrained Helfrich energy: the   axisymmetric case
  Since the pioneering work of Canham and Helfrich, variational formulations involving curvature-dependent functionals, like the classical Willmore functional, have proven useful for shape analysis of biomembranes. We address minimizers of the Canham-Helfrich functional defined over closed surfaces enclosing a fixed volume and having fixed surface area. By restricting attention to axisymmetric surfaces, we prove the existence of global minimizers. 
Rustum Choksi and Marco Veneroni
math.AP math.DG