ABSTRACT
In this chapter we give analytical representations of Willmore and generalized Willmore (Helfrich) surfaces. All cylindrical surfaces of that type are described. Some special families of axially symmetric surfaces providing local extrema to the corresponding functionals are presented. The foregoing surfaces are determined analytically, the components of the respective position vectors being represented either in explicit form or by quadratures in terms of suitable variables. Parametric equations of Dupin cyclides and other Willmore surfaces obtained by inversions of catenoids and Enneper’s surface are also presented.
