Inverted catenoid as a fluid membrane with two points pulled together

Under inversion in any (interior) point, a catenoid transforms into a deflated compact geometry which touches at two points (its poles). The catenoid is a minimal surface and, as such, is an equilibrium shape of a symmetric fluid membrane. The conformal symmetry of the Hamiltonian implies that inverted minimal surfaces are also equilibrium shapes. However, they will exhibit curvature singularities at their poles. Such singularities are the geometrical signature of the external forces required to pull the poles together. These forces will set up stresses in the inverted shapes. Tuning the force corresponds geometrically to the translation of the point of inversion. For any fixed surface area, there will be a maximum force. The associated shape is a symmetric discocyte. Lowering the external force will induce a transition from the discocyte to a cup-shaped stomatocyte.