  If a catenoid is inverted in any interior point, a deflated compact geometry
is obtained 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 exhibit curvature
singularities at their poles. These singularities are associated with external
forces pulling the poles together. Unlike the catenoid which is free of stress,
there will be stress within the inverted shapes. If the surface area is fixed,
reducing the external force induces a transition from a discocyte to a
cup-shaped stomatocyte.
