Axisymmetric Membranes under External Force

Motivated by recent experimental work on colloidal membranes, we calculate the shape and force vs. extension for an axisymmetric membrane of fixed area connecting two rigid rings. For sufficiently small area, the catenoid always solves the Euler-Lagrange equation for this problem, but the area constraint means that the shape is a catenoid only for a single value of the extension. At large extensions, structures can form where the two rings are connected by a thin cylindrical tether. We describe the conditions under which there is a maximum extension as in the soap film problem, and when there is no maximum limit to the extension.