  Minimal surface problems arise naturally in many soft matter systems whose
free energies are dominated by surface or interface energies. Of particular
interest are the shapes, stability and mechanical stresses of minimal surfaces
spanning specific geometric boundaries. The "catenoid" is the best-known
example where an analytical solution is known which describes the form and
stability of a minimal surface held between two parallel, concentric circular
frames. Here we extend this problem to non-axisymmetric, parallel frame shapes
of different orientations, by developing a perturbation approach around the
known catenoid solution. We show that the predictions of the perturbation
theory are in good agreement with experiments on soap films and finite element
simulations (Surface Evolver). Combining theory, experiment and simulation, we
analyse in depth how the shapes, stability and mechanical properties of the
minimal surfaces depend on the type and orientation of elliptical and
three-leaf clover shaped frames. In the limit of perfectly aligned
non-axisymmetric frames, our predictions show excellent agreement with a recent
theory established by Alimov et al (M. M. Alimov, A. V. Bazilevsky and K. G.
Kornev, Physics of Fluids, 2021, 33, 052104). Moreover, we put in evidence the
intriguing capacity of minimal surfaces between non-axisymmetric frames to
transmit a mechanical torque despite being completely liquid. These forces
could be interesting to exploit for mechanical self-assembly of soft matter
systems or as highly sensitive force captors.
