Consider minimizing the surface area of the surface spanning two parallel rings. It is well known that the global minimizer is either a catenoid, or the so-called Goldschmidt solution consisting of two disks, depending on the radius and separation distance of the rings.
My question is: is the Goldschmidt solution always a local minimum (even in cases where a catenoid is the global minimum)? Or does it become a saddle point for some values of the radius/distance?
Ordinarily I’d analyze this problem by looking at the second variations of the surface area functional, but I’m not sure how to do that here since the Goldschmidt solution is not continuous.