Say we have two loops given parametrically by $X^\mu(\sigma)$ and $Y^\mu(\sigma)$. We want to find the minimal possible area of a tube-like surface ending on these loops. It would be a functional $F[X,Y]$. And it also can't depend on the paramaterisation. Is this formula known or is there a way of finding this out?
I want to find an explict formula involving $X$ and $Y$ as an integral or summation. Because of the symmetries involved this should narrow down the possible formula.
We can think of this as two loops of wire and a bubble film joining them.
I suspect this is almost the same problem as the minimal area of a bubble on a single loop, since one can pinch a single loop to almost form two loops. So if one can solve the problem for one loop then this automatically solves the cases where the two loops are joined at a point.