Doughnut-shaped soap bubbles

Soap bubbles are thin liquid films enclosing a fixed volume of air. Since the surface tension is typically assumed to be the only responsible for conforming the soap bubble shape, the realized bubble surfaces are always minimal area ones. Here, we consider the problem of finding the axisymmetric minimal area surface enclosing a fixed volume $V$ and with a fixed equatorial perimeter $L$. It is well known that the sphere is the solution for $V=L^3/6\pi^2$, and this is indeed the case of a free soap bubble, for instance. Surprisingly, we show that for $V<\alpha L^3/6\pi^2$, with $\alpha\approx 0.21$, such a surface cannot be the usual lens-shaped surface formed by the juxtaposition of two spherical caps, but rather a toroidal surface. Practically, a doughnut-shaped bubble is known to be ultimately unstable and, hence, it will eventually lose its axisymmetry by breaking apart in smaller bubbles. Indisputably, however, the topological transition from spherical to toroidal surfaces is mandatory here for obtaining the global solution for this axisymmetric isoperimetric problem. Our result suggests that deformed bubbles with $V<\alpha L^3/6\pi^2$ cannot be stable and should not exist in foams, for instance.