  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.
