Uniqueness, stability and Hessian eigenvalues for two-dimensional bubble clusters

A recent conjecture on two-dimensional foams suggested that for fixed topology with given bubble areas there is a unique state of stable equilibrium, We present counter-examples, consisting of a ring of bubbles around a central one, which refute this conjecture. The discussion centres on a novel form of instability which causes symmetric clusters to become distorted. The stability of these bubble clusters is examined in terms of the Hessian of the energy.