I will consider thickness perturbations to a toy model (suggested by Frolov) for studying transitions with a change of Euclidean topology.
The model consists of a bulk N-dimensional black hole and a test D-dimensional brane in it. The black hole is spherically symmetric, static and can be neutral or charged. The unperturbed brane is infinitely thin, therefore it can be described by the Dirac-Nambu-Goto equation. It is also static, spherically symmetric and it is assumed that it reaches the asymptotic infinity where it has the form of a D-1 dimensional plane. Due to the gravitational attraction of the black hole the brane is deformed
and there are two types of equilibrium configurations. The brane either crosses the black hole horizon, or it lies totally outside of the black hole. In between the two types of configurations there exists a critical solution that separates the two phases. Frolov studied the transition between the so called subcritical (when the brane does not intersect the black hole horizon) and supercritical (when the brane crosses the horizon) phases and found a strong similarity with the Choptuik critical collapse phenomena and the merger transition in a black hole - black string system. The model is also very useful for the study of certain types of phase transitions of strongly interactive matter in QCD, due to the gauge/gravity correspondence.