Critical escape velocity of black holes from branes

In recent work we have shown that a black hole stacked on a brane escapes once it acquires a recoil velocity. This result was obtained in the {\it probe-brane} approximation, {\it i.e.}, when the tension of the brane is negligibly small. Therefore, it is not clear whether the effect of the brane tension may prevent the black hole from escaping for small recoil velocities. The question is whether a critical escape velocity exists. Here, we analyze this problem by studying the interaction between a Dirac-Nambu-Goto brane and a black hole assuming adiabatic (quasi-static) evolution. By describing the brane in a fixed black hole spacetime, which restricts our conclusions to lowest order effects in the tension, we find that the critical escape velocity does not exist for co-dimension one branes, while it does for higher co-dimension branes.