Stable black strings in anti–de Sitter space

In five-dimensional Einstein gravity with a negative cosmological constant in the presence or absence of a non-fine-tuned 3-brane, we investigate the classical stability of black string solutions which are foliations of four-dimensional AdS/dS-Schwarzschild black holes. Such black strings are generically unstable as in the well-known Gregory-Laflamme instability. For AdS black strings, however, it turns out that they become stable if the longitudinal size of the horizon is larger than the order of the ${\mathrm{AdS}}_{4}$ radius. Even in the case of unstable black strings, the AdS black strings have very different features of string fragmentation from those in the flat brane world. Some implications of our results for the Gubser-Mitra conjecture are also discussed.