Stabilizing homogeneous black strings in AdS

It is known that homogenous black strings in flat spacetime are unstable under long wavelength perturbations. Recently, by including scalar fields that depend on the extended coordinates only, homogenous black strings have been constructed in the presence of a negative cosmological constant, therefore implying that generic perturbations will contain both metric and scalar fluctuations. Here we show that for such generic perturbations homogeneous black strings with a negative cosmological constant are perturbatively stable at linear level, regardless the size of the Schwarzschild-AdS black hole located on the brane.