Perturbations of the Kerr black hole in the class of axisymmetric artificial black holes

Artificial black holes (called also acoustic or optical black holes) are the black holes for the linear wave equation describing the wave propagation in a moving medium. They attracted a considerable interest of physicists who study them to better understand the black holes in general relativity. We consider the case of stationary axisymmetric metrics and we show that the Kerr black hole is not stable under perturbations in the class of all axisymmetric metrics. We describe families of axisymmetric metrics having black holes that are the perturbations of the Kerr black hole. We also show that the ergosphere can be determined by boundary measurements and we prove the uniform boundness of the solution in the exterior of the black hole when the event horizon coincides with the ergosphere.