Abstract:  Based on a pure geometric description of Kerr black holes, we show that if the vacuum space-time is locally close to Kerr family, the geometry on the horizons must match the geoemtry on the asymptotic infinity. In particular, it shows that there is only one black hole. This generalizes the famous theorem of Bunting-Masood-ul-Alam on the static case. The work is joint with Willie Wai-Yeung Wong.