  We establish an attractor mechanism for the horizon metric of asymptotically
locally AdS4 supersymmetric black holes. The K\"ahler moduli of the metric are
not fixed in the asymptotic region, however supersymmetry dictates that the
near horizon metric must be the constant curvature one. We show how this
mechanism is realized in detail for four-dimensional N=2 gauged supergravity
coupled to vector multiplets by focusing on the STU model. A similar analysis
is performed for gauged supergravity theories in five, six, and seven
dimensions where we establish the same mechanism by extending previous results
on holographic uniformization.
