  It has recently been shown that the dynamics of black holes in large number
of dimensions D can be recast as the dynamics of a probe membrane propagating
in the background spacetime which solves Einstein equations without matter. The
equations of motion of this membrane are simply the statement of conservation
of the stress tensor and charge current defined on this membrane. In this paper
we obtain the effective equations of motion for stationary membranes in any
empty background both in presence and absence of charge. It turns out that the
thermodynamic quantities associated with the stationary membranes that satisfy
these effective equations also satisfy the first law of black hole
thermodynamics. These stationary membrane equations have some interesting
solutions such as charged rotating black holes in flat and AdS backgrounds as
well as black ring solutions in large D.
