  We study $SO(d+1)$ invariant solutions of the classical vacuum Einstein
equations in $p+d+3$ dimensions. In the limit $d \to \infty$ with $p$ held
fixed we construct a class of solutions labelled by the shape of a membrane
(the event horizon), together with a `velocity' field that lives on this
membrane. We demonstrate that our metrics can be corrected to nonsingular
solutions at first sub-leading order in $\frac{1}{d}$ if and only if the
membrane shape and `velocity' field obey equations of motion which we
determine. These equations define a well posed initial value problem for the
membrane shape and this `velocity' and so completely determinethe dynamics of
the black hole. They may be viewed as governing the non-linear dynamics of the
light quasi normal modes of Emparan, Suzuki and Tanabe.
