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Black Holes as Incompressible Fluids on the Sphere

We consider finite deformations of the p+2-dimensional Schwarzschild geometry which obey the vacuum Einstein equation, preserve the mean curvature and induced conformal metric on a sphere a distance $λ$ from the horizon and are regular on the future horizon. We show perturbatively that in the limit $λ$ approaches 0 the deformations are given by solutions of the nonlinear incompressible Navier-Stokes equation on the p-sphere. This relation provides a link between global existence for p-dimensional incompressible Navier-Stokes fluids and a novel form of cosmic censorship in p+2-dimensional general relativity.