We use the AdS/CFT correspondence to argue that large rotating black holes in global AdS D spaces are dual to stationary solutions of the relativistic Navier-Stokes equations on S D−2 . Reading off the equation of state of this fluid from the thermodynamics of non-rotating black holes, we proceed to construct the nonlinear spinning solutions of fluid mechanics that are dual to rotating black holes. In all known examples, the thermodynamics and the local stress tensor of our solutions are in precise agreement with the thermodynamics and boundary stress tensor of the spinning black holes. Our fluid dynamical description applies to large non-extremal black holes as well as a class of large non-supersymmetric extremal black holes, but is never valid for supersymmetric black holes. Our results yield predictions for the thermodynamics of all large black holes in all theories of gravity on AdS spaces, for example, string theory on AdS 5 × S 5 and M theory on AdS 4 × S 7 and AdS 7 × S 4 .

Mikhailov has constructed an infinite family of 1 8 BPS D3-branes in AdS 5 × S 5 . We regulate Mikhailov's solution space by focussing on finite dimensional submanifolds. Our submanifolds are topologically complex projective spaces with symplectic form cohomologically equal to 2πN times the Fubini-Study Kähler class. Upon quantization and removing the regulator we find the Hilbert Space of N noninteracting Bose particles in a 3d Harmonic oscillator, a result previously conjectured by Beasley.This Hilbert Space is isomorphic to the classical chiral ring of 1 8 BPS states in N = 4 Yang-Mills theory. We view our result as evidence that the spectrum of 1 8 BPS states in N = 4 Yang Mills theory, which is known to jump discontinuously from zero to infinitesimal coupling, receives no further renormalization at finite values of the 't Hooft coupling.

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We construct solutions to the relativistic Navier-Stokes equations that describe the long wavelength collective dynamics of the deconfined plasma phase of N = 4 Yang Mills theory compactified down to d = 3 on a Scherk-Schwarz circle and higher dimensional generalisations. Our solutions are stationary, axially symmetric spinning balls and rings of plasma. These solutions, which are dual to (yet to be constructed) rotating black holes and black rings in Scherk-Schwarz compactified AdS 5 and AdS 6 , and have properties that are qualitatively similar to those of black holes and black rings in flat five dimensional supergravity. .More generally, one would expect a finite lump of plasma that evolves according to the relativistic Navier-Stokes equations map in the bulk to a 'black hole' that evolves according to the Einstein equations. Provided all length scales in the plasma solution are 1 Notice that, at large u, T x (u) ≃ 1, so (1) reduces to AdS d+2 in Poincaré-patch coordinates, with u as the radial scale coordinate, and with one of the spatial boundary coordinates, θ, compactified on a circle (the remaining boundary coordinates, w i and t, remain non-compact).2 T c = 1/2π in the dimensionless units of (3)