Black Rings in Large $D$ Membrane Paradigm at the First Order

Black rings are the black objects found in $D$ spacetime dimensional gravity when $D \geq 5$. These have event horizon topology $S^{D-3}\times S^1$. In this work the solutions of the large $D$ membrane paradigm dual to stationary black rings in Einstein-Maxwell theory with or without cosmological constant are studied. It is shown that the first order membrane equations can only admit static asymptotically flat black rings, and the equilibrium angular velocity for the asymptotically AdS black rings at large $D$ was obtained. The thermodynamic and dynamic stability of the asymptotically flat black ring solutions is studied. The apparent shortcomings of some of these results are argued to be curable within the large $D$ membrane paradigm framework.