Abstract: Given a closed hyperbolic manifold M of dimension at least 3, let G be its fundamental group. G acts by the regular representation on the unit sphere S in the space of L^2 functions on G. There is a natural homotopy class of Lipschitz maps from M to the quotient space S/G. The infimum of the volume of the images of M by such maps is known. I will talk about the following uniqueness property for Plateau solutions: given any image of M whose volume is close to the infimum, that image is close to (a rescaling of) the hyperbolic metric in the intrinsic flat topology of Sormani-Wenger. I will then explain how this uniqueness result applies to the question of stability for the volume entropy inequality of Besson-Courtois-Gallot.