Hyperbolic groups and spherical minimal surfaces

Let M be a closed, oriented, negatively curved, n-dimensional manifold with fundamental group. Let S^ be the unit sphere in ² (), on which acts by the regular representation. The spherical volume of M is a topological invariant introduced by Besson-Courtois-Gallot. We show that it is equal to the area of an n-dimensional area-minimizing minimal surface inside the ultralimit of S^/, in the sense of Ambrosio-Kirchheim. Our proof combines the theory of metric currents with a study of limits of the regular representation of torsion-free hyperbolic groups.