Volume rigidity for finite volume manifolds

We extend to finite volume manifolds some volume rigidity results of Besson-Courtois-Gallot. Our main result is a Volume Theorem which shows that proper maps from finite volume manifolds with Ricci curvature bounded from below to finite volume manifolds of pinched negative curvature decrease volume. As a consequence we deduce a lower bound on the minimal volume of finite volume manifolds possessing a proper map to a finite volume manifold of pinched negative curvature, and we show asymptotic isometry holds when the minimal volume is attained. Finally, we prove that finite, rank one, locally symmetric manifolds minimize normalized entropy rigidity in their homotopy class.