Smooth Volume Rigidity for Manifolds with Negatively Curved Targets

We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the degree is one. The conditions hold when the volumes or entropy-volumes of the two manifolds differ by less than a uniform constant after an appropriate normalization of the metrics. The results are qualitatively sharp in the sense that all dependencies are necessary. We present a number of corollaries including a corresponding finiteness result. Notably, the method of proof does not rely on a Cα or Gromov-Hausdorff precompactness result nor on surgery technology.