2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/139396We 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 the dependencies are necessary. We give a number of corollaries.Updated from 2006 versionDifferential GeometryGeometric Topology53C20,53C23,53C24text