2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/152072We prove that Riemannian metrics with a uniform weak norm can be smoothed to having arbitrarily high regularity. This generalizes all previous smoothing results. As a consequence we obtain a generalization of Gromov's almost flat manifold theorem. A uniform Betti number estimate is also obtained.18 pages, LatexDifferential Geometrytext