Local mollification of metrics with small curvature concentration

In this work, we establish a local-smoothing result on metrics with small curvature concentration with respect to Sobolev constants and volume growth. As an application, we prove that the Gromov–Hausdorff limit of compact manifolds with bounded curvature concentration, Ahlfors n -regularity, and bounded Sobolev constants is a topological manifold away from finitely many points. In the complete non-compact case, we show that manifolds with Euclidean-type Sobolev inequality, Euclidean volume growth, and small curvature concentration are necessarily diffeomorphic to Euclidean spaces. In contrast with all previous works, we remove the Ricci-curvature condition.