In this note, we give a diffeomorphism (to Rⁿ) criterion via long-time Ricci flow and show some applications. In particular, we provide an affirmative answer that the conclusion in Manifolds with small curvature concentration, Ann. PDE, 2024 by Chan, Lee and the first named author and Removing scalar curvature assumption for Ricci flow smoothing, Bull. Lond. Math. Soc. , 2025 by A. Martens about manifolds with small curvature concentration can be improved to diffeomorphism in dimension 4.
Huang et al. (Sat,) studied this question.