Bi-Lipschitz Smoothing under Ricci and Injectivity Bounds

We prove that a complete Riemannian manifold with a positive uniform lower bound on injectivity radius and a positive uniform lower bound on Ricci curvature admits an $L^{\mathrm{\infty }}$-close (bi-Lipschitz) smooth metric with two-sided Ricci curvature bounds and a uniform positive lower bound on injectivity radius. This answers Question 2 in the Morgan–Pansu list of open problems from the conference Modern Trends in Differential Geometry (São Paulo, 2018), proposed by L. Bandara. In the proof, we rely on controlled smoothing with Croke’s universal local volume lower bound and the Cheeger–Gromov–Taylor injectivity radius estimate.