Regularity and structure of spaces with synthetic Ricci bounds and positive injectivity radius

Abstract

In the paper, we develop a structure theory for metric measure spaces with synthetic lower Ricci curvature bounds, known as RCD spaces. Under a positive injectivity-radius assumption, we recover a smooth differential structure and prove a regularity result: such spaces are $W^{1,p}_{\mathrm{loc}}\cap C^{0,α}_{\mathrm{loc}}$-Riemannian manifolds whose weight functions also belong to $W^{1,p}_{\mathrm{loc}}\cap C^{0,α}_{\mathrm{loc}}$, in both distance and harmonic charts for all $p<\infty$ and $α\in (0,1)$. We also establish a quantitative lower bound for the harmonic radius, together with compactness theorems and quantitative geometric bounds. As a byproduct, we develop a comprehensive elliptic regularity theory in this setting. These results apply, in particular, to smooth weighted Riemannian manifolds and to RCD spaces satisfying a synthetic curvature upper bound, or CBA condition. Even in these settings, the resulting regularity statements are new. In the latter case, both the Riemannian metric and the weight function are shown to be locally Lipschitz. As further applications, we establish fibration theorems and use metric smoothing to confirm, in a synthetic framework, a conjecture of V. Kapovitch concerning almost flat manifolds with mixed curvature bounds.