Local bounded covering geometry in the non-smooth RCD setting

Extend the metric-smoothing and collapsing-structure techniques from bounded covering geometry to non-smooth RCD spaces with locally bounded covering geometry.

Background

The paper proves smoothing and fibration results for RCD spaces with a uniform lower bound on the injectivity radius of the universal cover, called bounded covering geometry. A weaker local bounded covering geometry condition is natural in collapsed geometry because local universal covers may have controlled injectivity radius even when the global universal cover remains collapsed.

For smooth spaces, the authors indicate that local Riemannian techniques can extend the results to this weaker setting. The unresolved point is whether those techniques, including the required smoothing and convergence arguments, can be justified without an a priori smooth structure.