Precompactness of domains with lower Ricci curvature bound under Gromov–Hausdorff topology

Abstract Based on a quantitative version of the classical Hopf–Rinow theorem in terms of the doubling property, we prove new precompactness principles in the (pointed) Gromov–Hausdorff topology for domains in (maybe incomplete) Riemannian manifolds with a lower Ricci curvature bound, which are applicable to those with weak regularities considered in PDE theory, and the covering spaces of balls naturally appear in the study of local geometry and topology of manifolds with lower curvature bounds. All the new principles are more general than those earlier known for manifolds with smooth boundary, and improve those for manifolds with non-smooth boundary.