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

TL;DR

This paper establishes new precompactness principles in the Gromov-Hausdorff topology for domains with lower Ricci curvature bounds, applicable to irregular geometries and covering spaces, extending previous results for smooth and non-smooth boundaries.

Contribution

It introduces a quantitative Hopf-Rinow-based approach to prove precompactness for domains with lower Ricci bounds, generalizing earlier results to less regular settings.

Findings

New precompactness principles for domains with Ricci bounds

Applicable to irregular geometries and covering spaces

Improves previous results for non-smooth boundaries

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 improves those for manifolds with non-smooth boundary.