Precompactness of domains with lower Ricci curvature bound under Gromov–Hausdorff topology
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Shicheng Xu
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.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11821101
Award Identifier / Grant number: 12271372
Funding statement: Supported in part by NSFC 11821101 and 12271372.
Acknowledgements
The author is grateful to Professor Xiaochun Rong for proposing the problem on the precompactness of covering spaces of metric balls, and Professor Christina Sormani for pointing out earlier works on the same topic and valuable comments that led to improvement of the comparison with earlier results.
References
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