## The cut-off covering spectrum

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- by Christina Sormani and Guofang Wei PDF
- Trans. Amer. Math. Soc.
**362**(2010), 2339-2391 Request permission

## Abstract:

We introduce the $R$ cut-off covering spectrum and the cut-off covering spectrum of a metric space or a Riemannian manifold. The spectra measure the sizes of localized holes in the space and are defined using covering spaces called $\delta$ covers and $R$ cut-off $\delta$ covers. They are investigated using $\delta$ homotopies which are homotopies via grids whose squares are mapped into balls of radius $\delta$.

On locally compact spaces, we prove that these new spectra are subsets of the closure of the length spectrum. We prove the $R$ cut-off covering spectrum is almost continuous with respect to the pointed Gromov-Hausdorff convergence of spaces and that the cut-off covering spectrum is also relatively well behaved. This is not true of the covering spectrum defined in our earlier work, which was shown to be well behaved on compact spaces. We close by analyzing these spectra on Riemannian manifolds with lower bounds on their sectional and Ricci curvature and their limit spaces.

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## Additional Information

**Christina Sormani**- Affiliation: Graduate School and University Center, CUNY, New York, New York 10016 – and – Lehman College, CUNY, Bronx, New York 10468
- MR Author ID: 637216
- ORCID: 0000-0002-2295-2585
- Email: sormanic@member.ams.org
**Guofang Wei**- Affiliation: Department of Mathematics, University of California, Santa Barbara, Santa Barbara, California 93106
- MR Author ID: 252129
- Email: wei@math.ucsb.edu
- Received by editor(s): October 31, 2007
- Published electronically: December 3, 2009
- Additional Notes: The first author was partially supported by a grant from the City University of New York PSC-CUNY Research Award Program

The second author was partially supported by NSF Grant # DMS-0505733 - © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**362**(2010), 2339-2391 - MSC (2010): Primary 54E45, 53C20
- DOI: https://doi.org/10.1090/S0002-9947-09-04916-2
- MathSciNet review: 2584603