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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The cut-off covering spectrum


Authors: Christina Sormani and Guofang Wei
Journal: Trans. Amer. Math. Soc. 362 (2010), 2339-2391
MSC (2010): Primary 54E45, 53C20
Published electronically: December 3, 2009
MathSciNet review: 2584603
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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
Email: sormanic@member.ams.org

Guofang Wei
Affiliation: Department of Mathematics, University of California, Santa Barbara, Santa Barbara, California 93106
Email: wei@math.ucsb.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04916-2
PII: S 0002-9947(09)04916-2
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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.