The cutoff covering spectrum
Authors:
Christina Sormani and Guofang Wei
Journal:
Trans. Amer. Math. Soc. 362 (2010), 23392391
MSC (2010):
Primary 54E45, 53C20
Published electronically:
December 3, 2009
MathSciNet review:
2584603
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Abstract: We introduce the cutoff covering spectrum and the cutoff 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 covers and cutoff covers. They are investigated using homotopies which are homotopies via grids whose squares are mapped into balls of radius . On locally compact spaces, we prove that these new spectra are subsets of the closure of the length spectrum. We prove the cutoff covering spectrum is almost continuous with respect to the pointed GromovHausdorff convergence of spaces and that the cutoff 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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 J. Cheeger, D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413443. MR 0309010 (46:8121)
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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/S0002994709049162
PII:
S 00029947(09)049162
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 PSCCUNY Research Award Program
The second author was partially supported by NSF Grant # DMS0505733
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
