The Cut-off Covering Spectrum

Details The Cut-off Covering Spectrum The Cut-off Covering Spectrum

2007-05-25

arxiv.org/abs/0705.3822v3

Transactions of the American Mathematical Society 362 (2010), 2339-2391.

Mathematics

Metric Geometry

44 pages, 4 figures; v3 numbered as will be published, minor clarifications of some statements and details added to some proof

Scientific paper

We introduce the $R$ cut-off covering spectrum and the cut-off covering spectrum of a complete length space or 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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