# The Cut-off Covering Spectrum - Mathematics > Metric Geometry

Abstract: We introduce the $R$ cut-off covering spectrum and the cut-off coveringspectrum of a complete length space or Riemannian manifold. The spectra measurethe sizes of localized holes in the space and are defined using covering spacescalled $\delta$ covers and $R$ cut-off $\delta$ covers. They are investigatedusing $\delta$ homotopies which are homotopies via grids whose squares aremapped into balls of radius $\delta$.On locally compact spaces, we prove that these new spectra are subsets of theclosure of the length spectrum. We prove the $R$ cut-off covering spectrum isalmost continuous with respect to the pointed Gromov-Hausdorff convergence ofspaces 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 wasshown to be well behaved on compact spaces. We close by analyzing these spectraon Riemannian manifolds with lower bounds on their sectional and Riccicurvature and their limit spaces.

Author: Christina Sormani, Guofang Wei

Source: https://arxiv.org/