A Spectra comparison theorem and its applications

TL;DR

This paper establishes a precise spectral comparison between two Riemannian manifolds under broad conditions, including volume and approximation criteria, without requiring curvature restrictions.

Contribution

It introduces a sharp spectral comparison theorem for manifolds with bounded geometry and Gromov-Hausdorff approximations, without curvature constraints.

Findings

Spectral bounds depend on volume and approximation quality.

No curvature assumptions are necessary for the comparison.

Results apply to manifolds with bounded geometry and non-zero degree approximations.

Abstract

We give a sharp comparison between the spectra of two Riemannian manifolds (Y,g) and (X,g_0) under the following assumptions: (X,g_0) has bounded geometry, (Y,g) admits a continuous Gromov-Hausdorff {\epsilon}-approximation onto (X,g_0) of non zero absolute degree, and the volume of (Y,g) is almost smaller than the volume of (X,g_0). These assumption imply no restrictions on the local topology or geometry of (Y,g) in particular no curvature assumption is supposed or infered.