Quantitative partitioned index theorem and noncompact band-width

TL;DR

This paper develops a quantitative partitioned manifold index theory applicable to noncompact hypersurfaces, leading to a generalized Gromov's band-width estimate for noncompact Riemannian bands with scalar curvature bounds.

Contribution

It introduces a new quantitative index theory for noncompact hypersurfaces and extends Gromov's band-width estimate to noncompact Riemannian bands.

Findings

Established a quantitative version of partitioned manifold index theory.

Proved a generalized Gromov's band-width estimate for noncompact bands.

Extended scalar curvature bounds to noncompact hypersurfaces.

Abstract

Gromov's band-width conjecture gives a precise upper bound for the width of a compact Riemannian band with positive scalar curvature lower bound, assuming that the cross-section of the band admits no positive scalar curvature metrics. Versions of this were proved by Cecchini and by Zeidler. In this paper, we develop a quantitative version of partitioned manifold index theory, which applies to noncompact hypersurfaces. Using this, we prove a version of Gromov's band-width estimate for possibly noncompact Riemannian bands.