Macroscopic band width inequalities

TL;DR

This paper establishes new band width inequalities for Riemannian manifolds with specific topological properties, introducing the concept of filling enlargeable manifolds and linking geometric bounds to topological invariants.

Contribution

It introduces the class of filling enlargeable manifolds and proves band width inequalities based on volume conditions in the universal cover.

Findings

Filling enlargeable manifolds satisfy specific width bounds.

Enlargeable and aspherical manifolds are filling enlargeable.

Width bounds depend on the volume of unit balls in the universal cover.

Abstract

Inspired by Gromov's work on 'Metric inequalities with scalar curvature' we establish band width inequalities for Riemannian bands of the form $(V=M\times[0,1],g)$, where $M^{n-1}$ is a closed manifold. We introduce a new class of orientable manifolds we call filling enlargeable and prove: If $M$ is filling enlargeable and all unit balls in the universal cover of $(V,g)$ have volume less than a constant $\frac{1}{2}\varepsilon_n$, then $width(V,g)\leq1$. We show that if a closed orientable manifold is enlargeable or aspherical, then it is filling enlargeable. Furthermore we establish that whether a closed orientable manifold is filling enlargeable or not only depends on the image of the fundamental class under the classifying map of the universal cover.