# Macroscopic band width inequalities

Source: [Macroscopic band width inequalities](<https://www.sota2.com/research/paper/68fa59441e209131ce9b598e>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-28T13:56:01.050Z

Authors: Daniel R\\"ade

Published: 2019-11-29T08:40:50.000Z

## 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1911.13000>)

[PDF](<https://arxiv.org/pdf/1911.13000.pdf>)