Volumes of tubes in hyperbolic 3-manifolds
type: Item
instance of: scholarly article
title: Volumes of tubes in hyperbolic 3-manifolds
zbMATH Open document ID: 1029.57014
author: David Gabai
author: G. Robert Meyerhoff
author: Peter Milley
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: ``We give the first explicit lower bound for the length of a geodesic in a closed orientable hyperbolic 3-manifold \(M\) of lowest volume. We also give an upper bound for the tube radius of any shortest geodesic in \(M\). We explain how these results might be the first steps towards a rigorous computer assisted effort to determine the least volume closed orientable hyperbolic 3-manifold(s).'' \smallskip The main theorem states that a maximal tube of radius \(r\) about a geodesic \(\gamma\) in a complete orientable hyperbolic 3-manifold has volume at least 0.95 if either length\((\gamma) \leq 0.069\) or \(r \geq 1.483\). Noting that the smallest known volume of a complete hyperbolic 3-manifold (that of the ''Fomenko-Matveev-Weeks manifold'') is less than 0.95, this gives a lower and upper bound as above for a smallest volume manifold. These bounds have been improved in several preprints mentioned in the paper.
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 1782673
zbMATH Keywords: geodesics and tubes in hyperbolic 3-manifolds
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090348088
OpenAlex ID: W1517052668
Wikidata QID: Q115202922
DOI: 10.4310/JDG/1090348088
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description: scientific article; zbMATH DE number 1782673