Cones embedded in hyperbolic manifolds
type: Item
instance of: scholarly article
title: Cones embedded in hyperbolic manifolds
zbMATH Open document ID: 1050.57013
author: Andrew Przeworski
published in: Journal of Differential Geometry
publication date: 2004-03-08T00:00:00Z
review text: In was shown by \textit{D. Gabai, R. Meyerhoff} and \textit{N. Thurston} [Ann. Math. (2) 157, 335--431 (2003; Zbl 1052.57019)], that every closed orientable hyperbolic 3-manifold except Vol3 contains an embedded tube of radius at least \(0.52959\ldots\) about the shortest geodesic, and if the shortest geodesic has length at most \(1.0595\ldots\), there is a tube of radius at least \({\log 3} \over {2}\) about it. In the paper under review the author proves that if there is a maximal embedded tube of radius \(r\), then there is another embedded region \(W\), which is basically the union of two cones, whose shapes are determined by \(r\). Also, it is shown that a certain portion of the region \(W\) lies outside of the tube of radius \(r\), and the volume of this portion is estimated. Moreover, since the author's techniques are not specific to three dimensions, some results in arbitrary dimensions hold. The obtained results implies a lot of nice applications: (1) If a hyperbolic manifold contains a tube of radius \(r\) about a geodesic, then it also contains an embedded ball of radius \(\sinh^{-1}({1 \over 2} \tanh r)\). (2) Every closed orientable hyperbolic 3-manifold contains a ball of radius at least \(\sinh^{-1} {1 \over 4} = 0.24746\ldots\). (3) If \(M\) is a closed hyperbolic \(n\)-manifold then \(\text{Vol}(M)\geq V_n(1) ( \frac{\sigma(k+1)} {\pi^k} (\frac {\sqrt{2}-1} {4}) ^{\frac {k+1}{2}})^n\), where \(V_n(1)\) is the volume of an \(n\)-dimensional Euclidean ball of radius \(1\), \(k = [{n-1 \over 2}]\), and \(\sigma(p) = \int_0^{\pi /2} \sin^p x \,dx\). (4) Every closed orientable hyperbolic 3-manifold has volume at least \(0.28\). (5) The shortest geodesic in the smallest volume orientable hyperbolic 3-manifold has length at least \(0.09\). (6) If an orientable noncompact hyperbolic 3-manifold \(M\) has Betti number at least \(4\) then \(\text{Vol}(M) \geq \pi (\log{4 \over 3} + {3 \over 4})\).
reviewed by: Andrei Vesnin
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 2052877
zbMATH Keywords: hyperbolic manifold
zbMATH Keywords: shortest geodesic
zbMATH Keywords: minimal volume
Wikidata QID: Q115202645
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090348325
OpenAlex ID: W1595633376
DOI: 10.4310/JDG/1090348325
description: scientific article; zbMATH DE number 2052877