Cones embedded in hyperbolic manifolds
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 \(\text{Vol}(M) \geq \pi (\log{4 \over 3} + {3 \over 4})\).
scientific article; zbMATH DE number 2052877