Volumes of tubes in hyperbolic 3-manifolds
``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.
geodesics and tubes in hyperbolic 3-manifolds
scientific article; zbMATH DE number 1782673