Minimum Volume Cusped Hyperbolic Three-manifolds

This corollary extends work of Cao and Meyerhoff who had earlier shown that m003 and m004 were the smallest volume cusped manifolds. Also, the above list agrees with the SnapPea census of one-cusped manifolds produced by Jeff Weeks ([W]), whose initial members are conjectured to be an accurate list of small-volume cupsed manifolds. Let N be a closed hyperbolic 3-manifold with simple closed geodesic γ and let Nγ denote the manifold N \ γ. Agol ([Ago]) discovered a formula relating Vol(N) to Vol(Nγ) and the tube radius of γ. Assuming certain results of Perelman, Agol and Dunfield (see [AST]) have further strengthened that result. A straightforward calculation (see [ACS]) using this stronger result, the log(3)/2 theorem of [GMT], plus bounds on the density of hyperbolic tube packings by Przeworksi, shows that a compact hyperbolic manifold with volume less than that of the Weeks manifold must be obtainable by Dehn filling on a cusped manifold with volume less than or equal to 2.848. The paper [MM] rigorously shows that the Weeks manifold is the unique compact hyperbolic 3-manifold of smallest volume obtained by filling any of the 10 manifolds listed in Corollary 1.2. We therefore obtain,