Volume growth of some uniformly contractible Riemannian manifolds

Abstract

It is shown that if a uniformly contractible Riemannian n-manifoldwhere ω n is the volume of the unit Euclidean ball. In particular, if M is uniformly contractible and d GH ((M, d g ), (V n , · )) < ∞, then M has at least Euclidean volume growth. This corollary covers an earlier result by Burago and Ivanov. Our results are motivated by a volume growth theorem contained in Gromov's book [Gromov in Progress in Mathematics, vol. 152, Birkhäuser, Boston, 1999, p. 256], we give a detailed proof of this theorem. Using the same argument, we also derive a generalization of the theorem which is pointed out by Gromov.

Zero asymptotic Lipschitz distance and finite Gromov-Hausdorff distance

“…The construction of the metric in Proposition 1.4 and Theorem 1.5 is motivated by a suggestion of the referee of a recent paper [7] in which the author proves the same conclusion in one dimension case. In two dimension case, the author attempted to prove Theorem 1.5 only by inspecting the behavior of nets under the Hausdorff approximations, but he was not successful.…”