Volume growth on manifolds with more than one end
The volume growth function of a Riemannian manifold of bounded geometry is a function having bounded growth of derivative, called a bgd-function. Grimaldi and Pansu showed that any one-ended manifold of finite type admits a metric of bounded geometry such that the volume growth function has the growth type of any given bgd-function. In the case of infinite type, this holds for any superlinear bgd-function. We generalize this result to manifolds with finitely many ends and show that a manifold with infinitely many ends does not admit a metric with some bgd-functions as volume growth, including polynomials. We also prove that on certain manifolds with countably many ends, there exist metrics such that the volume growth functions have a growth type arbitrarily close to that of any predetermined bgd-function.