Volume growth and asymptotic cones of manifolds with nonnegative Ricci curvature

Abstract

Let $M$ be an open (i.e. complete and noncompact) manifold with nonnegative Ricci curvature. In this paper, we study whether the volume growth order of $M$ is always greater than or equal to the dimension of some (or every) asymptotic cone of $M$. Our first main result asserts that, under the conic at infinity condition, if the infimum of the volume growth order of $M$ equals $k$, then there exists an asymptotic cone of $M$ whose upper box dimension is at most $k$. In particular, this yields a complete affirmative answer to our problem in the setting of nonnegative sectional curvature. In the subsequent part of the paper, we extend or partially extend Sormani's results concerning $M$ with linear volume growth to more relaxed volume growth conditions. Our approach also allows us to present a new proof of Sormani's sublinear diameter growth theorem for open manifolds with $\mathrm{Ric}\geq 0$ and linear volume growth. Finally, we construct an example of an open $n$-manifold $M$ with $\mathrm{sec}_M\geq0$ whose volume growth order oscillates between 1 and $n$.