Volume growth and the topology of Gromov-Hausdorff limits

TL;DR

This paper investigates how volume growth in Gromov-Hausdorff limit spaces of nonnegatively curved manifolds influences their topological properties, establishing new estimates and conditions for triviality of homotopy groups.

Contribution

It generalizes previous work to relate volume growth bounds to the triviality of homotopy groups in Gromov-Hausdorff limit spaces, with explicit constants and new excess estimates.

Findings

Proves an Abresch-Gromoll type excess estimate for limit spaces.

Shows volume growth lower bounds imply triviality of certain homotopy groups.

Provides explicit constants depending on dimension and homotopy level.

Abstract

We examine topological properties of pointed metric measure spaces $(Y, p)$ that can be realized as the pointed Gromov-Hausdorff limit of a sequence of complete, Riemannian manifolds $\{(M^n_i, p_i)\}_{i=1}^{\infty}$ with nonnegative Ricci curvature. Cheeger and Colding \cite{ChCoI} showed that given such a sequence of Riemannian manifolds it is possible to define a measure $\nu$ on the limit space $(Y, p)$. In the current work, we generalize previous results of the author to examine the relationship between the topology of $(Y, p)$ and the volume growth of $\nu$. In particular, we prove a Abresch-Gromoll type excess estimate for triangles formed by limiting geodesics in the limit space. Assuming explicit volume growth lower bounds in the limit, we show that if $\lim_{r \to \infty} \frac{\nu(B_p(r))}{\omega_n r^n} > \alpha(k,n)$, then the $k$-th group of $(Y,p)$ is trivial. The constants $\alpha(k,n)$ are explicit and depend only on $n$, the dimension of the manifolds $\{(M^n_i, p_i)\}$, and $k$, the dimension of the homotopy in $(Y,p)$.