Volume growth and the topology of Gromov-Hausdorff limits

الأصل · EN

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 {(Mⁿᵢ, pᵢ)}ᵢ₌₁∞ with nonnegative Ricci curvature. Cheeger and Colding ChCoI showed that given such a sequence of Riemannian manifolds it is possible to define a measure ν 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 ν. 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 ᵣ → ∞ ν(Bₚ(r))/ωₙ rⁿ > α(k,n), then the k-th group of (Y,p) is trivial. The constants α(k,n) are explicit and depend only on n, the dimension of the manifolds {(Mⁿᵢ, pᵢ)}, and k, the dimension of the homotopy in (Y,p).