Volume growth and the topology of manifolds with nonnegative Ricci curvature

2007/12/05 by Michael Munn, Munn, Michael
Mathematics · #53C21 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0712.0827

openalex publication_date 2007/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Mⁿ be a complete, open Riemannian manifold with \Ric ≥ 0. In 1994, Grigori Perelman showed that there exists a constant δ_n>0, depending only on the dimension of the manifold, such that if the volume growth satisfies α_M := lim_{r → ∞} (\Vol(B_p(r)))/(ω_n rⁿ) ≥ 1-δ_n, then Mⁿ is contractible. Here we employ the techniques of Perelman to find specific lower bounds for the volume growth, α(k,n), depending only on k and n, which guarantee the individual k-homotopy group of Mⁿ is trivial.

Citations

• Collapsing of Riemannian manifolds and eigenvalues of Laplace operator

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