A finiteness theorem for Ricci curvature in dimension three (Q688600)

The purpose of this paper is to generalize Cheeger's finiteness theorem to an assumption on the Ricci curvature instead of sectional curvature. The author proves the following result: Theorem 1: There are only finitely many homotopy types in the class of three-dimensional Riemannian manifolds \(M\) satisfying \(\text{Ric}(M)\geq -H^ 2\), \(\text{Diam}(M)\leq D\), \(\text{Vol}(M)\geq V\). Theorem 2: Let \(M^ 3\) be a complete open three-manifold satisfying \(\text{Ric}\geq 0\), \(\text{Vol}(B_ p(r))\geq cr^ 3\). Then \(M\) is contractible.