Open manifolds with nonnegative Ricci curvature and large volume growth
This work is devoted to study complete open Riemannian manifolds with nonnegative Ricci curvature. It is known that if \(M\) is an \(n\)-dimensional complete Riemannian manifold with nonnegative sectional curvature, then it has finite topological type, i.e. it is homeomorphic to the interior of a compact manifold with boundary [see \textit{S. Zhu}, Math. Sci. Res. Inst. Publ. 30, 221-262 (1997; Zbl 0896.53036)]. Then, it is natural to ask if it is possible to get this conclusion when one replaces nonnegative sectional curvature by nonnegative Ricci curvature. However, there are examples due to \textit{J.-P. Sha} and \textit{D.-G. Yang} [J. Differ. Geom. 29, 95-103 (1989; Zbl 0633.53064)] showing that this needs not to be the case. The author presents three finiteness results for this problem additionally assuming that: 1) For any minimal geodesic issuing from a given point \(p\), all sectional curvatures of the planes tangent to this geodesic are bounded from below by a negative quantity \(k >-\infty\); 2) \(M\) has large volume growth (i.e., the quantity \(\alpha_M = \lim_{r \to 0}\frac{\text{vol}(B_r(p))}{w_n r^n}\) is strictly positive, where \(B_r(p)\) denotes the geodesic \(r\)-ball in \(M\) with center \(p\), and \(w_n\) is the volume of the unit disk in the \(n\)-dimensional Euclidean space. \(\alpha_M\) is a global geometric invariant of \(M\)). These additional conditions, which allow to obtain topological finiteness, appear studying the relationship between Ricci curvature and volume growth of geodesic balls. Under all these general conditions, Theorems 1.1 and 1.3 assert that, when the volume growth of geodesic balls around the given point \(p \in M\) is not too far from that of the balls in \(\mathbb{R}^n\), then \(M\) is diffeomorphic to the Euclidean space. Theorem 1.2 asserts that if the limit of the difference \((\frac{\text{vol}(B_r(p))}{w_n r^n} -\alpha_M) r^{n-1}\), when \(r\) goes to infinity, is bounded from above by a quantity involving \(k\) and \(\alpha_M\) then \(M\) has finite topological type. The proofs are based on the study of the critical points (in the sense of Grove-Shiohama) of the distance function \(d_p\) from the given point \(p\in M\), using some properties of the excess function \(e_p\) and the Isotopy Lemma.
scientific article; zbMATH DE number 1375332