A topological rigidity theorem on open manifolds with nonnegative Ricci curvature
Let \(M\) be an \(n\)-dimensional complete Riemannian manifold with nonnegative Ricci curvature, \(B(p,r)\) the open geodesic ball of radius r at \(p \in M\), and vol\(B(p,r)\) its volume. Let \(\omega_n\) be the volume of the unit ball in the Euclidean space \(\mathbb R^n\), and denote by \(\alpha_M\) the \(\lim_{r\rightarrow + \infty}\frac{\operatorname{vol}B(p,r)}{\omega_n r^n}\). It is well-known that \(\alpha_M\) is well-defined, i.e. the limit exists and it does not depend on the choice of the point \(p\). The manifold \(M\) is said to have large volume growth if \(\alpha_M > 0.\) Such manifolds have been studied extensively. In [Fields Inst. Monogr. 4, 87--115 (1996; Zbl 0843.53031)], \textit{P. Petersen} put forward the following conjecture: An \(n\)-dimensional complete Riemannian manifold with nonnegative Ricci curvature and \(\alpha_M > \frac{1}{2}\) is diffeomorphic to \(\mathbb{R}^n\). In the paper, the author proves the following topological rigidity theorem which supports Petersen's conjecture and contains no condition on the sectional curvature. Theorem. Given \(\alpha\in (\frac{1}{2},1), \rho_0 >0\) and an integer \(n\geq 2\), there exist positive constants \(r_0 = r_0(\alpha, \rho_0, n)\) and \(\varepsilon = \varepsilon (\alpha, \rho_0, n)\) such that any complete Riemannian \(n\)-manifold \(M\) with Ricci curvature \(\text{Ricci}_M \geq 0\), \(\alpha_M \geq \alpha\), conjugate radius \(\text{conj}_M \geq \rho_0\) and \[ \frac{\operatorname{vol}B(p,r)}{\omega_n r^n} - \alpha_M \leq \frac{\varepsilon}{r^{n-2+1/n}} \] for some \(p \in M\) and all \(r\geq r_0,\) is diffeomorphic to \(\mathbb R^n\).
scientific article; zbMATH DE number 2103857
scientific article