Manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type
Let \((M^n,g)\) be a noncompact Riemannian manifold and \(p_0\in M\) a reference point. Denoting by \(B(p_0,t)\) the ball with centre \(p_0\) and radius \(t\) in \(M\), define \[ K_{p_0}(t) := \inf_{M\setminus B_{p_0}(t)} K\,, \] where \(K\) is the sectional curvature of \((M,g)\). The manifold \((M,g)\) is said to have \textit{quadratically asymptotically nonnegative curvature}, if there exists a \(K_0 > 0\) such that \[ K_{p_0}(t) \geq -\frac{K_0}{1+t^2} \quad \text{for all}\quad t\geq 0\,. \] The manifold \(M\) is said to be of \emph{finite topological type}, if it is homeomorphic to the interior of a compact manifold with boundary, and of \emph{infinite topological type} otherwise. The main result of the article, Theorem 1.1, asserts the existence of a complete noncompact 6-dimensional Riemannian manifold of positive Ricci curvature, quadratically asymptotically nonnegative curvature and infinite topological type. Moreover, this manifold satisfies \[ \lim_{t\to\infty} \frac{\operatorname{vol} B(p_0,t)}{t^6} = 0\quad\text{and}\quad \operatorname{diam}(p_0;t) = O(t)\,, \] where \(\operatorname{diam}(p_0;t)\) denotes a certain diameter function introduced in [\textit{U. Abresch} and \textit{D. Gromoll}, J. Am. Math. Soc. 3, No. 2, 355--374 (1990; Zbl 0704.53032)]. This answers in the negative a question raised in [\textit{J. Sha} and \textit{Z. Shen}, Am. J. Math. 119, No. 6, 1399--1404 (1997; Zbl 0901.53023)]. The manifold in Theorem 1.1 is explicitely constructed, starting from a certain doubly warped product and performing an infinite number of surgeries.
quadratically asymptotically nonnegative curvature
scientific article; zbMATH DE number 7451605