Manifolds with asymptotically nonnegative minimal radial curvature

Abstract

In this paper we extend results on the geometry of manifolds with asymptotically nonnegative curvature to manifolds with asymptotically nonnegative minimal radial curvature, showing that most of the results obtained by U. Abresh [1], A. Kasue [28] and S. Zhu [48] hold in a more general context. Particularly, we show that there exists one and only one tangent cone at infinity to each such manifold, in contrast with the class of manifolds of nonnegative Ricci curvature. A. IntroductionIn several independent papers, it has been shown that results concerning topological finiteness and rigidity of open manifolds with nonnegative sectional curvature can be extended to manifolds with nonnegative sectional curvature outside a compact set (see for instance [4], [20], [29]), [30]); the next step was to ask if those results would still hold for manifolds of nonnegative curvature at infinity (and how to define this notion). In case the curvatures decay faster then quadratically, one possible way to handle this problem, can be found in the paper "Lower curvature bounds, Toponogov's Theorem, and bounded topology" of U. Abresch [1], where he studied manifolds of asymptotically nonnegative curvature, which generalize in an interesting way the notion of a manifold with nonnegative sectional curvature at infinity; also a generalized Toponogov Comparison Theorem is proved for this class of manifolds. After Abresch's paper, some results were obtained on the geometry of this class of manifolds, for instance: in [28], there is a comprehensive discussion on the ideal boundaries of this class of manifolds; in [48], a volume comparison theorem is proved. On the other hand, a careful reading of Abresch, Kasue and Zhu * supported by CNPq, UNICAMP-Brazil. 332Newton L. Santos results shows that the condition related to the monotonicity of the sectional curvature can be reduced, without loss of generality, to a condition on the monotonicity of the minimal radial curvatures, because a substantial part of the information one can obtain on the geometry of a manifold with base point (M n , o) is given by the behavior of the geodesic spray, from o (that is, a substantial amount of information of the manifold is contained in the behavior of its radial Jacobi fields). For instance, the Bonnet-Myers Theorem and the Bishop-Gromov Comparison Theorem hold for manifolds with minimal radial curvature bounded below. Also, the Rauch and Berger Comparison Theorems admit a version for minimal radial curvature Most of the results which will be developed next are, actually, obvious reformulations of results presented in Abresch [1], Kasue [28] and Zhu [48], that is, the following results have already been developed and proved by the above mentioned mathematicians, in a context of less generality.Recall that the minimal radial curvatures K min o (p) at a point p in M n are just the sectional curvatures restricted to planes tangent to minimizing geodesics departing from a base point o in M n . A.1. Definition. A complete open Riemannian manifold (M n , o, g) wit...