The asymptotic cones of manifolds of roughly non-negative radial curvature

Abstract

We prove that the asymptotic cone of every complete, connected, non-compact Riemannian manifold of roughly non-negative radial curvature exists, and it is isometric to the Euclidean cone over their Tits ideal boundaries.Obviously, the Gaussian curvature G( p) at p ∈ M is equal to K d M ( õ, p) . We say that the radial sectional curvature at o of M is bounded below by

Gravitational instantons with faster than quadratic curvature decay. I

“…Theorem 3.6. ( [27], [14], [32]) Let M be a manifold with asymptotically nonnegative curvature. Two rays σ and γ starting from o are called equivalent if…”

Gravitational instantons with faster than quadratic curvature decay (II)

“…Theorem 3.6. ( [25] [12] [30]) Let M be a manifold with asymptotically nonnegative curvature. Two rays σ and γ starting from o are called equivalent if lim t→∞ dist(σ(t), γ(t))/t = 0.…”

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. II

“…Therefore, by Sector Theorem and Main Theorem, M has a finitely generated fundamental group. ✷ Remark 5.5 Under the assumption in Theorem 5.3, or Corollary 5.4, it follows from [MNO,Theorem 0.1] that (M, p) admits the asymptotic cone via rescaling argument, i.e., the pointed Gromov-Hausdorff limit space of ((1/t)M, p) exists as t → ∞, and the space is, naturally, isometric to a Euclidean cone (see [G2,Definition 3.14] for a definition of the pointed Gromov-Hausdorff convergence). However, one should notice again that our models in Theorem 5.3 and Corollary 5.4 have been constructed from any complete open Riemannian manifold with an arbitrary given point as a base point, and that the metrics (5.9) in Theorem 5.3 and Corollary 5.4 are not always differentiable around their base points.…”

“…However, one should notice again that our models in Theorem 5.3 and Corollary 5.4 have been constructed from any complete open Riemannian manifold with an arbitrary given point as a base point, and that the metrics (5.9) in Theorem 5.3 and Corollary 5.4 are not always differentiable around their base points. In particular, our Main Theorem has a wider class of metrics than those described in [MNO,Theorem 0.1].…”