Islands at infinity on manifolds of asymptotically nonnegative curvature (Q839960)

Let \((M,g)\) be a complete Riemannian manifold. If \(P\in M\), let \(B_P(r)\) be the geodesic ball of radius \(r\) centered at \(P\) and let \(K(P)\) be the infimum of the sectional curvatures at \(P\). The authors define the \textit{\(R\) eccentricity at \(P\)} to be the number: \[ h_R(P):=\sup_{r\in(0,R]}\big\{r^2\inf\{K(q):q\in B_P(r)\}\big\}\,. \] This is invariant under a constant rescaling of the metric. The authors show: Theorem 1. Let \((M,g)\) be a complete manifold with asymptotically nonnegative curvature. Then for any constant \(R>0\), \(\lim_{P\rightarrow\infty}h_R(P)=0\). They then obtain Corollary 2. Let \(M\) be a complete and noncompact manifold with asymptotically nonnegative curvature. Assume that \(M\) contains a point with positive sectional cuvature. Then the group of isometries of \((M,g)\) is compact. Furthermore, if one considers the sequence \(\{f^k\}\) where \(f\) is an isometry of \((M,g)\), then for any point \(P\) of positive curvature, there exists a convergent subsequence \(f^{k_i}\rightarrow g\) such that \(g(P)=P\). The paper examines ``islands'' (geodesic balls with all sectional curvatures bounded from below by a positive constant) at infinity and was motivated by the intuition that a complete manifold with asymptotically nonnegative curvature should have, in some sense, sectional curvatures close to zero at infinity.