Excess functions of rays on complete noncompact manifolds (Q1411407)

Let \(M^n\) be an \(n\)-dimensional complete noncompact Riemannian manifold. In this paper, the authors study excess functions associated to geodesic rays in \(M^n\). We recall that if \(\gamma:[0,\infty[\to M^n\) is a geodesic ray satisfying \(\gamma(0)=p\) then, the excess function associated to \(\gamma\) is defined as \[ E^{\gamma}(x)=\lim_{t\to\infty}(d(p,x)+d(x,\gamma(t))-t). \] The paper contains some estimates for excess functions in terms of the curvature of the manifold. For instance, if the sectional curvature of \(M\) is nonnegative, then one has the inequality \[ E^{\gamma}(x)\leq {d(x,\gamma)^2\over d(x,p)}. \] The authors give also an upper bound for \(E^{\gamma}(x)\) in the case where the sectional curvature is \(\geq -k\) for some positive constant \(K\). As an application, they provide a proof of a result of \textit{C. Xia} [Am. J. Math 122, 745--755 (2000; Zbl 0993.53011)] which states that under some curvature conditions, a complete Riemannian manifold is diffeomorphic to \(\mathbb{R}^n\). Reviewer's remark: The excess function is, up to a constant, the Busemann function associated to the given ray.