Compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below
Let \(M\) be an \(n\)-dimensional complete non-compact Riemannian manifold and \(p\in M\) a point such that the radial sectional curvature along any minimizing geodesic emanating from \(p\) is bounded below by a function \(K:[0,\infty[\to\mathbb R\). Assume further that \(\widetilde M^2\) is a Riemannian surface whose metric in terms of the geodesic polar coordinates \((t,\theta)\) around a fixed point \(\widetilde p\in\widetilde M^2\) is rotational symmetric, i.e. given by \(\widetilde g_2=dt^2+f(t)^2d\theta^2\) with a positive smooth function \(f\) such that \(K\) is the Gaussian curvature of \( \widetilde M^2\), hence \(f''+Kf=0\) and \(f(0)=0\), \(f'(0)=1\). The surface \(\widetilde M^2\) is then called a model surface for \(M\). The main result is as follows: If, in this situation, (a) \(\int_1^\infty dt/f(t)^2=\infty\) or (b) \(\liminf_{t\to\infty}f(t)=0\) then the ideal boundary \(M(\infty)\) of \(M\) consists of a single point and \(M\) is diffeomorphic to \(\mathbb R^{n}\) in case (b), resp. isometric in case (a) to the \(n\)-dimensional model \(\widetilde M^n\), defined in analogy to \(\widetilde M^2\) by the metric \(\widetilde g_n=dt^2+f(t)^2 g_{S^{n-1}}(\Theta)\), the last term being the standard metric of the unit \((n-1)\)-sphere. The proof proceeds in two steps: First, several asymptotic properties of the model are derived from (a) or (b). Then, for the transition from the model to \(M\) itself, a generalized Toponogov comparison theorem announced in a preprint of Itokawa-Machigashira-Shiohama is applied.
scientific article; zbMATH DE number 1837387