Complete metrics of negative Ricci curvature (Q1179715)

In this paper the authors prove the following Theorem. Let \(M\) be a compact, connected manifold of dimension \(n\geq 3\), and let \(p\) be a point of \(M\). Then \(M-\{p\}\) admits a complete metric of negative Ricci curvature. To begin the proof one fixes a triangulation of \(M\). Starting with a metric of negative Ricci curvature on a neighborhood of the 0- skeleton the authors use conformal metric changes to extend it inductively from a neighborhood of the \(d\)-skeleton to a neighborhood of the \((d+1)\)-skeleton. One finally obtains a metric of negative Ricci curvature on a neighborhood \(U\) of the \((n-1)\)-skeleton, which one may arrange so that \(U\) is diffeomorphic to a set obtained by deleting a finite number of disjoint open balls from \(M\). By connecting these balls one obtains a manifold \(N\subseteq U\) such that \(N\) is diffeomorphic to the complement of a single open ball in \(M\). After restricting the metric on \(U\) to \(N\) and making another conformal metric change the authors obtain a complete metric of negative Ricci curvature on \(N-\partial N\), which is diffeomorphic to \(M\) with a point removed.