The existence of negatively Ricci curved metrics on three manifolds (Q1082619)

The main theorem of this paper is that on any compact 3-manifold M without boundary there exists a Riemannian metric with negative Ricci curvature. The authors also prove that any noncompact 3-manifold that is diffeomorphic to the interior of a compact 3-manifold with boundary admits a complete metric of negative Ricci curvature. The first step in the proof of the main theorem is to construct on \(S^ 2\times S^ 1\) a metric g of negative Ricci curvature and a simple closed geodesic \(\gamma\) such that the complement in \(S^ 2\times S^ 1\) of a small tubular neighborhood of \(\gamma\) is diffeomorphic to a solid torus \(D^ 2\times S^ 1\) (Theorem 3). The proof of this step involves a technique that the authors also use to give a simplified proof of an earlier result (Theorem 2) from [\textit{L. Z. Gao}, Math. Ann. 271, 185-208 (1985; Zbl 0542.53024)], which proves the existence of metrics of negative Ricci curvature on certain connected sums of 3-manifolds that admit metrics of negative Ricci curvature. A result of Thurston says that every compact 3-manifold is obtained from \(S^ 3\) by Dehn surgery along some link L whose complement in \(S^ 3\) admits a complete metric g with finite volume and sectional curvature \(K\equiv -1\). The ends of such a hyperbolic manifold \((S^ 3-L,g)\) having neighborhoods isometric to a warped product \(T^ 2\times (0,\infty)\), where \(T^ 2\) is a flat torus and the warping function is \(f(t)=e^ t\) for \(t\in (0,\infty)\). Finally the authors use this fact and Theorem 3 to construct a metric of negative Ricci curvature on any manifold obtained from \(S^ 3\) by Dehn surgery along a link L.