A construction of metrics of negative Ricci curvature (Q1093181)

This paper gives a construction of metrics of negative Ricci curvature on all compact 3-manifolds. The existence of such a metric was previously demonstrated by Gao and Yau. The construction given here relies on the existence of tubular neighborhoods of specified width about short geodesics in hyperbolic 3-manifolds, and also makes use of the Hilden- Lozano-Montesinos theory of universal knots. It is shown that there are positive constants a and b such that every 3-manifold carries a metric whose Ricci curvatures are pinched between -a and -b. A generalization to some manifolds of higher dimension is also given.