We first summarize very briefly the topology of 3-manifolds and the approach of Thurston towards their geometrization. After discussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston's Geometrization Conjecture for closed oriented 3-manifolds. The final two sections present evidence for the validity of these conjectures and outline an approach toward their proof.

Introduction

In the late seventies and early eighties Thurston proved a number of very remarkable results on the existence of geometric structures on 3-manifolds. These results provide strong support for the profound conjecture, formulated by Thurston, that every compact 3-manifold admits a canonical decomposition into domains, each of which has a canonical geometric structure.

For simplicity, we state the conjecture only for closed, oriented 3-manifolds.

GEOMETRIZATION CONJECTURE [Thurston 1982], Let M be a closed, oriented, prime S-manifold. Then there is a finite collection of disjoint, embedded tori in M, such that each component of the complement admits a geometric structure, i.e., a complete, locally homogeneous Riemannian metric.

A more detailed description of the conjecture and the terminology will be given in Section 1. A complete Riemannian manifold Ñ is locally homogeneous if the universal cover Ñ is a complete homogenous manifold, that is, if the isometry group Isomi Ñ acts transitively on Ñ. It follows that Nis isometric to N/Γ, where Γ is a discrete subgroup of Isomi Ñ, which acts freely and properly discontinuously on Ñ.