Summary: Comparison Geometry
Volume 30, 1997
Scalar Curvature and Geometrization
Conjectures for 3-Manifolds
MICHAEL T. ANDERSON
Abstract. We first summarize very briefly the topology of 3-manifolds
and the approach of Thurston towards their geometrization. After dis-
cussing 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.
In the late seventies and early eighties Thurston proved a number of very re-
markable results on the existence of geometric structures on 3-manifolds. These
results provide strong support for the profound conjecture, formulated by Thur-
ston, that every compact 3-manifold admits a canonical decomposition into do-
mains, each of which has a canonical geometric structure.
For simplicity, we state the conjecture only for closed, oriented 3-manifolds.