Summary: Comparison Geometry
Volume 30, 1997
Scalar Curvature and Geometrization
Conjectures for 3Manifolds
MICHAEL T. ANDERSON
Abstract. We first summarize very briefly the topology of 3manifolds
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 3manifolds. 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 3manifolds. These
results provide strong support for the profound conjecture, formulated by Thur
ston, that every compact 3manifold 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 3manifolds.