Summary: SCALAR CURVATURE AND THE EXISTENCE OF GEOMETRIC
STRUCTURES ON 3-MANIFOLDS, I.
MICHAEL T. ANDERSON
0. Introduction. 1
1. Background Material. 7
2. Geometrization of Tame 3-Manifolds. 12
3. Metric Surgery on Spheres in Asymptotically Flat Ends. 30
4. Asymptotically Flat Ends and Annuli. 36
Abstract. This paper analyses the convergence and degeneration of sequences of metrics on a
3-manifold, and relations of such with Thurston's geometrization conjecture. The sequences are
minimizing sequences for a certain (optimal) scalar curvature-type functional and their degeneration
is related to the sphere and torus decompositions of the 3-manifold under certain conditions.
This paper and its sequel are concerned with the limiting behavior of minimizing sequences
for certain curvature integrals on the space of metrics on a 3-manifold M , and the relations of
such behavior with the geometrization conjecture of Thurston . First, recall the statement of
Thurston's conjecture, in the case of closed, oriented 3-manifolds.
Geometrization Conjecture (Thurston).
Let M be a closed, oriented 3-manifold. Then M admits a canonical decomposition into domains,