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SCALAR CURVATURE AND THE EXISTENCE OF GEOMETRIC STRUCTURES ON 3-MANIFOLDS, I.
 

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
References 43
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.
0. Introduction.
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 [37]. 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,

  

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

 

Collections: Mathematics