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SCALAR CURVATURE, METRIC DEGENERATIONS AND THE STATIC VACUUM EINSTEIN EQUATIONS ON 3-MANIFOLDS, I.
 

Summary: SCALAR CURVATURE, METRIC DEGENERATIONS AND THE STATIC
VACUUM EINSTEIN EQUATIONS ON 3-MANIFOLDS, I.
MICHAEL T. ANDERSON
Contents
0. Introduction. 1
1. Background Material. 6
2. Initial Global Estimates for Yamabe Metrics. 17
3. Existence of Non-Flat Blow-Ups. 26
4. Remarks on the Hypotheses. 53
5. Completeness of the Blow-up Limits. 58
6. Construction of Yamabe Sequences with Singular Limits. 65
7. Palais-Smale Sequences for Scalar Curvature Functionals. 84
8. Appendix. 88
References 93
0. Introduction
In this paper, we prove that degenerations of sequences of Yamabe metrics on 3-manifolds are
modeled or described by solutions to the static vacuum Einstein equations. One underlying moti-
vation to understand such degenerations is the question of existence of constant curvature metrics
on 3-manifolds, in other words with the geometrization conjecture of Thurston [Th2]. An approach
towards resolving this conjecture via study of Yamabe metrics is outlined in [An1].

  

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

 

Collections: Mathematics