Scalar Curvature, Metric Degenerations and the Static Vacuum Einstein Equations on 3-manifolds, I
نویسنده
چکیده
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 motivation 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]. Let M denote the space of all smooth Riemannian metrics on a closed, oriented 3-manifoldM, and M1 the subset of metrics satisfying volgM = 1. Define the total scalar curvature or Einstein-Hilbert action S : M → R by
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