Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds, I

Abstract

We study Yamabe metrics, and the moduli space of Yamabe metrics, on an arbitrary closed 3-manifold M. The main focus is on the boundary behavior of the moduli space, i.e. the behavior of degenerating sequences of unit volume Yamabe metrics on M. It is proved that such degenerations, when non-trivial in a certain sense, are described by solutions of the static vacuum Einstein equations. Natural conditions are given for the non-triviality of degenerating sequences and relations with Palais-Smale sequences for the Einstein-Hilbert functional are explored. A number of new examples, both trivial and non-trivial,of degenerating sequences are constructed. It is also proved that the only complete static vacuum solution without horizon is the flat metric, generalizing a classical result of Lichnerowicz.