Well-Posedness of the Ambient Metric Equations and Stability of Even Dimensional Asymptotically de Sitter Spacetimes

Abstract

AbstractThe vanishing of the Fefferman–Graham obstruction tensor was used by Anderson and Chruściel to show stability of the asymptotically de Sitter spaces in even dimensions. However, the existing proofs of hyperbolicity of this equation contain gaps. We show in this paper that it is indeed a well-posed hyperbolic system with unique up to diffeomorphism and conformal transformations smooth development for smooth Cauchy data. Our method applies also to equations defined by various versions of the Graham–Jenne–Mason–Sparling operators. In particular, we use one of these operators to propagate Gover’s condition of being almost Einstein (basically conformal to Einsteinian metric). This allows us to study initial data also for Cauchy surfaces which cross the conformal boundary. As a by-product we show that on globally hyperbolic manifolds one can always choose a conformal factor such that Branson Q-curvature vanishes.