Well-posedness of the ambient metric equations and stability of even dimensional asymptotically de Sitter spacetimes

TL;DR

This paper proves the well-posedness and hyperbolicity of the ambient metric equations in even-dimensional asymptotically de Sitter spacetimes, ensuring stable evolution of smooth initial data and extending previous results with rigorous mathematical foundations.

Contribution

It establishes the hyperbolic nature and well-posedness of the ambient metric equations, filling gaps in prior proofs and applying to various Graham-Jenne-Mason-Sparling operators.

Findings

Confirmed hyperbolicity of ambient metric equations.

Proved existence of smooth developments for Cauchy data.

Showed conformal factor can be chosen to nullify Branson Q-curvature.

Abstract

Vanishing of the Fefferman-Graham obstruction tensor was used by Andersson and Chru{\'s}ciel to show stability of the asymptotically de Sitter spaces in even dimensions. However, 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 Graham-Jenne-Mason-Sparling operators. In particular, we use one of these operators to propagate Gover's condition of being almost conformally Einstein. This allows to study initial data also for Cauchy surfaces which cross conformal boundary. As a by-product we show that on globally hyperbolic manifolds one can always choose conformal factor such that Branson Q-curvature vanishes.