Lorentzian Einstein metrics with prescribed conformal infinity

Original Title: “Lorentzian Einstein metrics with prescribed conformal infinity”

Authors: Alberto Enciso (ICMAT-CSIC) y Niky Kamran

Source: Journal of Differential Geometry 112 (2019), 505554

Date of online publication: 2019

Link: https://projecteuclid.org/euclid.jdg/1563242472


A highly topical subject in the general theory of relativity is the analysis of asymptotically antide Sitter (AdS) spacetimes, given their role in the AdS/CFT duality conjectured by Juan Maldacena. Their mathematical interest is focused on the study of the stability or instability of this space when subject to small perturbations, in the context of the Einstein equations. The existence and uniqueness theorem established by Yvonne Choquet-Bruhat and Robert Geroch is not applied to these spacetimes, which satisfy the Einstein equations with a negative cosmological constant. This is because on the coordinates (known as “wave” coordinates) at which the local existence of solutions can be demonstrated, an AdS-type metric critically diverges: as the inverse square of the distance to the conformal boundary of the spacetime. This singularity is too strong to be included in any standard functional space on which to establish the local existence of solutions. According to Helmut Friedrich, in dimension four, when the Cauchy data (which must satisfy the so-called constraint equations) and the contour data fulfil an additional technical condition – which is generally not satisfied – the Einstein equations with hyperboloidal data (that is, of AdS-type) are well posed. In 1998, Edward Witten, on the basis of the classical results obtained by Robin Graham and Jack Lee in Riemannian signature, noted that in general this should actually occur. This is known as the holographic principle for the Einstein equations.

In this paper, Alberto Enciso and Niky Kamran show that the holographic principle is indeed generally valid for the Einstein equations with sufficiently regular AdS-type data, without additional hypotheses and in any dimension. Their approach to local existence, which is completely different from the conformal geometry methods employed by Friedrich, is purely analytical. It is based on three principles: the systematic use of estimates with singular weights and derivatives with twist; the development of intrinsically vector function space scales based on a polyhomogeneous regularity, adapted to the AdS geometry at infinity; the construction of a symbolic peeling calculus, that is, a partially algebraic procedure that enables the approximate study of the evolution of the asymptotically AdS metrics at infinity, generating only weak errors.