Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 5

A density theorem for asymptotically hyperbolic initial data

Pages: 1669 – 1710

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n5.a3


Mattias Dahl (Institutionen för Matematik, Kungliga Tekniska Högskolan, Stockholm, Sweden)

Anna Sakovich (Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Potsdam, Germany; and Matematiska Institutionen, Uppsala University, Uppsala, Sweden)


When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data sets with such properties is dense in the set of physically reasonable asymptotically hyperbolic initial data sets. More specifically, we show that an asymptotically hyperbolic initial data set with nonnegative local energy density can be approximated by an initial data set with strictly positive local energy density and a simple structure at infinity, while changing the mass arbitrarily little. This is achieved by suitably modifying the argument used by Eichmair, Huang, Lee and Schoen in the asymptotically Euclidean case.

Received 30 July 2020

Accepted 18 April 2021

Published 26 January 2022