Smoothly compactifiable shear-free hyperboloidal data is dense in the physical topology

TL;DR

This paper proves that shear-free smoothly conformally compact vacuum initial data are dense in the space of polyhomogeneous asymptotically hyperbolic solutions to Einstein's constraints, enabling better approximation of physical solutions.

Contribution

It establishes the density of shear-free conformally compact data within the class of polyhomogeneous asymptotically hyperbolic solutions for Einstein's vacuum constraints.

Findings

Shear-free conformally compact data can approximate any polyhomogeneous asymptotically hyperbolic solution.

The approximation is arbitrarily close in Hölder norms determined by the physical metric.

The result enhances understanding of the structure of initial data in general relativity.

Abstract

We show that any polyhomogeneous asymptotically hyperbolic constant-mean-curvature solution to the vacuum Einstein constraint equations can be approximated, arbitrarily closely in H\"older norms determined by the physical metric, by shear-free smoothly conformally compact vacuum initial data.