Spacelike initial data for black hole stability

TL;DR

This paper develops a method to construct initial data for black hole stability analysis, specifically perturbations of Kerr data with boundary conditions that exclude certain symmetries, aiding the proof of the Kerr stability conjecture.

Contribution

It introduces a new perturbative construction of initial data with boundary conditions suitable for Kerr stability, excluding Killing initial data sets.

Findings

Constructed initial data with decay properties near the black hole horizon.

Designed boundary conditions to exclude Killing initial data.

Applicable to asymptotically flat initial data with specific topology.

Abstract

We construct initial data suitable for the Kerr stability conjecture, that is, solutions to the constraint equations on a spacelike hypersurface with boundary entering the black hole horizon that are arbitrarily decaying perturbations of a Kerr initial data set. This results from a more general perturbative construction on any asymptotically flat initial data set with the topology of $\mathbb{R}^3\setminus\{r<1\}$ enjoying some analyticity near and at the boundary. In particular, we design a suitable mixed boundary condition for the elliptic operator of the conformal method in order to exclude the Killing initial data sets (KIDS).